C ALGORITHM 835, COLLECTED ALGORITHMS FROM ACM. C THIS WORK PUBLISHED IN TRANSACTIONS ON MATHEMATICAL SOFTWARE, C VOL. 30, NO. 2, June, 2004, P. 218--236. #! /bin/sh # This is a shell archive, meaning: # 1. Remove everything above the #! /bin/sh line. # 2. Save the resulting text in a file. # 3. Execute the file with /bin/sh (not csh) to create the files: # Doc/ # Doc/cmip.ps # Doc/readme.txt # Matlab/ # Matlab/Drivers/ # Matlab/Drivers/bt01.m # Matlab/Drivers/bt02.m # Matlab/Drivers/bt03.m # Matlab/Drivers/bt04.m # Matlab/Drivers/farloi01.m # Matlab/Drivers/fib.m # Matlab/Drivers/fib05.m # Matlab/Drivers/fib10.m # Matlab/Drivers/fib100.m # Matlab/Drivers/fib15.m # Matlab/Drivers/fib150.m # Matlab/Drivers/fib20.m # Matlab/Drivers/fib30.m # Matlab/Drivers/fib50.m # Matlab/Drivers/fibsq04.m # Matlab/Drivers/fibsq08.m # Matlab/Drivers/fibsq16.m # Matlab/Drivers/fibsq24.m # Matlab/Drivers/fibsq32.m # Matlab/Drivers/fibsq48.m # Matlab/Drivers/fl.m # Matlab/Drivers/fl01.m # Matlab/Drivers/fl02.m # Matlab/Drivers/fl03.m # Matlab/Drivers/fl04.m # Matlab/Drivers/fl05.m # Matlab/Drivers/fl06.m # Matlab/Drivers/fl07.m # Matlab/Drivers/henrici.m # Matlab/Drivers/igyp00.m # Matlab/Drivers/igyp01.m # Matlab/Drivers/igyp02a.m # Matlab/Drivers/igyp02b.m # Matlab/Drivers/igyp02c.m # Matlab/Drivers/igyp03.m # Matlab/Drivers/iliev00.m # Matlab/Drivers/iliev01.m # Matlab/Drivers/iliev02.m # Matlab/Drivers/iliev03.m # Matlab/Drivers/inex01.m # Matlab/Drivers/inex02.m # Matlab/Drivers/inex03.m # Matlab/Drivers/inex04.m # Matlab/Drivers/jt01a.m # Matlab/Drivers/jt01b.m # Matlab/Drivers/jt02.m # Matlab/Drivers/jt03.m # Matlab/Drivers/jt04.m # Matlab/Drivers/jt05.m # Matlab/Drivers/jt06.m # Matlab/Drivers/jt07a.m # Matlab/Drivers/jt07b.m # Matlab/Drivers/jt07c.m # Matlab/Drivers/jt07d.m # Matlab/Drivers/jt08.m # Matlab/Drivers/jt09.m # Matlab/Drivers/jt10a.m # Matlab/Drivers/jt10b.m # Matlab/Drivers/jt10c.m # Matlab/Drivers/jt11a.m # Matlab/Drivers/jt11b.m # Matlab/Drivers/jt11c.m # Matlab/Drivers/large01.m # Matlab/Drivers/large02.m # Matlab/Drivers/large03.m # Matlab/Drivers/large04.m # Matlab/Drivers/large05.m # Matlab/Drivers/lgd.m # Matlab/Drivers/lgd05.m # Matlab/Drivers/lgd10.m # Matlab/Drivers/lgd100.m # Matlab/Drivers/lgd15.m # Matlab/Drivers/lgd20.m # Matlab/Drivers/lgd24.m # Matlab/Drivers/lgd50.m # Matlab/Drivers/miyak00.m # Matlab/Drivers/miyak02.m # Matlab/Drivers/miyak04.m # Matlab/Drivers/miyak08.m # Matlab/Drivers/near01.m # Matlab/Drivers/near02.m # Matlab/Drivers/near03.m # Matlab/Drivers/petk01.m # Matlab/Drivers/petk02.m # Matlab/Drivers/petk03.m # Matlab/Drivers/petk04.m # Matlab/Drivers/petk05.m # Matlab/Drivers/petk06.m # Matlab/Drivers/petk07.m # Matlab/Drivers/toh01.m # Matlab/Drivers/toh02.m # Matlab/Drivers/toh03.m # Matlab/Drivers/toh04.m # Matlab/Drivers/toh05.m # Matlab/Drivers/toh06a.m # Matlab/Drivers/toh06b.m # Matlab/Drivers/toh06c.m # Matlab/Drivers/triple.m # Matlab/Drivers/triple01.m # Matlab/Drivers/triple02.m # Matlab/Drivers/triple03.m # Matlab/Drivers/triple04.m # Matlab/Drivers/twin01.m # Matlab/Drivers/twin02.m # Matlab/Drivers/twin03.m # Matlab/Drivers/twin04.m # Matlab/Drivers/uhlig01.m # Matlab/Drivers/uhlig02.m # Matlab/Drivers/uhlig03.m # Matlab/Drivers/uhlig04.m # Matlab/Drivers/uhlig05.m # Matlab/Drivers/uhlig06.m # Matlab/Src/ # Matlab/Src/backsub.m # Matlab/Src/cauchymt.m # Matlab/Src/diagproc.m # Matlab/Src/fixend # Matlab/Src/forsub.m # Matlab/Src/gcdroot.m # Matlab/Src/hessqr.m # Matlab/Src/hqrt.m # Matlab/Src/lsqdiv.m # Matlab/Src/mroot.m # Matlab/Src/multroot.m # Matlab/Src/pejroot.m # Matlab/Src/rtsinfo.m # Matlab/Src/scalsq.m # Matlab/Src/spcond.m # Matlab/Src/sylmat.m # Matlab/Src/sylup.m # Matlab/Src/sylves.m # Matlab/Src/uminsv.m # Matlab/Src/zgcdgn0.m # Matlab/Src/zminsv.m # This archive created: Wed Aug 25 14:15:03 2004 export PATH; PATH=/bin:$PATH if test ! 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All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMCSC10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMCSC10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 80 /P put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 89 /Y put dup 90 /Z put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 121 /y put dup 122 /z put dup 127 /dieresis put readonly def /FontBBox{14 -250 1077 750}readonly def /UniqueID 5000772 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA0529731C99A784CCBE85B4993B2EEBDE 3B12D472B7CF54651EF21185116A69AB1096ED4BAD2F646635E019B6417CC77B 532F85D811C70D1429A19A5307EF63EB5C5E02C89FC6C20F6D9D89E7D91FE470 B72BEFDA23F5DF76BE05AF4CE93137A219ED8A04A9D7D6FDF37E6B7FCDE0D90B 986423E5960A5D9FBB4C956556E8DF90CBFAEC476FA36FD9A5C8175C9AF513FE D919C2DDD26BDC0D99398B9F4D03D5993DFC0930297866E1CD0A30EB76029337 900ECFB1390CA5C0C3A04528044F266BA17BE487C79B94FAC6D6484684C5BFEA 87BCCC77D40AD11552035E95E3007126418ED49B68468B38A14E88E68A267B98 076F1C9769A5AFBC285E5B158EAC9F926F1D6C0B8F1D57D9C31D25AE27123518 9D2CD92E5689E0213089BD268DA5E47525CB8EABAA4B78A15AEA34705889AB3A FFB8953B5B3482E52BFA0940630ADF8C0AC2177D907324299EE980E850F203CD B627962F43D5A678C44243CDE97853BDC6AB45FD5C09AD274DAF89929F583CC9 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cleartomark %%EndFont %%BeginFont: CMBX9 %!PS-AdobeFont-1.1: CMBX9 1.0 %%CreationDate: 1991 Aug 20 16:36:25 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMBX9) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMBX9 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 117 /u put dup 118 /v put dup 119 /w put readonly def /FontBBox{-58 -250 1195 750}readonly def /UniqueID 5000767 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMSY10 %!PS-AdobeFont-1.1: CMSY10 1.0 %%CreationDate: 1991 Aug 15 07:20:57 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put dup 1 /periodcentered put dup 2 /multiply put dup 3 /asteriskmath put dup 6 /plusminus put dup 17 /equivalence put dup 19 /reflexsuperset put dup 20 /lessequal put dup 21 /greaterequal put dup 24 /similar put dup 26 /propersubset put dup 27 /propersuperset put dup 33 /arrowright put dup 50 /element put dup 54 /negationslash put dup 74 /J put dup 102 /braceleft put dup 103 /braceright put dup 104 /angbracketleft put dup 105 /angbracketright put dup 106 /bar put dup 107 /bardbl put dup 112 /radical put dup 120 /section put readonly def /FontBBox{-29 -960 1116 775}readonly def /UniqueID 5000820 def currentdict end currentfile eexec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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMTT10 %!PS-AdobeFont-1.1: CMTT10 1.00B %%CreationDate: 1992 Apr 26 10:42:42 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMTT10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch true def end readonly def /FontName /CMTT10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 39 /quoteright put dup 40 /parenleft put dup 41 /parenright put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 47 /slash put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 58 /colon put dup 65 /A put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 73 /I put dup 74 /J put dup 78 /N put dup 80 /P put dup 82 /R put dup 83 /S put dup 84 /T put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put readonly def /FontBBox{-4 -235 731 800}readonly def /UniqueID 5000832 def currentdict end currentfile eexec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F476925663CFE76A309EB9E433093897 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMTT8 %!PS-AdobeFont-1.1: CMTT8 1.0 %%CreationDate: 1991 Aug 20 16:46:05 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMTT8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch true def end readonly def /FontName /CMTT8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 40 /parenleft put dup 41 /parenright put dup 43 /plus put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 47 /slash put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 91 /bracketleft put 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All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY5) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY5 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put readonly def /FontBBox{21 -944 1448 791}readonly def /UniqueID 5000815 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052F09F9C8ADE9D907C058B87E9B6964 7D53359E51216774A4EAA1E2B58EC3176BD1184A633B951372B4198D4E8C5EF4 A213ACB58AA0A658908035BF2ED8531779838A960DFE2B27EA49C37156989C85 E21B3ABF72E39A89232CD9F4237FC80C9E64E8425AA3BEF7DED60B122A52922A 221A37D9A807DD01161779DDE7D5FC1B2109839E5B52DFBAF552B11EFFB6A16C F03FB920C15AE724EFDF0CCBF00A838D34440FF9FED532F44036AD22561184C5 283722DDFA7285E62754372D716D704AC0E00B2F6AB67154241C7449AA047833 94CEDB08E8C92907FE72A0B05AE36A7B9226ACD6E7890A0B528FDDE84A950FC6 801DE75CF2E739E9121149CCB8B1C87A106822648D84A3D3FBF295EE6C4BF403 BBE9A1C1F6DAEDD1E642ACC486E609703D7612BFFD10C324F5DC710811F7F614 3691B400E3773987424C0D2B0D8A736873C6371DDB2442F05E018A2B5CA9A4AA 17AABB95D09E5890CFFFED5AC01495D89A53D3C89F3D8DBAC85E06561646557A B16BAE67D1C3D5F6EC51F05178B6CBC34F9F491056E5ADE1256277DB904A7F75 6105D774449D6CBB70F2F539D739D011BDBE27A9674A5CA3D5BD42B7FBFBBA35 89966597639DE60F2A4812AFAEFBBBB26C9E24F72439F3AF2A7D5C0D2AFB5146 58112FC60E9307B9FE29090D81441A1C3B7E371656A3321A0031038D54BB6670 0594C9C72FDDE3DE50EE1CFBDEEE5B6DDBA17B1D6328651FE1EFD1E2B8376949 5353E35A988064FDB6CE37ED4570B49903AF26304DE40164903476EC2F4830D6 4995CEE9B37B 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR5 %!PS-AdobeFont-1.1: CMR5 1.00B %%CreationDate: 1992 Feb 19 19:55:02 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR5) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR5 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 49 /one put dup 50 /two put readonly def /FontBBox{-341 -250 1304 965}readonly def /UniqueID 5000788 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMMI5 %!PS-AdobeFont-1.1: CMMI5 1.100 %%CreationDate: 1996 Aug 02 08:21:10 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI5) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI5 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 106 /j put dup 107 /k put readonly def /FontBBox{37 -250 1349 750}readonly def /UniqueID 5087380 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMSY7 %!PS-AdobeFont-1.1: CMSY7 1.0 %%CreationDate: 1991 Aug 15 07:21:52 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY7) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY7 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put dup 2 /multiply put dup 62 /latticetop put readonly def /FontBBox{-15 -951 1252 782}readonly def /UniqueID 5000817 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMMI7 %!PS-AdobeFont-1.1: CMMI7 1.100 %%CreationDate: 1996 Jul 23 07:53:53 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI7) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI7 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 14 /delta put dup 96 /lscript put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 115 /s put dup 116 /t put readonly def /FontBBox{0 -250 1171 750}readonly def /UniqueID 5087382 def currentdict end currentfile eexec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84022E6418CCCE16F5EFC2CDE75F0CF46F11F18654B372C373FE029C8610FC08 B3036311B6FD85E2489BBB347F2931B76A3FE89E447439FC19C504BF8149EA60 A86BA817F6D38E5EA3A628F0545B78CB87EE15A40F39B082FDC23CA730F19B31 BA7B38224D5C0A5C71D213C7F20BFD80661004F4DC1A136B0AA7AD3AA109F80A BDF5D9876B62DD8D66F340E18CDB403A616658768F72A3F5E707B3A98C275195 AFAD7B44971EA2366F6E98990CAA18BDB5DA370A948A28E044A2055B2B330AAB 64B8B599957EEE4DF3441EB678D8C4428B96F053E9640F5258E3C97F18E1A6B6 C86BE2F6CC209887CC6B0A346B2A1AC720BAAC366B82003A7C769023ACFD6F1C AAA4834154DCD3BBFD86071DAF04378A7BA75453B41A334DB8F386B8A58A4552 1A76A65C7211BBCBB53B70A2C671363AE5030B4B58FDC2A3DC8B909B0C949CF9 08C2583A05EA089AC7F9799D8328883A0FFBF8DE08A6E0D81E175E2367125E28 A4AAF546891BC3052DB8D73ADACC5D37AAB55E0D11D9CE64AD464DF62167F556 22A3549F3B48B10E62CEC7F8CAF4FF40EB6EA855C686C21A290203904023BA4E 2FD2153F60E0171E17A2352EAA55E1CFBE7E6A512195D9DB61F2978D1E80C07B E68714E680B5FFAEFEE6A301A654D5782EB6AA5A1927CAD6A725D9235B 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR7 %!PS-AdobeFont-1.1: CMR7 1.0 %%CreationDate: 1991 Aug 20 16:39:21 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR7) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR7 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 40 /parenleft put dup 41 /parenright put dup 43 /plus put dup 48 /zero put dup 49 /one put dup 50 /two put dup 56 /eight put readonly def /FontBBox{-27 -250 1122 750}readonly def /UniqueID 5000790 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMMI10 %!PS-AdobeFont-1.1: CMMI10 1.100 %%CreationDate: 1996 Jul 23 07:53:57 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 11 /alpha put dup 13 /gamma put dup 14 /delta put dup 15 /epsilon1 put dup 18 /theta put dup 20 /kappa put dup 21 /lambda put dup 26 /rho put dup 27 /sigma put dup 28 /tau put dup 30 /phi put dup 33 /omega put dup 34 /epsilon put dup 37 /rho1 put dup 38 /sigma1 put dup 58 /period put dup 59 /comma put dup 60 /less put dup 61 /slash put dup 62 /greater put dup 64 /partialdiff put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 70 /F put dup 71 /G put dup 73 /I put dup 74 /J put dup 76 /L put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 87 /W put dup 96 /lscript put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 122 /z put readonly def /FontBBox{-32 -250 1048 750}readonly def /UniqueID 5087385 def currentdict end currentfile eexec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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMI9 %!PS-AdobeFont-1.1: CMMI9 1.100 %%CreationDate: 1996 Jul 23 07:53:55 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI9) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI9 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 58 /period put dup 59 /comma put dup 74 /J put dup 87 /W put dup 96 /lscript put dup 97 /a put dup 98 /b put dup 105 /i put dup 112 /p put dup 120 /x put readonly def /FontBBox{-29 -250 1075 750}readonly def /UniqueID 5087384 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA0529731C99A784CCBE85B4993B2EEBDE 3B12D472B7CF54651EF21185116A69AB1096ED4BAD2F646635E019B6417CC77B 532F85D811C70D1429A19A5307EF63EB5C5E02C89FC6C20F6D9D89E7D91FE470 B72BEFDA23F5DF76BE05AF4CE93137A219ED8A04A9D7D6FDF37E6B7FCDE0D90B 986423E5960A5D9FBB4C956556E8DF90CBFAEC476FA36FD9A5C8175C9AF513FE 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMBXTI10 %!PS-AdobeFont-1.1: CMBXTI10 1.0 %%CreationDate: 1991 Aug 18 17:46:30 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMI6 %!PS-AdobeFont-1.1: CMMI6 1.100 %%CreationDate: 1996 Jul 23 07:53:52 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI6 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 27 /sigma put dup 87 /W put dup 97 /a put dup 105 /i put dup 106 /j put dup 107 /k put dup 109 /m put dup 110 /n put dup 120 /x put readonly def /FontBBox{11 -250 1241 750}readonly def /UniqueID 5087381 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA0529731C99A784CCBE85B4993B2EEBDE 3B12D472B7CF54651EF21185116A69AB1096ED4BAD2F646635E019B6417CC77B 532F85D811C70D1429A19A5307EF63EB5C5E02C89FC6C20F6D9D89E7D91FE470 B72BEFDA23F5DF76BE05AF4CE93137A219ED8A04A9D7D6FDF37E6B7FCDE0D90B 986423E5960A5D9FBB4C956556E8DF90CBFAEC476FA36FD9A5C8175C9AF513FE D919C2DDD26BDC0D99398B9F4D03D6A8F05B47AF95EF28A9C561DBDC98C47CF5 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cleartomark %%EndFont %%BeginFont: CMBX8 %!PS-AdobeFont-1.1: CMBX8 1.0 %%CreationDate: 1991 Aug 20 16:36:07 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMBX8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMBX8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 49 /one put dup 50 /two put dup 51 /three put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 67 /C put dup 113 /q put dup 114 /r put dup 117 /u put dup 120 /x put dup 122 /z put readonly def /FontBBox{-59 -250 1235 750}readonly def /UniqueID 5000766 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758 469A6C558785B226332152109871A9883487DD7710949204DDCF837E6A8708B8 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMEX10 %!PS-AdobeFont-1.1: CMEX10 1.00 %%CreationDate: 1992 Jul 23 21:22:48 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMEX10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMEX10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /parenleftbig put dup 1 /parenrightbig put dup 12 /vextendsingle put dup 13 /vextenddouble put dup 16 /parenleftBig put dup 17 /parenrightBig put dup 18 /parenleftbigg put dup 19 /parenrightbigg put dup 32 /parenleftBigg put dup 33 /parenrightBigg put dup 34 /bracketleftBigg put dup 35 /bracketrightBigg put dup 40 /braceleftBigg put dup 48 /parenlefttp put dup 49 /parenrighttp put dup 50 /bracketlefttp put dup 51 /bracketrighttp put dup 52 /bracketleftbt put dup 53 /bracketrightbt put dup 54 /bracketleftex put dup 55 /bracketrightex put dup 56 /bracelefttp put dup 57 /bracerighttp put dup 58 /braceleftbt put dup 59 /bracerightbt put dup 60 /braceleftmid put dup 61 /bracerightmid put dup 62 /braceex put dup 64 /parenleftbt put dup 65 /parenrightbt put dup 66 /parenleftex put dup 67 /parenrightex put dup 80 /summationtext put dup 81 /producttext put dup 88 /summationdisplay put dup 89 /productdisplay put dup 104 /bracketleftBig put dup 105 /bracketrightBig put dup 110 /braceleftBig put dup 111 /bracerightBig put dup 116 /radicalbt put dup 117 /radicalvertex put dup 118 /radicaltp put dup 122 /bracehtipdownleft put dup 123 /bracehtipdownright put dup 124 /bracehtipupleft put dup 125 /bracehtipupright put readonly def /FontBBox{-24 -2960 1454 772}readonly def /UniqueID 5000774 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758 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All Rights Reserved) readonly def /FullName (CMSY9) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY9 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put dup 6 /plusminus put dup 24 /similar put dup 33 /arrowright put readonly def /FontBBox{-30 -958 1146 777}readonly def /UniqueID 5000819 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMR6 %!PS-AdobeFont-1.1: CMR6 1.0 %%CreationDate: 1991 Aug 20 16:39:02 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR6 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 43 /plus put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put readonly def /FontBBox{-20 -250 1193 750}readonly def /UniqueID 5000789 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMR8 %!PS-AdobeFont-1.1: CMR8 1.0 %%CreationDate: 1991 Aug 20 16:39:40 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 14 /ffi put dup 39 /quoteright put dup 40 /parenleft put dup 41 /parenright put dup 43 /plus put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 61 /equal put dup 91 /bracketleft put dup 93 /bracketright put dup 95 /dotaccent put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 105 /i put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put readonly def /FontBBox{-36 -250 1070 750}readonly def /UniqueID 5000791 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758 469A6C558785B226332152109871A9883487DD7710949204DDCF837E6A8708B8 2BDBF16FBC7512FAA308A093FE5CF4E9D2405B169CD5365D6ECED5D768D66D6C 68618B8C482B341F8CA38E9BB9BAFCFAAD9C2F3FD033B62690986ED43D9C9361 3645B82392D5CAE11A7CB49D7E2E82DCD485CBA1772CE422BB1D7283AD675B65 48A7EA0069A883EC1DAA3E1F9ECE7586D6CF0A128CD557C7E5D7AA3EA97EBAD3 9619D1BFCF4A6D64768741EDEA0A5B0EFBBF347CDCBE2E03D756967A16B613DB 0FC45FA2A3312E0C46A5FD0466AB097C58FFEEC40601B8395E52775D0AFCD7DB 8AB317333110531E5C44A4CB4B5ACD571A1A60960B15E450948A5EEA14DD330F EA209265DB8E1A1FC80DCD3860323FD26C113B041A88C88A21655878680A4466 FA10403D24BB97152A49B842C180E4D258C9D48F21D057782D90623116830BA3 9902B3C5F2F2DD01433B0D7099C07DBDE268D0FFED5169BCD03D48B2F058AD62 D8678C626DC7A3F352152C99BA963EF95F8AD11DB8B0D351210A17E4C2C55AD8 9EB64172935D3C20A398F3EEEEC31551966A7438EF3FEE422C6D4E05337620D5 ACC7B52BED984BFAAD36EF9D20748B05D07BE4414A63975125D272FAD83F76E6 10FFF8363014BE526D580873C5A42B70FA911EC7B86905F13AFE55EB0273F582 83158793B8CC296B8DE1DCCF1250FD57CB0E035C7EDA3B0092ED940D37A05493 2EC54E09B984FCA4AB7D2EA182BCF1263AA244B07EC0EA901C077A059F709F30 4384CB5FA748F2054FAD9A7A43D4EA427918BD414F766531136B60C3477C6632 BEFE3897B58C19276A301926C2AEF2756B367319772C9B201C49B4D935A8267B 041D6F1783B6AEA4DAC4F5B3507D7032AA640AAB12E343A4E9BDCF419C04A721 3888B25AF4E293AACED9A6BDC78E61DA1C424C6503CC1885F762BFC1C033CB0F C3FAF5BEB2A12E58F8AF586A3E0C18B5EA359157AA22C689820759DB617B7715 8B000A31A83375C41CAE4C09A334F8D832EC683E246AED7136A93D880326D334 8845ABFFAEF3E1737F79EFAC7AE679FA8A21297A7D42B42BFBF136A35FBF4BD4 1C94DD1F923610208CBD1433C3B43004ADB30D24DAF76F6C2D9A4AF194544776 AD0E5547825589A51C6E1E44B55C6C33ED3FCA8DFE8AC371C60DBC4905804BE5 98FD6BEA9149E80DBA9A071E7C80FE42FE9068F648C95E70C8500CCBD66E3068 C44DE0A90978568D3BFAFCEE224909FF43E466EB6B5A139557CE32438A967C37 8A754D3C0D31B528A4BB843AA8A12C8260CF6077902A2633CEEFD5E80ADC712E C54D5A511E1EF2B3615275E329F9B9EB86968B3ACFC39C663083C8FDFBD742A4 4E2A34DD296AEB14C4943ABFA26BB6427610F52494C5F2F36C4B654E007BBDA8 6A8165A2ED676418CBBFB424EE60A05F19BABC1453544B82C900F75A0C0160F3 2D6C27221102758D5E6D96957E6CFD2D64F9C49A6A8E329FBBE2CB2DDEAFC99A 24431C4DC5E0840225F15F160AA5BBE49ED97712248CF920C772E9829C9C876D E559BEABBC2A5CB640034DFB5330A4939DD3D915F35A56A9A498EB6A81D30E59 0404F6F8FD44F208C9EEABCF966E7D77E51AC27DDE4047A322C2DD4F36FF3914 7DA6D24D2D3AC6EDDE1C61E45E7C245D04F68045616B19EF5B25E7F12EC90457 35A9AC43CEB17EB89F7F401C3C8A580396865C7AD4E57D0C03544B2141CC6566 FA3AD16B10246A7F5BB75CE9CEDE117A6CDE56B4669DD6310217546C79407624 3216B4C476889ACD0768480096E6F164BFA0DA5F8DF5BC9DEC25310161221A0F 20943AB70E5D0CC4546BA1E2BE23B89CC28D91D59297506C77574572ACEC73F1 5236379A8146DF11881BD128B41F5E9FE96DF1B66BE7412BCE26E5470473D551 EBB532B8E71F8140CDBFF8989C2C75A111289AC376304B474DB57457E7FD0E88 BE3C7442001A30F6CD5CD80AA3D29F0C07DC44BE642CB417018232DC52032D04 E3A3B93F3F8F087D35A7349A1F22C601E339ACB9FEB7127AE8BD75F7656595A4 15B8C76296E0A296E8DBB80B40833E090FCB3B342A8B318E8D5635B54D8E79C3 FD168A798F1E3FBA79F74D62A52057B7A39E529850DE51D37FABD32D5241795A CA91A86BC58D03A90EBC38F6D979D554FAB656B3A2F49FA673A5679361A9FA6F 7E0C6C20380B5EB89FE02708D27F3AEFD9BC6A4F2C720ADBADBCDB40FEC5E2A8 8A06830DF5DB9FF5A52599FB856D7CBFBD75CEAA6BB23D6E7EE66499445F4061 D76BAB05293F6C0330E7F822DCAA18A2934F626A07EC7F4638C31E583440E9D1 7B749BB0CB0B339EA4339881F1B7DDCC384596CE85C7192E97F35584102BCB65 11BFD3B09C355037487C744ADCB4B3FE1338592D74280B7B4A83C7E74E2FC9DC 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMTT9 %!PS-AdobeFont-1.1: CMTT9 1.0 %%CreationDate: 1991 Aug 20 16:46:24 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMTT9) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch true def end readonly def /FontName /CMTT9 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 46 /period put dup 47 /slash put dup 58 /colon put dup 64 /at put dup 80 /P put dup 97 /a put dup 100 /d put dup 101 /e put dup 103 /g put dup 104 /h put dup 105 /i put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 119 /w put dup 122 /z put readonly def /FontBBox{-6 -233 542 698}readonly def /UniqueID 5000831 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR9 %!PS-AdobeFont-1.1: CMR9 1.0 %%CreationDate: 1991 Aug 20 16:39:59 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR9) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR9 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 11 /ff put dup 12 /fi put dup 40 /parenleft put dup 41 /parenright put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 61 /equal put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 72 /H put dup 73 /I put dup 76 /L put dup 77 /M put dup 78 /N put dup 80 /P put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 87 /W put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 121 /y put dup 122 /z put readonly def /FontBBox{-39 -250 1036 750}readonly def /UniqueID 5000792 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMSY6 %!PS-AdobeFont-1.1: CMSY6 1.0 %%CreationDate: 1991 Aug 15 07:21:34 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY6 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put dup 3 /asteriskmath put dup 121 /dagger put readonly def /FontBBox{-4 -948 1329 786}readonly def /UniqueID 5000816 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMBX10 %!PS-AdobeFont-1.1: CMBX10 1.00B %%CreationDate: 1992 Feb 19 19:54:06 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMBX10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMBX10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 11 /ff put dup 12 /fi put dup 34 /quotedblright put dup 45 /hyphen put dup 46 /period put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 58 /colon put dup 65 /A put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 73 /I put dup 76 /L put dup 78 /N put dup 80 /P put dup 82 /R put dup 83 /S put dup 84 /T put dup 92 /quotedblleft put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put readonly def /FontBBox{-301 -250 1164 946}readonly def /UniqueID 5000768 def currentdict end currentfile eexec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All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR12 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 44 /comma put dup 48 /zero put dup 50 /two put dup 51 /three put dup 52 /four put dup 83 /S put dup 90 /Z put dup 97 /a put dup 98 /b put dup 101 /e put dup 103 /g put dup 104 /h put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 114 /r put dup 116 /t put readonly def /FontBBox{-34 -251 988 750}readonly def /UniqueID 5000794 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR17 %!PS-AdobeFont-1.1: CMR17 1.0 %%CreationDate: 1991 Aug 20 16:38:24 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR10 %!PS-AdobeFont-1.1: CMR10 1.00B %%CreationDate: 1992 Feb 19 19:54:52 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 1 /Delta put dup 5 /Pi put dup 10 /Omega put dup 11 /ff put dup 12 /fi put dup 13 /fl put dup 14 /ffi put dup 19 /acute put dup 33 /exclam put dup 34 /quotedblright put dup 39 /quoteright put dup 40 /parenleft put dup 41 /parenright put dup 43 /plus put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 47 /slash put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 61 /equal put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 89 /Y put dup 91 /bracketleft put dup 92 /quotedblleft put dup 93 /bracketright put dup 94 /circumflex put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put dup 123 /endash put dup 126 /tilde put readonly def /FontBBox{-251 -250 1009 969}readonly def /UniqueID 5000793 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMI8 %!PS-AdobeFont-1.1: CMMI8 1.100 %%CreationDate: 1996 Jul 23 07:53:54 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMBX12 %!PS-AdobeFont-1.1: CMBX12 1.0 %%CreationDate: 1991 Aug 20 16:34:54 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMBX12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMBX12 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 11 /ff put dup 12 /fi put dup 14 /ffi put dup 45 /hyphen put dup 46 /period put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 58 /colon put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 70 /F put dup 71 /G put dup 73 /I put dup 76 /L put dup 78 /N put dup 80 /P put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 90 /Z put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put readonly def /FontBBox{-53 -251 1139 750}readonly def /UniqueID 5000769 def currentdict end currentfile eexec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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont TeXDict begin 40258431 52099146 1000 600 600 (zroot4.dvi) @start /Fa 132[42 34 40 39 55 38 45 28 34 35 38 42 42 47 68 21 38 25 25 42 38 25 38 42 38 38 42 6[51 3[62 62 59 47 61 64 56 64 62 74 52 2[32 1[64 54 56 63 59 1[62 1[42 4[25 42 1[42 4[42 42 42 1[25 30 25 4[25 64 17[42 5[73 49 47 12[{}62 83.022 /CMTI10 rf /Fb 133[38 5[44 1[45 1[42 47 9[42 47 44 16[60 10[64 71[{}10 74.7198 /CMCSC10 rf /Fc 133[33 5[39 1[40 1[37 42 4[28 4[37 41 39 16[53 1[49 8[57 71[{}12 66.4176 /CMCSC10 rf /Fd 136[64 47 49 117[{}3 74.7198 /CMBX9 rf /Fe 222[172 32[134{}2 172.188 /CMSY10 rf /Ff 137[48 1[48 48 48 2[48 48 1[48 2[48 48 1[48 48 48 48 14[48 48 2[48 9[48 48 18[48 48 49[{}20 90.9091 /CMTT10 rf /Fg 131[35 1[35 35 1[35 35 35 35 35 35 1[35 35 35 35 35 35 35 35 35 35 35 35 35 35 35 35 3[35 1[35 3[35 35 35 35 35 35 1[35 35 35 35 35 35 1[35 35 35 35 35 35 35 35 35 6[35 35 35 35 35 35 35 35 35 35 35 35 35 35 35 35 1[35 35 40[{}66 66.4176 /CMTT8 rf /Fh 255[45{}1 41.511 /CMSY5 rf /Fi 205[28 28 49[{}2 41.511 /CMR5 rf /Fj 148[31 24 106[{}2 41.511 /CMMI5 rf /Fk 193[52 59[52 1[52{}3 58.1154 /CMSY7 rf /Fl 139[25 31 4[41 59 21 35 27 23 8[28 81[30 14[{}10 58.1154 /CMMI7 rf /Fm 148[42 3[42 42 47[0 3[55 25[65 2[65 19[23 65{}9 83.022 /CMSY10 rf /Fn 199[33 5[33 33 33 4[51 1[26 26 40[{}7 58.1154 /CMR7 rf /Fo 133[39 1[47 59 40 48 30 39 2[42 40 50 73 25 43 34 29 1[40 41 39 43 36 2[35 8[78 3[51 3[63 4[46 2[65 3[59 63 62 4[65 23 23 19[30 43 6[49 1[36 1[43 4[48 2[39 3[37 14[{}40 83.022 /CMMI10 rf /Fp 133[44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 12[44 44 44 1[44 1[44 3[44 44 1[44 2[44 44 1[44 6[44 4[44 44 44 44 44 1[44 44 44 44 2[44 44 44 39[{}50 83.022 /CMTT10 rf /Fq 135[44 7[39 6[26 6[33 41 32 8[73 12[42 14[21 21 58[{}10 74.7198 /CMMI9 rf /Fr 134[51 2[48 56 35 44 46 1[54 54 59 86 27 1[32 32 2[36 48 54 48 48 54 82[95 14[{}20 90.9091 /CMBXTI10 rf /Fs 135[35 9[38 54 1[33 25 22 7[34 9[57 59[36 27[{}9 49.8132 /CMMI6 rf /Ft 133[36 1[43 2[45 2[34 43 45[59 10[41 41 41 41 1[41 41 41 49[{}13 66.4176 /CMBX8 rf /Fu 130[37 37 37 37 3[88 88 88 4[55 55 4[39 39 14[106 120 6[78 88 12[73 73 73 73 1[74 74 74 74 74 74 74 55 55 55 55 55 55 73 73 7[67 4[48 48 66 66 12[61 61 50 50 2[46 28 10[38 38{}47 83.022 /CMEX10 rf /Fv 222[77 8[60 17[60 5[60{}4 74.7198 /CMSY9 rf /Fw 203[30 30 30 30 30 4[47 43[{}6 49.8132 /CMR6 rf /Fx 135[37 51 37 39 27 28 28 37 39 35 39 59 20 37 1[20 1[35 22 31 39 31 39 35 1[20 1[20 1[20 29[55 3[35 35 35 35 35 35 35 35 35 35 1[20 24 20 55 1[27 27 20 24[59 14[{}44 66.4176 /CMR8 rf /Fy 135[40 7[76 4[45 25 35 35 45 45 27[62 23[61 16[91 5[71 71 1[71 2[71 71 71 1[71 10[71 3[71 25 71{}22 90.9091 /CMSY10 rf /Fz 133[39 2[39 1[39 39 39 39 1[39 39 39 39 39 2[39 39 39 1[39 39 2[39 16[39 15[39 5[39 10[39 39 46[{}22 74.7198 /CMTT9 rf /FA 133[34 41 1[55 41 43 30 30 30 1[43 38 43 64 21 41 23 21 43 38 23 34 43 34 43 38 9[79 1[58 55 43 57 1[52 1[58 70 48 2[28 58 1[50 52 59 55 5[60 2[21 38 38 38 38 38 38 38 38 38 38 1[21 26 21 2[30 30 27[43 45 11[{}58 74.7198 /CMR9 rf /FB 134[29 117[32 2[48{}3 49.8132 /CMSY6 rf /FC 132[46 37 44 42 60 42 49 30 37 38 42 46 46 51 74 23 42 28 28 46 42 28 42 46 42 42 46 11[68 65 51 66 70 2[68 1[57 1[48 35 68 70 59 2[65 1[68 5[28 2[46 46 1[46 1[46 46 46 46 1[28 33 28 2[37 37 25[80 1[51 12[{}56 90.9091 /CMTI10 rf /FD 133[42 50 3[53 37 38 39 2[48 1[80 27 2[27 53 48 1[44 53 42 53 46 16[65 12[69 1[72 19[32 45[{}21 83.022 /CMBX10 rf /FE 134[31 13[35 44[55 7[0 3[47 71 19 14[71 28[55 1[55 20 55{}12 66.4176 /CMSY8 rf /FF 139[38 1[38 1[54 49 54 81 4[54 49 1[43 2[54 49 6[60 6[54 30[49 49 49 1[49 3[27 44[{}18 99.6264 /CMR12 rf /FG 252[50 3[{}1 99.6264 /CMSY10 rf /FH 134[70 70 2[73 51 52 51 1[73 66 73 111 36 2[36 1[66 40 58 1[58 1[66 29[96 67[{}18 143.462 /CMR17 rf /FI 133[46 56 2[56 56 54 42 55 1[51 58 56 1[47 1[39 27 1[58 1[51 57 54 1[56 13[56 73 1[67 2[90 5[77 1[67 69[{}24 90.9091 /CMCSC10 rf /FJ 128[46 4[42 51 2[51 51 49 38 50 1[46 53 51 62 43 53 35 25 51 53 44 46 52 49 48 51 6[56 68 1[92 68 68 65 51 66 1[62 1[68 82 57 70 47 34 68 71 59 62 69 65 64 68 18[27 31 27 44[{}50 83.022 /CMCSC10 rf /FK 132[42 37 44 44 60 44 46 32 33 33 44 46 42 46 69 23 44 25 23 46 42 25 37 46 37 46 42 2[42 23 1[23 1[62 1[85 62 62 60 46 61 1[57 1[62 76 52 1[43 30 62 65 54 57 63 60 59 62 3[65 3[42 42 42 42 42 42 42 42 42 42 42 23 28 23 65 1[32 32 23 24[69 1[46 48 9[69 1[{}74 83.022 /CMR10 rf /FL 133[33 35 40 51 34 1[25 33 2[36 1[43 62 21 37 29 24 1[34 34 33 3[37 29 8[67 4[53 56 8[58 6[53 37 4[20 20 20[36 2[33 6[40 8[33 3[31 14[{}32 66.4176 /CMMI8 rf /FM 133[42 1[52 65 44 52 33 43 41 41 46 44 55 80 27 47 37 31 52 43 45 42 47 39 39 48 38 8[86 3[56 69 72 58 69 2[62 1[50 40 1[71 58 1[75 65 1[68 48 1[71 45 71 25 25 19[33 47 2[42 57 2[54 1[40 52 47 5[52 1[43 2[37 40 47 1[58 11[{}60 90.9091 /CMMI10 rf /FN 129[45 3[40 48 48 66 48 51 35 36 36 48 51 45 51 76 25 48 28 25 51 45 28 40 51 40 51 45 2[45 25 45 25 1[68 1[93 68 68 66 51 67 71 62 71 68 83 57 71 47 33 68 71 59 62 69 66 64 68 3[71 2[25 45 45 45 45 45 45 45 45 45 45 45 25 30 25 71 1[35 35 25 4[45 25 13[45 4[76 51 51 53 66 4[68 3[76 1[{}85 90.9091 /CMR10 rf /FO 133[46 55 55 76 55 58 41 41 43 55 58 52 58 87 29 55 1[29 58 52 32 48 58 46 58 51 4[55 7[73 58 78 1[71 1[82 1[63 2[40 1[82 66 69 80 76 1[79 6[29 1[52 52 52 52 52 52 52 52 52 1[29 35 10[55 21[58 61 11[{}54 90.9091 /CMBX10 rf /FP 134[59 59 81 59 62 44 44 46 59 62 56 62 93 31 59 34 31 62 56 34 51 62 50 62 54 11[86 78 62 84 3[88 4[42 1[88 2[86 81 80 85 10[56 56 56 56 56 56 2[31 37 30[93 1[62 65 11[{}47 99.6264 /CMBX12 rf /FQ 134[71 1[97 71 75 52 53 55 1[75 67 75 112 37 2[37 75 67 41 61 75 60 75 65 6[82 5[94 1[100 1[92 1[105 1[81 2[50 2[85 2[97 1[102 6[37 4[67 67 67 67 67 3[45 32[75 12[{}39 119.552 /CMBX12 rf /FR 134[85 85 2[90 63 64 66 1[90 81 90 134 45 2[45 1[81 49 74 1[72 1[78 29[117 67[{}18 143.462 /CMBX12 rf end 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b(Computing)30 b(the)g(m)m(ultiplicit)m(y)j(structure)d (and)f(initial)j(ro)s(ot)f(estimates)65 b(.)46 b(.)g(.)f(.)h(.)g(.)f(.) h(.)g(.)117 b(24)231 4232 y(4.4)94 b(Con)m(trol)31 b(parameters)38 b(.)45 b(.)h(.)g(.)f(.)h(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)f(.)h(.)g(.) g(.)f(.)h(.)g(.)f(.)h(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)f(.)h(.)g(.)117 b(25)231 4345 y(4.5)94 b(Remarks)30 b(on)g(the)h(con)m(v)m(ergence)i (of)d(Algorithm)h(I)s(I)25 b(.)46 b(.)g(.)g(.)f(.)h(.)g(.)f(.)h(.)g(.)f (.)h(.)g(.)g(.)f(.)h(.)g(.)f(.)h(.)g(.)117 b(26)231 4458 y(4.6)94 b(Numerical)31 b(results)f(for)g(Algorithm)h(I)s(I)74 b(.)46 b(.)f(.)h(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)f(.)h(.)g(.)f(.)h(.) g(.)g(.)f(.)h(.)g(.)f(.)h(.)g(.)117 b(27)94 4662 y FO(5)85 b(Numerical)35 b(results)g(for)g(the)g(com)m(bined)h(metho)s(d)1475 b(28)231 4775 y FN(5.1)94 b(The)30 b(e\013ect)i(of)e(inexact)i(co)s (e\016cien)m(ts)d(.)46 b(.)g(.)g(.)f(.)h(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.) g(.)f(.)h(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)f(.)h(.)g(.)117 b(28)231 4888 y(5.2)94 b(The)30 b(e\013ect)i(of)e(nearb)m(y)g(m)m (ultiple)i(ro)s(ots)60 b(.)46 b(.)f(.)h(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g (.)f(.)h(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)f(.)h(.)g(.)117 b(29)231 5001 y(5.3)94 b(A)30 b(large)i(inexact)g(problem)60 b(.)45 b(.)h(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.) g(.)f(.)h(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)f(.)h(.)g(.)117 b(30)94 5204 y FO(References)3126 b(30)p eop %%Page: 2 2 2 1 bop 94 99 a FQ(List)46 b(of)f(Figures)231 301 y FN(1)164 b(The)30 b(Gauss-Newton)h(iteration)i(.)46 b(.)f(.)h(.)g(.)g(.)f(.)h(.) g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)f(.)h(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g (.)f(.)h(.)g(.)162 b(6)231 414 y(2)i(P)m(ejorativ)m(e)33 b(manifolds)66 b(.)46 b(.)g(.)f(.)h(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)f (.)h(.)g(.)g(.)f(.)h(.)g(.)f(.)h(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)f(.) h(.)g(.)162 b(9)231 527 y(3)i(The)30 b(sensitivit)m(y)i(and)d(ro)s(ot)i (m)m(ultiplicities)84 b(.)46 b(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)f(.)h (.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)f(.)h(.)g(.)117 b(14)231 640 y(4)164 b(Pseudo-co)s(des)30 b(for)g(ev)-5 b(aluating)32 b FM(G)1647 655 y FL(`)1681 640 y FN(\()p FO(z)p FN(\))f(and)f FM(J)9 b FN(\()p FO(z)p FN(\))58 b(.)46 b(.)g(.)g(.)f(.)h(.)g(.)f(.)h (.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)f(.)h(.)g(.)117 b(14)231 753 y(5)164 b(Pseudo-co)s(de)30 b(of)h(Algorithm)g(I)64 b(.)46 b(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)f(.) h(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)f(.)h(.)g(.)117 b(15)231 866 y(6)164 b(Ro)s(ots)31 b(of)f(a)h(p)s(olynomial)g(with)f(clustered)g (m)m(ultiple)i(ro)s(ots)e(calculated)i(b)m(y)f(Matlab)71 b(.)46 b(.)g(.)117 b(17)231 979 y(7)164 b(Matlab)32 b(result)e(for)g(a) h(p)s(olynomial)f(of)h(large)g(degree)h(and)d(high)h(m)m(ultiplicities) 36 b(.)46 b(.)g(.)f(.)h(.)g(.)117 b(18)231 1092 y(8)164 b(Sparsit)m(y)30 b(of)h(the)f(Jacobian)58 b(.)45 b(.)h(.)g(.)f(.)h(.)g (.)g(.)f(.)h(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)f(.)h(.)g(.)f(.)h(.)g(.) g(.)f(.)h(.)g(.)f(.)h(.)g(.)117 b(24)231 1205 y(9)164 b(Pseudo-co)s(de)30 b(of)h(Algorithm)g(I)s(I)d(.)46 b(.)f(.)h(.)g(.)g (.)f(.)h(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)f(.)h(.)g(.)f(.)h(.)g(.)g(.) f(.)h(.)g(.)f(.)h(.)g(.)117 b(26)231 1318 y(10)i(MPSolv)m(e)32 b(results)e(for)g(the)g(p)s(olynomial)h(\(32\))h(using)e(m)m (ultiprecision)80 b(.)46 b(.)g(.)g(.)f(.)h(.)g(.)f(.)h(.)g(.)117 b(27)94 1604 y FQ(List)46 b(of)f(T)-11 b(ables)231 1807 y FN(1)164 b(Comparison)30 b(with)g(the)g(F)-8 b(armer-Loizou)32 b(third)e(order)g(iteration)i(in)e(lo)m(w)h(m)m(ultiplicit)m(y)h(case) 53 b(16)231 1920 y(2)164 b(Comparison)26 b(with)g(the)h(F)-8 b(armer-Loizou)29 b(third)d(order)g(iteration)i(in)e(high)h(m)m (ultiplicit)m(y)h(case)52 b(16)231 2033 y(3)164 b FK(Comparison)26 b(with)i(m)n(ultiprecision)f(ro)r(ot-\014nders)f FJ(MPSol)-7 b(ve)26 b FK(and)i FJ(Eigensol)-7 b(ve)53 b FN(.)45 b(.)h(.)g(.)f(.)h (.)g(.)117 b(16)231 2146 y(4)164 b(P)m(artial)32 b(list)f(of)g(m)m (ultiple)g(ro)s(ots)f(on)g(di\013eren)m(t)h(p)s(ejorativ)m(e)h (manifolds)69 b(.)46 b(.)g(.)g(.)f(.)h(.)g(.)f(.)h(.)g(.)117 b(17)231 2259 y(5)164 b(The)30 b(comparison)g(b)s(et)m(w)m(een)h(the)g (conditions)g(of)f(\(26\))i(and)e(\(27\))i(for)e FM(v)s FN(\()p FM(x)p FN(\))c(=)f FM(x)20 b FN(+)g(25.)89 b(.)46 b(.)117 b(23)231 2372 y(6)164 b(A)30 b(n)m(umerical)h(comparison)g(b)s (et)m(w)m(een)g(long)g(division)f(and)g(least)i(squares)e(division)i(.) 45 b(.)h(.)g(.)117 b(23)231 2484 y(7)164 b(Ro)s(ots)31 b(of)f FM(p)p FN(\()p FM(x)p FN(\))h(in)f(\(32\))i(computed)e(in)g(t)m (w)m(o)i(stages)73 b(.)46 b(.)g(.)f(.)h(.)g(.)f(.)h(.)g(.)f(.)h(.)g(.)g (.)f(.)h(.)g(.)f(.)h(.)g(.)117 b(27)231 2597 y(8)164 b(Comparison)30 b(b)s(et)m(w)m(een)h FI(pzer)n(o)e FN(and)g FI(GcdR)m(oot)49 b FN(.)d(.)g(.)g(.)f(.)h(.)g(.)f(.)h(.)g(.)f(.)h(.)g (.)g(.)f(.)h(.)g(.)f(.)h(.)g(.)117 b(28)231 2710 y(9)164 b(E\013ect)31 b(of)g(co)s(e\016cien)m(t)h(error)e(on)g(computed)h(ro)s (ots)40 b(.)46 b(.)g(.)g(.)f(.)h(.)g(.)f(.)h(.)g(.)f(.)h(.)g(.)g(.)f(.) h(.)g(.)f(.)h(.)g(.)117 b(28)231 2823 y(10)i(E\013ect)31 b(of)g(decreasing)g(ro)s(ot)g(gap)f(on)h(computed)f(ro)s(ots)94 b(.)45 b(.)h(.)g(.)f(.)h(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)f(.)h(.)g(.) 117 b(29)231 2936 y(11)i(E\013ect)31 b(of)g(tin)m(y)g(ro)s(ot)f(gap)63 b(.)46 b(.)f(.)h(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)f(.)h(.)g(.)g(.)f(.) h(.)g(.)f(.)h(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)f(.)h(.)g(.)117 b(29)p eop %%Page: 1 3 1 2 bop 521 463 a FH(Computing)43 b(m)l(ultiple)g(ro)t(ots)f(of)h (inexact)g(p)t(olynomials)3388 411 y FG(\003)1602 716 y FF(Zhonggang)32 b(Zeng)2307 679 y FE(y)1537 919 y FF(Septem)m(b)s(er) j(24,)d(2003)1772 1253 y FD(Abstract)446 1401 y FK(W)-7 b(e)35 b(presen)n(t)e(a)h(com)n(bination)f(of)h(t)n(w)n(o)g(algorithms) e(that)j(accurately)d(calculate)i(m)n(ultiple)g(ro)r(ots)322 1501 y(of)i(general)f(p)r(olynomials.)63 b(Algorithm)36 b(I)h(transforms)e(the)i(singular)e(ro)r(ot-\014nding)h(in)n(to)g(a)g (regular)322 1601 y(nonlinear)26 b(least)g(squares)f(problem)i(on)f(a)h (p)r(ejorativ)n(e)f(manifold,)h(and)g(calculates)e(m)n(ultiple)j(ro)r (ots)e(si-)322 1700 y(m)n(ultaneously)d(from)i(a)f(giv)n(en)f(m)n (ultiplicit)n(y)i(structure)f(and)g(initial)h(ro)r(ot)f(appro)n (ximations.)33 b(T)-7 b(o)24 b(ful\014ll)322 1800 y(the)36 b(input)g(requiremen)n(t)f(of)g(Algorithm)h(I,)f(w)n(e)h(dev)n(elop)e (a)i(n)n(umerical)e(GCD-\014nder)i(con)n(taining)f(a)322 1899 y(successiv)n(e)e(singular)h(v)-5 b(alue)35 b(up)r(dating)g(and)f (an)h(iterativ)n(e)f(GCD)i(re\014nemen)n(t)e(as)g(the)i(main)f(engine) 322 1999 y(of)c(Algorithm)g(I)r(I)h(that)g(calculates)e(the)i(m)n (ultiplicit)n(y)g(structure)f(and)g(the)h(initial)g(ro)r(ot)e(appro)n (xima-)322 2099 y(tion.)51 b(While)32 b(limitations)g(of)h(our)e (algorithm)g(exist)h(in)h(iden)n(tifying)f(the)h(m)n(ultiplicit)n(y)f (structure)g(in)322 2198 y(certain)27 b(situations,)h(the)g(com)n (bined)g(metho)r(d)g(calculates)f(m)n(ultiple)i(ro)r(ots)e(with)h(high) g(accuracy)e(and)322 2298 y(consistency)i(in)h(practice)f(without)h (using)f(m)n(ultiprecision)h(arithmetic)f(ev)n(en)h(if)g(the)g(co)r (e\016cien)n(ts)f(are)322 2398 y(inexact.)68 b(This)39 b(is)f(p)r(erhaps)f(the)i(\014rst)f(blac)n(kb)r(o)n(x-t)n(yp)r(e)f(ro)r (ot-\014nder)g(with)h(suc)n(h)g(capabilities.)69 b(T)-7 b(o)322 2497 y(measure)32 b(the)h(sensitivit)n(y)f(of)h(the)h(m)n (ultiple)f(ro)r(ots,)g(a)g(structure-preserving)d(condition)j(n)n(um)n (b)r(er)f(is)322 2597 y(prop)r(osed)d(and)h(error)f(b)r(ounds)h(are)f (established.)45 b(According)29 b(to)i(our)e(computational)h(exp)r (erimen)n(ts)322 2696 y(and)24 b(error)f(analysis,)h(a)g(p)r(olynomial) f(b)r(eing)i(ill-conditioned)f(in)h(the)g(con)n(v)n(en)n(tional)e (sense)h(can)g(b)r(e)h(w)n(ell)322 2796 y(conditioned)k(with)g(the)h(m) n(ultiplicit)n(y)f(structure)g(b)r(eing)g(preserv)n(ed,)f(and)g(its)i (m)n(ultiple)f(ro)r(ots)f(can)h(b)r(e)322 2896 y(computed)f(with)g (high)f(accuracy)-7 b(.)94 3278 y FQ(1)135 b(In)l(tro)t(duction)94 3505 y FN(In)34 b(this)g(pap)s(er,)g(w)m(e)g(presen)m(t)g(a)h(com)m (bination)g(of)f(t)m(w)m(o)h(n)m(umerical)g(algorithms)g(for)f (computing)g(m)m(ultiple)94 3618 y(ro)s(ots)24 b(and)f(the)h(m)m (ultiplicit)m(y)i(structures)d(of)h(p)s(olynomials.)39 b(According)24 b(to)g(our)g(extensiv)m(e)h(computational)94 3731 y(exp)s(erimen)m(ts)37 b(and)e(error)h(estimates,)k(the)d(metho)s (d)e(accurately)j(calculates)h(p)s(olynomial)e(ro)s(ots)f(of)h(non-)94 3843 y(trivial)23 b(m)m(ultiplicities)h(without)e(using)f(m)m (ultiprecision)h(arithmetic,)j(ev)m(en)e(if)e(the)h(co)s(e\016cien)m (ts)i(are)e(inexact.)94 4051 y(P)m(olynomial)31 b(ro)s(ot-\014nding)e (is)g(among)h(the)f(classical)j(problems)c(with)h(longest)i(and)d(ric)m (hest)i(history)-8 b(.)41 b(One)94 4164 y(of)27 b(the)f(most)h (di\016cult)f(issues)f(in)h(ro)s(ot-\014nding)g(is)g(computing)g(m)m (ultiple)h(ro)s(ots.)39 b(In)26 b(addition)g(to)h(requiring)94 4277 y FC(exact)d 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b(Multiprecision)35 b(soft)m(w)m(ares)h(suc)m(h)f (as)g([1])g(are)g(a)m(v)-5 b(ailable.)56 b(Ho)m(w)m(ev)m(er,)38 b(when)94 5067 y(p)s(olynomial)27 b(co)s(e\016cien)m(ts)g(are)f (truncated,)h(m)m(ultiple)g(ro)s(ots)f(turn)f(in)m(to)h(clusters,)h (and)f(extending)g(mac)m(hine)94 5180 y(precision)g(will)h(nev)m(er)f (rev)m(erse)g(clusters)g(bac)m(k)h(to)f(m)m(ultiple)h(ro)s(ots.)39 b(In)25 b(the)h(absence)h(of)f(accurate)h(metho)s(ds)p 94 5258 1488 4 v 197 5312 a FB(\003)233 5343 y FA(Mathematics)g(Sub)t (ject)e(Classi\014cation)80 b(65F35,)27 b(65H05)200 5405 y FB(y)233 5437 y FA(Dept)e(of)h(Math,)g(Northeastern)g(Illinois)h (Univ)n(ersit)n(y)-6 b(,)25 b(Chicago,)i(IL)f(60625,)79 b(email:)35 b Fz(zzeng@neiu.edu)p FA(.)1932 5686 y FN(1)p eop %%Page: 2 4 2 3 bop 94 99 a FN(that)41 b(are)g(indep)s(enden)m(t)e(of)h(m)m (ultiprecision)h(tec)m(hnology)-8 b(,)46 b(m)m(ultiple)41 b(ro)s(ots)f(of)h(p)s(erturb)s(ed)c(p)s(olynomials)94 211 y(w)m(ould)31 b(indeed)f(b)s(e)f(in)m(tractable.)94 419 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y(Ho)m(w)m(ev)m(er,)32 b(the)c(barrier)g(of)h (\\attainable)i(accuracy")f(ma)m(y)f(prev)m(en)m(t)g(those)g(ro)s (ot-\014nders)e(from)h(calculating)94 3431 y(m)m(ultiple)i(ro)s(ots)e (accurately)j(when)c(the)i(p)s(olynomials)g(are)g(inexact)h(\(e.g.)41 b(see)29 b(Figure)g(10)h(in)e Fy(x)p FN(4.6\))i(ev)m(en)f(if)94 3543 y(m)m(ultiprecision)d(is)f(used.)38 b(Our)23 b(algorithms)j(pro)m (vide)f(an)g(option)g(of)g(reac)m(hing)h(high)e(accuracy)i(on)f(m)m (ultiple)94 3656 y(ro)s(ots)31 b(at)g(higher)f(computing)h(cost)g(of)g FM(O)s FN(\()p FM(n)1617 3623 y Fx(3)1656 3656 y FN(\))f(whic)m(h)h(ma) m(y)g(not)f(b)s(e)g(a)h(loft)m(y)g(price)g(to)g(pa)m(y)-8 b(.)94 3864 y(The)39 b(idea)h(of)g(exploiting)h(the)e(p)s(ejorativ)m(e) i(manifold)e(and)g(the)g(problem)g(structure)g(has)g(b)s(een)g(applied) 94 3977 y(extensiv)m(ely)23 b(for)d(ill-conditioned)i(problems.)37 b(Besides)21 b(Kahan's)f(pioneering)h(w)m(ork)g(30)g(y)m(ears)h(ago,)i (theories)94 4090 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b(en)m(tries)h(in)f(a)g(matrix)g(are)g(\014lled)g(with)g (zeros.)40 b(The)25 b(notation)p 94 5285 1488 4 v 198 5338 a Fw(1)233 5370 y Fz(http://www.neiu.edu/)p Fv(\030)p Fz(zzeng/multro)q(ot.h)q(tm)1932 5686 y FN(3)p eop %%Page: 4 6 4 5 bop 94 99 a FN(\()p Fy(\001)p FN(\))189 66 y FE(>)280 99 y FN(represen)m(ts)31 b(the)f(transp)s(ose)g(of)h(\()p Fy(\001)p FN(\),)h(and)e(\()p Fy(\001)p FN(\))1801 66 y FL(H)1900 99 y FN(the)h(Hermitian)g(adjoin)m(t)g(\(i.e.)43 b(conjugate)32 b(transp)s(ose\))94 211 y(of)39 b(\()p Fy(\001)p FN(\).)67 b(When)38 b(w)m(e)i(use)e(a)h(\(lo)m(w)m(er)h (case\))h(letter,)h(sa)m(y)d FM(p)p FN(,)i(to)e(denote)h(a)f(p)s (olynomial)g(of)g(degree)g FM(n)p FN(,)i(then)94 324 y FM(p)140 338 y Fx(0)180 324 y FM(;)15 b(p)266 338 y Fx(1)305 324 y FM(;)g Fy(\001)g(\001)g(\001)i FM(;)e(p)553 338 y FL(n)661 324 y FN(are)30 b(its)h(co)s(e\016cien)m(ts)h(as)f(in) 1302 521 y FM(p)p FN(\()p FM(x)p FN(\))25 b(=)g FM(p)1637 535 y Fx(0)1677 521 y FM(x)1729 483 y FL(n)1796 521 y FN(+)20 b FM(p)1933 535 y Fx(1)1972 521 y FM(x)2024 483 y FL(n)p FE(\000)p Fx(1)2181 521 y FN(+)g Fy(\001)15 b(\001)g(\001)21 b FN(+)f FM(p)2535 535 y FL(n)2582 521 y FM(:)94 717 y FN(The)30 b(same)h(letter)h(in)e(b)s(oldface)g(\(e.g.) 43 b FO(p)p FN(\))30 b(denotes)h(the)g(co)s(e\016cien)m(t)h(\(column\)) f(v)m(ector)1532 913 y FO(p)25 b FN(=)g(\()15 b FM(p)1807 927 y Fx(0)1847 913 y FM(;)g(p)1933 927 y Fx(1)1973 913 y FM(;)g Fy(\001)g(\001)g(\001)h FM(;)f(p)2220 927 y FL(n)2283 913 y FN(\))2318 876 y FE(>)94 1109 y FN(unless)35 b(it)g(is)g(de\014ned)f(otherwise.)54 b(The)35 b(degree)g(of)g FM(p)g FN(is)g FM(deg)s FN(\()p FM(p)p FN(\).)56 b(F)-8 b(or)36 b(a)f(pair)f(of)h(p)s(olynomials)h FM(p)e FN(and)g FM(q)s FN(,)94 1222 y(their)d(greatest)h(common)f(divisor)f(\(GCD\))h (is)g(denoted)f(b)m(y)h FM(GC)7 b(D)s FN(\()p FM(p;)15 b(q)s FN(\).)94 1464 y FP(2.2)113 b(Basic)37 b(de\014nitions)i(and)g (lemmas)94 1636 y FO(De\014nition)d(2.1)46 b FC(L)-5 b(et)71 b FM(p)p FN(\()p FM(x)p FN(\))30 b(=)g FM(p)1302 1650 y Fx(0)1341 1636 y FM(x)1393 1603 y FL(n)1462 1636 y FN(+)22 b FM(p)1601 1650 y Fx(1)1640 1636 y FM(x)1692 1603 y FL(n)p FE(\000)p Fx(1)1852 1636 y FN(+)f Fy(\001)15 b(\001)g(\001)24 b FN(+)e FM(p)2212 1650 y FL(n)2329 1636 y FC(b)-5 b(e)35 b(a)h(p)-5 b(olynomial)38 b(of)e(de)-5 b(gr)g(e)g(e)36 b FM(n)p FC(.)49 b(F)-7 b(or)37 b(any)94 1749 y(inte)-5 b(ger)33 b FM(k)c Fy(\025)c FN(0)p FC(,)33 b(the)g(matrix)1411 2289 y FM(C)1476 2304 y FL(k)1518 2289 y FN(\()p FM(p)p FN(\))26 b(=)2063 1829 y FL(k)r Fx(+1)1834 1881 y Fu(z)p 1871 1881 219 10 v 219 w(}|)p 2164 1881 V 219 w({)1798 1953 y(2)1798 2099 y(6)1798 2149 y(6)1798 2199 y(6)1798 2249 y(6)1798 2299 y(6)1798 2349 y(6)1798 2398 y(6)1798 2448 y(6)1798 2498 y(6)1798 2548 y(6)1798 2601 y(4)1898 2019 y FM(p)1944 2033 y Fx(0)1898 2178 y FM(p)1944 2192 y Fx(1)2075 2120 y FC(.)2113 2145 y(.)2151 2170 y(.)1927 2272 y(.)1927 2305 y(.)1927 2338 y(.)2075 2280 y(.)2113 2305 y(.)2151 2330 y(.)2271 2338 y FM(p)2317 2352 y Fx(0)1894 2451 y FM(p)1940 2465 y FL(n)2271 2451 y FM(p)2317 2465 y Fx(1)2075 2553 y FC(.)2113 2577 y(.)2151 2603 y(.)2300 2544 y(.)2300 2577 y(.)2300 2611 y(.)2267 2724 y FM(p)2313 2738 y FL(n)2401 1953 y Fu(3)2401 2099 y(7)2401 2149 y(7)2401 2199 y(7)2401 2249 y(7)2401 2299 y(7)2401 2349 y(7)2401 2398 y(7)2401 2448 y(7)2401 2498 y(7)2401 2548 y(7)2401 2601 y(5)94 2879 y FC(is)33 b(c)-5 b(al)5 b(le)-5 b(d)34 b(the)f FM(k)s FC(-th)g(or)-5 b(der)34 b FO(con)m(v)m(olution)j(matrix)32 b FC(asso)-5 b(ciate)g(d)35 b(with)f FM(p)p FC(.)94 3128 y FO(Lemma)h(2.1)71 b FC(L)-5 b(et)26 b FM(f)35 b FC(and)26 b FM(g)j FC(b)-5 b(e)25 b(p)-5 b(olynomials)29 b(of)c(de)-5 b(gr)g(e)g(es)27 b FM(n)e FC(and)h FM(m)f FC(r)-5 b(esp)g(e)g(ctively) 27 b(with)f FM(h)p FN(\()p FM(x)p FN(\))g(=)f FM(f)10 b FN(\()p FM(x)p FN(\))p FM(g)s FN(\()p FM(x)p FN(\))p FC(.)94 3241 y(Then)66 b FO(h)f FC(is)33 b(the)g FO(con)m(v)m(olution)h FC(of)65 b FO(f)75 b FC(and)66 b FO(g)h FC(de\014ne)-5 b(d)34 b(by)1261 3354 y FO(h)25 b FN(=)g FM(conv)s FN(\()p FO(f)10 b FM(;)15 b FO(g)q FN(\))27 b(=)e FM(C)2018 3368 y FL(m)2085 3354 y FN(\()p FM(f)10 b FN(\))p FO(g)27 b FN(=)e FM(C)2450 3368 y FL(n)2496 3354 y FN(\()p FM(g)s FN(\))p FO(f)10 b FM(:)94 3582 y FO(Pro)s(of.)83 b FN(A)30 b(straigh)m(tforw)m(ard)h(v)m(eri\014cation.)1877 b(Q.E.D.)94 3810 y FO(De\014nition)36 b(2.2)46 b FC(L)-5 b(et)25 b FM(p)f FC(b)-5 b(e)25 b(a)g(p)-5 b(olynomial)28 b(of)c(de)-5 b(gr)g(e)g(e)26 b FM(n)e FC(and)i FM(p)2261 3777 y FE(0)2309 3810 y FC(b)-5 b(e)24 b(its)h(derivative.)40 b(F)-7 b(or)26 b FM(k)i FN(=)d(1)p FM(;)15 b FN(2)p FM(;)g Fy(\001)g(\001)g(\001)j FM(n)s Fy(\000)s FN(1)p FC(,)94 3923 y(the)33 b(matrix)h(of)f(size)g FN(\()p FM(n)20 b FN(+)g FM(k)s FN(\))h Fy(\002)f FN(\(2)p FM(k)k FN(+)c(1\))1320 4135 y FM(S)1376 4150 y FL(k)1419 4135 y FN(\()p FM(p)p FN(\))26 b(=)1657 4041 y Fu(h)1737 4135 y FM(C)1802 4150 y FL(k)1860 4135 y FN(\()p FM(p)1941 4102 y FE(0)1964 4135 y FN(\))2083 4037 y Fu(\014)2083 4087 y(\014)2083 4137 y(\014)2194 4135 y FM(C)2259 4150 y FL(k)r FE(\000)p Fx(1)2391 4135 y FN(\()p FM(p)p FN(\))2549 4041 y Fu(i)94 4342 y FC(is)33 b(c)-5 b(al)5 b(le)-5 b(d)34 b(the)f FM(k)s FC(-th)g FO(Sylv)m(ester)i(discriminan)m(t)h (matrix)p FC(.)94 4661 y FO(Lemma)f(2.2)46 b FC(L)-5 b(et)34 b FM(p)e FC(b)-5 b(e)34 b(a)f(p)-5 b(olynomial)36 b(of)e(de)-5 b(gr)g(e)g(e)34 b FM(n)f FC(and)h FM(p)2220 4628 y FE(0)2276 4661 y FC(b)-5 b(e)33 b(its)g(derivative)h(with)67 b FM(u)26 b FN(=)g FM(GC)7 b(D)s FN(\()p FM(p;)15 b(p)3728 4628 y FE(0)3751 4661 y FN(\))p FC(.)94 4774 y(F)-7 b(or)27 b FM(j)k FN(=)25 b(1)p FM(;)15 b Fy(\001)g(\001)g(\001)i FM(;)e(n)p FC(,)26 b(let)f FM(&)930 4788 y FL(j)992 4774 y FC(b)-5 b(e)25 b(the)h(smal)5 b(lest)26 b(singular)g(value)f(of)h FM(S)2306 4788 y FL(j)2342 4774 y FN(\()p FM(p)p FN(\))p FC(.)40 b(Then)25 b(the)h(fol)5 b(lowing)26 b(ar)-5 b(e)26 b(e)-5 b(quivalent)155 5045 y(\(a\))112 b FM(deg)s FN(\()p FM(u)p FN(\))27 b(=)e FM(m)p FC(,)160 5229 y(\(b\))111 b FM(p)65 b FC(has)h FM(k)28 b FN(=)d FM(n)20 b Fy(\000)g FM(m)65 b FC(distinct)33 b(r)-5 b(o)g(ots,)160 5414 y(\(c\))143 b FM(&)452 5428 y Fx(1)492 5414 y FM(;)40 b(&)590 5428 y Fx(2)630 5414 y FM(;)g Fy(\001)15 b(\001)g(\001)i FM(;)e(&)890 5429 y FL(k)r FE(\000)p Fx(1)1048 5414 y FM(>)25 b FN(0)p FM(;)168 b(&)1415 5429 y FL(k)1482 5414 y FN(=)25 b FM(&)1611 5429 y FL(k)r Fx(+1)1769 5414 y FN(=)g Fy(\001)15 b(\001)g(\001)27 b FN(=)e FM(&)2126 5428 y FL(n)2198 5414 y FN(=)g(0)p FC(.)1932 5686 y FN(4)p eop %%Page: 5 7 5 6 bop 94 99 a FO(Pro)s(of.)83 b FN(The)30 b(equiv)-5 b(alence)32 b(b)s(et)m(w)m(een)f(\(a\))h(and)e(\(b\))g(is)h(trivial)g (to)g(v)m(erify)-8 b(,)32 b(and)e(the)h(assertion)g(that)g(\(a\))g(is) 94 211 y(equiv)-5 b(alen)m(t)32 b(to)f(\(c\))h(is)e(part)g(of)h(Prop)s (osition)f(3.1)i(in)e([28)q(].)1458 b(Q.E.D.)94 441 y FO(Lemma)35 b(2.3)46 b FC(L)-5 b(et)37 b FM(p)f FC(b)-5 b(e)37 b(a)g(p)-5 b(olynomial)40 b(of)c(de)-5 b(gr)g(e)g(e)38 b FM(n)e FC(and)i FM(p)2251 408 y FE(0)2310 441 y FC(b)-5 b(e)37 b(its)g(derivative)g(with)g FM(u)c FN(=)f FM(GC)7 b(D)s FN(\()p FM(p;)15 b(p)3756 408 y FE(0)3779 441 y FN(\))94 554 y FC(and)34 b FM(deg)s FN(\()p FM(u)p FN(\))27 b(=)e FM(m)g FN(=)g FM(n)20 b Fy(\000)g FM(k)s FC(.)42 b(L)-5 b(et)33 b FM(v)j FC(and)d FM(w)i FC(b)-5 b(e)33 b(p)-5 b(olynomials)36 b(that)d(satisfy)1228 751 y FM(u)p FN(\()p FM(x)p FN(\))p FM(v)s FN(\()p FM(x)p FN(\))27 b(=)e FM(p)p FN(\()p FM(x)p FN(\))p FM(;)91 b(u)p FN(\()p FM(x)p FN(\))p FM(w)r FN(\()p FM(x)p FN(\))27 b(=)e FM(p)2510 714 y FE(0)2533 751 y FN(\()p FM(x)p FN(\))p FM(:)94 949 y FC(Then)155 1107 y(\(a\))79 b FM(v)36 b FC(and)e FM(w)h FC(ar)-5 b(e)33 b(c)-5 b(o-prime,)34 b(namely)f(they)g(have)h (no)f(c)-5 b(ommon)34 b(factors;)160 1292 y(\(b\))46 b(the)33 b(\(c)-5 b(olumn\))34 b(r)-5 b(ank)33 b(of)66 b FM(S)1262 1307 y FL(k)1304 1292 y FN(\()p FM(p)p FN(\))33 b FC(is)g(de\014cient)f(by)h(one;)160 1543 y(\(c\))46 b(the)39 b(normalize)-5 b(d)41 b(ve)-5 b(ctor)1220 1399 y Fu(")1401 1487 y FO(v)1310 1599 y Fy(\000)p FO(w)1499 1399 y Fu(#)1586 1543 y FC(is)38 b(the)h(right)g(singular)g(ve)-5 b(ctor)39 b(of)g FM(S)2865 1558 y FL(k)2907 1543 y FN(\()p FM(p)p FN(\))g FC(asso)-5 b(ciate)g(d)41 b(with)e(the)322 1721 y(smal)5 b(lest)34 b(\(zer)-5 b(o\))34 b(singular)f(value)g FM(&)1554 1736 y FL(k)1596 1721 y FC(;)155 1907 y(\(d\))47 b(if)39 b FO(v)i FC(is)f(known,)i(the)e(c)-5 b(o)g(e\016cient)41 b(ve)-5 b(ctor)40 b FO(u)g FC(of)g FM(u)e FN(=)g FM(GC)7 b(D)s FN(\()p FM(p;)15 b(p)2604 1874 y FE(0)2627 1907 y FN(\))40 b FC(is)g(the)g(solution)h(to)f(the)g(line)-5 b(ar)322 2020 y(system)98 b FM(C)749 2034 y FL(m)816 2020 y FN(\()p FM(v)s FN(\))p FO(u)26 b FN(=)f FO(p)p FM(:)94 2315 y FO(Pro)s(of.)80 b FN(Assertion)29 b(\(a\))h(is)f (trivial.)70 b FM(S)1482 2330 y FL(k)1525 2315 y FN(\()p FM(p)p FN(\))1656 2171 y Fu(")1837 2258 y FO(v)1746 2371 y Fy(\000)p FO(w)1935 2171 y Fu(#)2009 2315 y FN(=)25 b FM(C)2170 2330 y FL(k)2228 2315 y FN(\()p FM(p)2309 2282 y FE(0)2347 2315 y FN(\))16 b FO(v)i Fy(\000)f FM(C)2624 2330 y FL(k)r FE(\000)p Fx(1)2757 2315 y FN(\()p FM(p)p FN(\))p FO(w)27 b FN(=)d(0)58 b(b)s(ecause)29 b(it)g(is)g(the)94 2504 y(co)s(e\016cien)m(t)c(v)m(ector)f(of)45 b FM(p)940 2471 y FE(0)963 2504 y FM(v)8 b Fy(\000)t FM(pw)52 b Fy(\021)e FN(\()p FM(uw)r FN(\))p FM(v)8 b Fy(\000)t FN(\()p FM(uv)s FN(\))p FM(w)56 b Fy(\021)50 b FN(0.)38 b(Let)2392 2503 y(^)2387 2504 y FO(v)26 b Fy(2)f FO(C)2630 2471 y FL(k)r Fx(+1)2785 2504 y FN(and)2970 2503 y(^)2954 2504 y FO(w)h Fy(2)f FO(C)3218 2471 y FL(k)3305 2504 y FN(b)s(e)d(co)s(e\016cien)m(t)94 2617 y(v)m(ectors)30 b(of)f(p)s(olynomials)j(^)-49 b FM(v)32 b FN(and)46 b(^)-63 b FM(w)31 b FN(resp)s(ectiv)m(ely)e(that)g(also)h(satisfy)57 b FM(C)2597 2632 y FL(k)2655 2617 y FN(\()p FM(p)2736 2584 y FE(0)2774 2617 y FN(\))2830 2616 y(^)2825 2617 y FO(v)18 b Fy(\000)e FM(C)3050 2632 y FL(k)r FE(\000)p Fx(1)3182 2617 y FN(\()p FM(p)p FN(\))3315 2616 y(^)3298 2617 y FO(w)27 b FN(=)e(0.)40 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} repeat newfontname newfont definefont pop end } def /isovec [ 8#055 /minus 8#200 /grave 8#201 /acute 8#202 /circumflex 8#203 /tilde 8#204 /macron 8#205 /breve 8#206 /dotaccent 8#207 /dieresis 8#210 /ring 8#211 /cedilla 8#212 /hungarumlaut 8#213 /ogonek 8#214 /caron 8#220 /dotlessi 8#230 /oe 8#231 /OE 8#240 /space 8#241 /exclamdown 8#242 /cent 8#243 /sterling 8#244 /currency 8#245 /yen 8#246 /brokenbar 8#247 /section 8#250 /dieresis 8#251 /copyright 8#252 /ordfeminine 8#253 /guillemotleft 8#254 /logicalnot 8#255 /hyphen 8#256 /registered 8#257 /macron 8#260 /degree 8#261 /plusminus 8#262 /twosuperior 8#263 /threesuperior 8#264 /acute 8#265 /mu 8#266 /paragraph 8#267 /periodcentered 8#270 /cedilla 8#271 /onesuperior 8#272 /ordmasculine 8#273 /guillemotright 8#274 /onequarter 8#275 /onehalf 8#276 /threequarters 8#277 /questiondown 8#300 /Agrave 8#301 /Aacute 8#302 /Acircumflex 8#303 /Atilde 8#304 /Adieresis 8#305 /Aring 8#306 /AE 8#307 /Ccedilla 8#310 /Egrave 8#311 /Eacute 8#312 /Ecircumflex 8#313 /Edieresis 8#314 /Igrave 8#315 /Iacute 8#316 /Icircumflex 8#317 /Idieresis 8#320 /Eth 8#321 /Ntilde 8#322 /Ograve 8#323 /Oacute 8#324 /Ocircumflex 8#325 /Otilde 8#326 /Odieresis 8#327 /multiply 8#330 /Oslash 8#331 /Ugrave 8#332 /Uacute 8#333 /Ucircumflex 8#334 /Udieresis 8#335 /Yacute 8#336 /Thorn 8#337 /germandbls 8#340 /agrave 8#341 /aacute 8#342 /acircumflex 8#343 /atilde 8#344 /adieresis 8#345 /aring 8#346 /ae 8#347 /ccedilla 8#350 /egrave 8#351 /eacute 8#352 /ecircumflex 8#353 /edieresis 8#354 /igrave 8#355 /iacute 8#356 /icircumflex 8#357 /idieresis 8#360 /eth 8#361 /ntilde 8#362 /ograve 8#363 /oacute 8#364 /ocircumflex 8#365 /otilde 8#366 /odieresis 8#367 /divide 8#370 /oslash 8#371 /ugrave 8#372 /uacute 8#373 /ucircumflex 8#374 /udieresis 8#375 /yacute 8#376 /thorn 8#377 /ydieresis] def /Times-Roman /Times-Roman-iso isovec ReEncode /Palatino-BoldItalic /Palatino-BoldItalic-iso isovec ReEncode /Times-Bold /Times-Bold-iso isovec ReEncode /NewCenturySchlbk-BoldItalic /NewCenturySchlbk-BoldItalic-iso isovec ReEncode /Times-BoldItalic /Times-BoldItalic-iso isovec ReEncode /DrawEllipse { /endangle exch def /startangle exch def /yrad exch def /xrad exch def /y exch def /x exch def /savematrix mtrx currentmatrix def x y tr xrad yrad sc 0 0 1 startangle endangle arc closepath savematrix setmatrix } def /$F2psBegin {$F2psDict begin /$F2psEnteredState save def} def /$F2psEnd {$F2psEnteredState restore end} def $F2psBegin 10 setmiterlimit 0.06299 0.06299 sc % % Fig objects follow % /Times-Roman-iso ff 300.00 scf sf 4050 4185 m gs 1 -1 sc 290.0 rot (new iterate) col0 sh gr % Arc 15.000 slw n 7125.9 13621.6 10021.9 -124.9 -59.9 arc gs col0 s gr [90] 0 sd % Ellipse n 8415 3645 64 64 0 360 DrawEllipse gs 0.00 setgray ef gr gs col0 s gr [] 0 sd [90] 0 sd % Ellipse n 4455 1845 64 64 0 360 DrawEllipse gs 0.00 setgray ef gr gs col0 s gr [] 0 sd [90] 0 sd % Ellipse n 4275 3105 64 64 0 360 DrawEllipse gs 0.00 setgray ef gr gs col0 s gr [] 0 sd [90] 0 sd % Ellipse n 4302 3992 64 64 0 360 DrawEllipse gs 0.00 setgray ef gr gs col0 s gr [] 0 sd [90] 0 sd % Ellipse n 4815 3870 64 64 0 360 DrawEllipse gs 0.00 setgray ef gr gs col0 s gr [] 0 sd % Polyline n 990 2700 m 13136 4259 l gs 0.50 setgray ef gr gs col0 s gr % Polyline [90 36 15 27 15 27 15 36 ] 0 sd n 4455 1890 m 4275 3150 l gs 0.50 setgray ef gr gs col0 s gr [] 0 sd % Polyline 7.500 slw n 4320 2880 m 4095 2835 l 4050 3060 l gs col0 s gr % Polyline [60] 0 sd n 4500 1890 m 4815 3870 l gs col0 s gr [] 0 sd % Polyline n 4770 3600 m 5040 3555 l 5085 3825 l gs col0 s gr /Times-Roman-iso ff 300.00 scf sf 8415 3870 m gs 1 -1 sc 290.0 rot (initial iterate) col0 sh gr /Times-Bold-iso ff 300.00 scf sf 720 5589 m gs 1 -1 sc 30.0 rot (The manifold ) col0 sh gr /Symbol ff 390.00 scf sf 2396 4621 m gs 1 -1 sc 30.0 rot (P) col0 sh gr /NewCenturySchlbk-BoldItalic-iso ff 390.00 scf sf 8266 4463 m gs 1 -1 sc 290.0 rot (u = G\(z \)) col0 sh gr /Times-BoldItalic-iso ff 225.00 scf sf 8262 4712 m gs 1 -1 sc 290.0 rot (0) col0 sh gr /Times-BoldItalic-iso ff 225.00 scf sf 8735 5881 m gs 1 -1 sc 290.0 rot (0) col0 sh gr /Times-BoldItalic-iso ff 390.00 scf sf 10533 3237 m gs 1 -1 sc 354.0 rot (P) col0 sh gr /Times-BoldItalic-iso ff 225.00 scf sf 10707 3300 m gs 1 -1 sc 354.0 rot (0) col0 sh gr /Times-Bold-iso ff 300.00 scf sf 8071 2978 m gs 1 -1 sc 354.0 rot (The tangent plane ) col0 sh gr /NewCenturySchlbk-BoldItalic-iso ff 390.00 scf sf 7935 3416 m gs 1 -1 sc 354.0 rot (u=G\(z \)+J\(z \)\(z-z \)) col0 sh gr /NewCenturySchlbk-BoldItalic-iso ff 225.00 scf sf 9089 3628 m gs 1 -1 sc 354.0 rot (0) col0 sh gr /NewCenturySchlbk-BoldItalic-iso ff 225.00 scf sf 10257 3705 m gs 1 -1 sc 354.0 rot (0) col0 sh gr /NewCenturySchlbk-BoldItalic-iso ff 225.00 scf sf 11242 3809 m gs 1 -1 sc 354.0 rot (0) col0 sh gr /NewCenturySchlbk-BoldItalic-iso ff 390.00 scf sf 4523 3060 m gs 1 -1 sc 30.0 rot (u=G\(z \)+J\(z \)\(z -z \)) col0 sh gr /NewCenturySchlbk-BoldItalic-iso ff 225.00 scf sf 5581 2553 m gs 1 -1 sc 30.0 rot (0) col0 sh gr /NewCenturySchlbk-BoldItalic-iso ff 225.00 scf sf 6555 1990 m gs 1 -1 sc 30.0 rot (0) col0 sh gr /NewCenturySchlbk-BoldItalic-iso ff 225.00 scf sf 7529 1428 m gs 1 -1 sc 30.0 rot (0) col0 sh gr /NewCenturySchlbk-BoldItalic-iso ff 225.00 scf sf 7101 1675 m gs 1 -1 sc 30.0 rot (1) col0 sh gr /NewCenturySchlbk-BoldItalic-iso ff 390.00 scf sf 4478 2982 m gs 1 -1 sc 30.0 rot (^) col0 sh gr /Times-Bold-iso ff 300.00 scf sf 4804 2377 m gs 1 -1 sc 30.0 rot (project to the tangent plane) col0 sh gr /NewCenturySchlbk-BoldItalic-iso ff 390.00 scf sf 3865 4694 m gs 1 -1 sc 290.0 rot (u = G\(z \)) col0 sh gr /NewCenturySchlbk-BoldItalic-iso ff 225.00 scf sf 3857 4936 m gs 1 -1 sc 290.0 rot (1) col0 sh gr /NewCenturySchlbk-BoldItalic-iso ff 225.00 scf sf 4288 6120 m gs 1 -1 sc 290.0 rot (1) col0 sh gr /NewCenturySchlbk-BoldItalic-iso ff 390.00 scf sf 4879 4608 m gs 1 -1 sc 290.0 rot (u = G\( z \)) col0 sh gr /NewCenturySchlbk-BoldItalic-iso ff 390.00 scf sf 5064 4588 m gs 1 -1 sc 290.0 rot (~) col0 sh gr /NewCenturySchlbk-BoldItalic-iso ff 390.00 scf sf 5494 5772 m gs 1 -1 sc 290.0 rot (~) col0 sh gr /Times-Bold-iso ff 300.00 scf sf 5040 3960 m gs 1 -1 sc 290.0 rot (the solution) col0 sh gr /Palatino-BoldItalic-iso ff 390.00 scf sf 1491 5737 m gs 1 -1 sc 30.0 rot (u = G\(z\)) col0 sh gr /Times-Roman-iso ff 300.00 scf sf 1665 1440 m gs 1 -1 sc (The right-hand side) col0 sh gr /Times-Bold-iso ff 480.00 scf sf 4275 1710 m gs 1 -1 sc (a) col0 sh gr $F2psEnd rs %%EndDocument @endspecial 917 4884 a(Figure)31 b(1:)41 b(Illustration)31 b(of)g(the)f(Gauss-Newton)h(iteration)94 5077 y(The)h(Gauss-Newton)h (iteration)h(can)e(b)s(e)f(deriv)m(ed)h(as)h(follo)m(ws)g(\(see)g (Figure)f(1\).)47 b(T)-8 b(o)33 b(\014nd)d(a)j(least)g(squares)94 5189 y(solution)28 b FO(z)d FN(=)603 5188 y(~)602 5189 y FO(z)i FN(to)h(the)f(equation)g FM(G)p FN(\()p FO(z)p FN(\))g(=)e FO(a)p FN(,)i(w)m(e)h(lo)s(ok)f(for)f(the)h(p)s(oin)m(t) 2568 5188 y(~)2562 5189 y FO(u)e FN(=)g FM(G)p FN(\()2848 5188 y(~)2847 5189 y FO(z)q FN(\))i(that)g(is)g(the)g(orthogonal)94 5302 y(pro)5 b(jection)38 b(of)f FO(a)g FN(on)m(to)h(\005.)61 b(Let)38 b FO(u)1324 5316 y Fx(0)1399 5302 y FN(=)f FM(G)15 b FN(\()q FO(z)1675 5316 y Fx(0)1714 5302 y FN(\))38 b(in)e(\005)i(b)s(e)e(near)2350 5301 y(~)2344 5302 y FO(u)g FN(=)g FM(G)p FN(\()2652 5301 y(~)2651 5302 y FO(z)r FN(\).)61 b(W)-8 b(e)38 b(can)f(appro)m(ximate)h(the)94 5431 y(manifold)30 b(\005)h(with)e(the)h(tangen)m(t)i(plane)e FM(P)1562 5445 y Fx(0)1627 5431 y FN(=)1723 5337 y Fu(n)1794 5431 y FM(G)15 b FN(\()q FO(z)1962 5445 y Fx(0)2001 5431 y FN(\))20 b(+)f FM(J)9 b FN(\()p FO(z)2286 5445 y Fx(0)2327 5431 y FN(\))15 b(\()p FO(z)21 b Fy(\000)e FO(z)2615 5445 y Fx(0)2655 5431 y FN(\))2715 5333 y Fu(\014)2715 5383 y(\014)2715 5433 y(\014)2768 5431 y FO(z)26 b Fy(2)f FO(C)3002 5398 y FL(m)3093 5337 y Fu(o)3149 5431 y FN(.)40 b(Then)29 b(the)h(p)s(oin)m(t)1932 5686 y(6)p eop %%Page: 7 9 7 8 bop 94 99 a FO(a)27 b FN(is)f(orthogonally)j(pro)5 b(jected)27 b(on)m(to)g(the)g(tangen)m(t)h(plane)f FM(P)2159 113 y Fx(0)2225 99 y FN(at)2339 98 y(^)2332 99 y FO(u)f FN(=)f FM(G)p FN(\()p FO(z)2664 113 y Fx(0)2704 99 y FN(\))13 b(+)g FM(J)c FN(\()p FO(z)2976 113 y Fx(0)3016 99 y FN(\)\()p FO(z)3132 113 y Fx(1)3185 99 y Fy(\000)k FO(z)3315 113 y Fx(0)3354 99 y FN(\))27 b(b)m(y)g(solving)94 211 y(the)k(o)m(v)m(erdetermined)g(linear)g(system)661 410 y FM(G)15 b FN(\()q FO(z)829 424 y Fx(0)869 410 y FN(\))20 b(+)g FM(J)9 b FN(\()p FO(z)1155 424 y Fx(0)1195 410 y FN(\))15 b(\()p FO(z)22 b Fy(\000)e FO(z)1485 424 y Fx(0)1525 410 y FN(\))25 b(=)g FO(a)121 b FN(or)h FM(J)9 b FN(\()p FO(z)2196 424 y Fx(0)2236 410 y FN(\))15 b(\()p FO(z)22 b Fy(\000)d FO(z)2525 424 y Fx(0)2565 410 y FN(\))26 b(=)f Fy(\000)p FN([)p FM(G)15 b FN(\()q FO(z)2986 424 y Fx(0)3026 410 y FN(\))20 b Fy(\000)g FO(a)p FN(])450 b(\(3\))94 609 y(for)31 b(its)f(w)m(eigh)m(ted)i(least)g(squares)e (solution)348 835 y FO(z)394 849 y Fx(1)459 835 y FN(=)25 b FO(z)601 849 y Fx(0)661 835 y Fy(\000)752 741 y Fu(h)791 835 y FM(J)9 b FN(\()p FO(z)931 849 y Fx(0)971 835 y FN(\))1006 798 y Fx(+)1006 858 y Fs(W)1076 741 y Fu(i)1131 835 y FN([)p FM(G)p FN(\()p FO(z)1308 849 y Fx(0)1349 835 y FN(\))20 b Fy(\000)g FO(a)p FN(])61 b(with)90 b FM(J)9 b FN(\()p FO(z)2039 849 y Fx(0)2080 835 y FN(\))2115 798 y Fx(+)2115 858 y Fs(W)2210 835 y FN(=)2306 741 y Fu(h)2345 835 y FM(J)g FN(\()p FO(z)2485 849 y Fx(0)2525 835 y FN(\))2560 798 y FL(H)2628 835 y FM(W)2727 798 y Fx(2)2766 835 y FM(J)g FN(\()p FO(z)2906 849 y Fx(0)2946 835 y FN(\))2981 741 y Fu(i)3021 764 y FE(\000)p Fx(1)3115 835 y FM(J)g FN(\()p FO(z)3255 849 y Fx(0)3296 835 y FN(\))3331 798 y FL(H)3398 835 y FM(W)3497 798 y Fx(2)3536 835 y FM(:)137 b FN(\(4\))94 1055 y(As)44 b(long)h(as)f FM(J)9 b FN(\()p FO(z)718 1069 y Fx(0)758 1055 y FN(\))45 b(is)f(of)g(full)f(\(column\))i(rank,)i(the)d(pseudo)f(in)m(v)m(erse)89 b FM(J)9 b FN(\()p FO(z)2873 1069 y Fx(0)2913 1055 y FN(\))2948 1022 y Fx(+)2948 1077 y Fs(W)3106 1055 y FN(exists.)82 b(Therefore)94 1168 y FO(u)152 1182 y Fx(1)228 1168 y FN(=)35 b FM(G)p FN(\()p FO(z)486 1182 y Fx(1)527 1168 y FN(\))i(is)f(w)m(ell)i(de\014ned)d(and)h(is)h(exp)s(ected)g(to)g(b)s (e)f(a)h(b)s(etter)g(appro)m(ximation)g(to)3214 1167 y(~)3207 1168 y FO(u)f FN(=)f FM(G)p FN(\()3514 1167 y(~)3513 1168 y FO(z)r FN(\))i(than)94 1281 y FO(u)152 1295 y Fx(0)217 1281 y FN(=)25 b FM(G)p FN(\()p FO(z)465 1295 y Fx(0)506 1281 y FN(\).)39 b(The)22 b(Gauss-Newton)i(iteration)h (is)f(then)e(a)i(recursiv)m(e)g(application)g(of)g(\(4\))g(\(also)h (see)f([7,)g(10)q(]\).)94 1488 y(The)35 b(con)m(v)m(ergence)j(theory)d (of)h(the)f(Gauss-Newton)i(iteration)f(has)f(b)s(een)g(w)m(ell)h (established)g(for)f(o)m(v)m(erde-)94 1601 y(termined)c(systems)g(in)g (real)g(spaces)h([10)q(].)42 b(The)31 b(follo)m(wing)h(lemma)g(is)f(a)g (straigh)m(tforw)m(ard)h(generalization)94 1714 y(of)j(Theorem)g (10.2.1)h(in)f([10)q(])g(to)g(complex)g(spaces.)54 b(Since)35 b(the)g(lemma)g(itself)g(as)g(w)m(ell)h(as)e(the)h(pro)s(of)f(are)94 1827 y(nearly)d(iden)m(tical)h(to)f(those)g(in)f(the)h(real)g(case)g (in)f([10)q(],)h(w)m(e)g(shall)g(presen)m(t)f(the)h(lemma)g(without)f (pro)s(of.)94 2058 y FO(Lemma)35 b(2.6)206 b FC(L)-5 b(et)40 b FN(\012)f Fy(\032)f FO(C)1258 2025 y FL(m)1364 2058 y FC(b)-5 b(e)40 b(a)g(b)-5 b(ounde)g(d)42 b(op)-5 b(en)41 b(c)-5 b(onvex)40 b(set)h(and)f FM(F)67 b FN(:)54 b FM(D)42 b Fy(\032)c FO(C)3283 2025 y FL(m)3388 2058 y Fy(\000)-15 b(!)38 b FO(C)3649 2025 y FL(n)3735 2058 y FC(b)-5 b(e)94 2171 y(analytic)38 b(in)e(an)g(op)-5 b(en)37 b(set)f FM(D)e Fy(\033)p 1264 2098 66 4 v 32 w FN(\012)o FC(.)52 b(L)-5 b(et)36 b Fy(J)17 b FN(\()p FO(z)p FN(\))37 b FC(b)-5 b(e)36 b(the)g(Jac)-5 b(obian)37 b(of)f FM(F)13 b FN(\()p FO(z)p FN(\))p FC(.)53 b(Supp)-5 b(ose)38 b(that)f(ther)-5 b(e)37 b(exists)95 2282 y FN(~)94 2283 y FO(z)32 b Fy(2)f FN(\012)k FC(such)h(that)h Fy(J)17 b FN(\()879 2282 y(~)878 2283 y FO(z)q FN(\))960 2250 y FL(H)1027 2283 y FM(F)c FN(\()1134 2282 y(~)1133 2283 y FO(z)q FN(\))31 b(=)g(0)37 b FC(with)f Fy(J)17 b FN(\()1745 2282 y(~)1744 2283 y FO(z)p FN(\))37 b FC(ful)5 b(l)35 b(r)-5 b(ank.)53 b(L)-5 b(et)36 b FM(\033)j FC(b)-5 b(e)35 b(the)i(smal)5 b(lest)37 b(singular)f(value)g(of)94 2396 y Fy(J)17 b FN(\()209 2395 y(~)208 2396 y FO(z)q FN(\))p FC(.)42 b(L)-5 b(et)33 b FM(\016)c Fy(\025)c FN(0)33 b FC(b)-5 b(e)32 b(a)h(c)-5 b(onstant)35 b(such)d(that)835 2513 y Fu(\015)835 2563 y(\015)835 2613 y(\015)835 2663 y(\015)896 2542 y(h)950 2636 y Fy(J)17 b FN(\()p FO(z)p FN(\))k Fy(\000)f(J)d FN(\()1372 2635 y(~)1371 2636 y FO(z)p FN(\))1467 2542 y Fu(i)1507 2565 y FL(H)1574 2636 y FM(F)c FN(\()1681 2635 y(~)1680 2636 y FO(z)q FN(\))1777 2513 y Fu(\015)1778 2563 y(\015)1778 2613 y(\015)1778 2663 y(\015)1824 2717 y Fx(2)1889 2636 y Fy(\024)25 b FM(\016)2044 2538 y Fu(\015)2044 2588 y(\015)2044 2638 y(\015)2105 2636 y FO(z)20 b Fy(\000)2263 2635 y FN(~)2262 2636 y FO(z)2324 2538 y Fu(\015)2324 2588 y(\015)2324 2638 y(\015)2370 2692 y Fx(2)2518 2636 y FC(for)33 b(al)5 b(l)66 b FO(z)25 b Fy(2)g FN(\012)p FM(:)624 b FN(\(5\))94 2909 y FC(If)91 b FM(\016)53 b(<)48 b(\033)515 2876 y Fx(2)555 2909 y FC(,)h(then)d(for)g(any)91 b FM(c)49 b Fy(2)1435 2815 y Fu(\020)1498 2873 y Fx(1)p 1494 2888 43 4 v 1494 2940 a FL(\033)1547 2909 y FM(;)1597 2873 y FL(\033)p 1597 2888 V 1601 2940 a(\016)1650 2815 y Fu(\021)1700 2909 y FC(,)f(ther)-5 b(e)47 b(exists)f FM(")j(>)f FN(0)e FC(such)g(that)g(for)h(al)5 b(l)46 b FM(z)3341 2923 y Fx(0)3429 2909 y Fy(2)i FN(\012)e FC(with)94 3037 y Fy(k)p FO(z)185 3051 y Fx(0)246 3037 y Fy(\000)337 3036 y FN(~)337 3037 y FO(z)p Fy(k)428 3051 y Fx(2)494 3037 y FM(<)25 b(")p FC(,)32 b(the)h(se)-5 b(quenc)g(e)33 b(gener)-5 b(ate)g(d)34 b(by)e(the)h(Gauss-Newton)h (iter)-5 b(ation)234 3236 y FO(z)280 3251 y FL(k)r Fx(+1)438 3236 y FN(=)25 b FO(z)580 3251 y FL(k)643 3236 y Fy(\000)20 b(J)d FN(\()p FO(z)894 3251 y FL(k)937 3236 y FN(\))972 3199 y Fx(+)1032 3236 y FM(F)c FN(\()p FO(z)1184 3251 y FL(k)1227 3236 y FN(\))p FM(;)92 b(k)28 b FN(=)d(0)p FM(;)15 b FN(1)p FM(;)g Fy(\001)g(\001)g(\001)j FM(;)80 b FC(wher)-5 b(e)67 b Fy(J)16 b FN(\()p FO(z)2397 3251 y FL(k)2441 3236 y FN(\))2476 3199 y Fx(+)2560 3236 y FN(=)25 b([)p Fy(J)17 b FN(\()p FO(z)2841 3251 y FL(k)2884 3236 y FN(\))2919 3199 y FL(H)2987 3236 y Fy(J)f FN(\()p FO(z)3146 3251 y FL(k)3190 3236 y FN(\)])3250 3199 y FE(\000)p Fx(1)3345 3236 y Fy(J)g FN(\()p FO(z)3504 3251 y FL(k)3548 3236 y FN(\))3583 3199 y FL(H)3650 3236 y FM(;)94 3435 y FC(is)33 b(wel)5 b(l)33 b(de\014ne)-5 b(d)34 b(inside)e FN(\012)p FC(,)g(c)-5 b(onver)g(ges)34 b(to)1591 3434 y FN(~)1590 3435 y FO(z)q FC(,)e(and)i(satis\014es)1061 3574 y Fu(\015)1061 3624 y(\015)1061 3674 y(\015)1108 3672 y FO(z)1154 3687 y FL(k)r Fx(+1)1307 3672 y Fy(\000)1398 3671 y FN(~)1398 3672 y FO(z)1444 3574 y Fu(\015)1444 3624 y(\015)1444 3674 y(\015)1491 3728 y Fx(2)1555 3672 y Fy(\024)1661 3610 y FM(c\016)p 1661 3651 84 4 v 1675 3734 a(\033)1769 3574 y Fu(\015)1769 3624 y(\015)1769 3674 y(\015)1816 3672 y FO(z)1862 3687 y FL(k)1925 3672 y Fy(\000)2016 3671 y FN(~)2016 3672 y FO(z)2062 3574 y Fu(\015)2062 3624 y(\015)2062 3674 y(\015)2108 3728 y Fx(2)2168 3672 y FN(+)2269 3610 y FM(c\013\015)p 2269 3651 150 4 v 2294 3734 a FN(2)p FM(\033)2444 3574 y Fu(\015)2444 3624 y(\015)2444 3674 y(\015)2490 3672 y FO(z)2536 3687 y FL(k)2599 3672 y Fy(\000)2691 3671 y FN(~)2690 3672 y FO(z)2737 3574 y Fu(\015)2737 3624 y(\015)2737 3674 y(\015)2783 3601 y Fx(2)2783 3728 y(2)2822 3672 y FM(;)851 b FN(\(6\))94 3907 y FC(wher)-5 b(e)39 b FM(\013)c(>)e FN(0)38 b FC(is)f(the)h(upp)-5 b(er)39 b(b)-5 b(ound)38 b(of)75 b Fy(kJ)17 b FN(\()p FO(z)p FN(\))p Fy(k)1850 3921 y Fx(2)1966 3907 y FC(on)p 2138 3835 66 4 v 75 w FN(\012)p FC(,)38 b(and)h FM(\015)g(>)33 b FN(0)38 b FC(is)g(the)f(Lipschitz)i(c)-5 b(onstant)39 b(of)94 4036 y Fy(J)17 b FN(\()p FO(z)p FN(\))34 b FC(in)e FN(\012)p FC(,)g(namely,)934 3938 y Fu(\015)934 3988 y(\015)934 4038 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b(When)35 b(the)g(condition)94 4910 y(n)m(um)m(b)s(er)30 b(is)g(mo)s(derate,)h(the)g(m)m(ultiple)g(ro) s(ots)f(can)h(b)s(e)f(calculated)i(accurately)g(b)m(y)e(our)g (algorithm.)94 5153 y FP(3.1)113 b(Remarks)38 b(on)g(previous)g(w)m (ork)94 5324 y FN(In)e(P)m(art)h(I)s(I)e(of)h(the)h(T)-8 b(ec)m(hnical)37 b(Rep)s(ort)f([21)r(],)i(Kahan)d(discussed)g(the)i (sensitivit)m(y)g(of)g(p)s(olynomial)f(ro)s(ots)94 5437 y(with)27 b(enligh)m(tening)g(insigh)m(t,)h(and)e(p)s(oin)m(ted)g(out)h (that)g(it)g(ma)m(y)g(b)s(e)f(a)h(misconception)g(to)g(consider)g(m)m (ultiple)1932 5686 y(7)p eop %%Page: 8 10 8 9 bop 94 99 a FN(ro)s(ots)40 b(as)g(in\014nitely)g(ill)h (conditioned.)69 b(Kahan's)39 b(w)m(ork)h(on)g(ro)s(ots)g(of)g(a)g(p)s (olynomial)g FM(p)f FN(in)h([21)q(])g(can)g(b)s(e)94 211 y(brie\015y)23 b(summarized)h(as)g(follo)m(ws.)39 b(First,)26 b(Kahan)e(describ)s(es)f(the)h(\\p)s(ejorativ)m(e)h (manifold")f(of)g(p)s(olynomials)94 324 y(with)34 b(m)m(ultiple)g(ro)s (ots.)51 b(Secondly)-8 b(,)36 b(the)e(di\013eren)m(tiabilit)m(y)i(is)d (pro)m(v)m(ed)h(for)g(an)g FM(m)p FN(-fold)f(isolated)j(ro)s(ot)e(with) 94 437 y(resp)s(ect)i(to)f(co)s(e\016cien)m(ts)i(that)f(are)f (constrained)h(to)g(preserv)m(e)f(the)g(m)m(ultiplicit)m(y)i FM(m)e FN(of)g(that)h(ro)s(ot.)55 b(This)94 550 y(di\013eren)m (tiabilit)m(y)36 b(then)e(naturally)g(leads)g(to)g(the)g(existence)h (of)f(a)g(\014nite)g(lo)s(cal)h(condition)f(n)m(um)m(b)s(er)f(of)h(an) 94 663 y FC(isolate)-5 b(d)33 b FM(m)p FN(-fold)d(ro)s(ot)h(under)e (the)h(p)s(erturbation)f(that)i(is)g(constrained)f(to)h(preserv)m(e)g (the)f(m)m(ultiplicit)m(y)j FM(m)94 776 y FN(of)j(that)h(ro)s(ot.)57 b(Kahan)35 b(also)i(pro)m(v)m(es)f(the)g(existence)i(of)e(a)g(v)-5 b(anishing)35 b(p)s(oin)m(t)h(for)f FM(p)3009 743 y Fx(\()p FL(m)p FE(\000)p Fx(1\))3221 776 y FN(\()p FM(x)p FN(\))h(in)g(a)g (region)94 889 y(con)m(taining)i(a)e(cluster)h(of)f FM(m)g FN(ro)s(ots)g(of)g FM(p)p FN(.)57 b(Finally)-8 b(,)39 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b(imp)s(or-)94 1981 y(tan)m(tly)-8 b(,)29 b(w)m(e)e(emphasize)f(on)g(the)g(practical)h(computation)g(of)f(m)m (ultiple)h(ro)s(ots.)39 b(Our)25 b(main)g(con)m(tribution)i(in)94 2094 y(this)33 b(section)h(also)g(include)e(the)h(follo)m(wing:)47 b(First,)34 b(w)m(e)f(con)m(v)m(ert)h(the)f(singular)g(ro)s (ot-\014nding)f(problem)g(to)94 2207 y(a)g(least)h(squares)d(problem)h (and)g(pro)m(v)m(e)h(its)g(regularit)m(y)-8 b(.)45 b(Secondly)-8 b(,)32 b(w)m(e)f(establish)h(the)f(lo)s(cal)i(con)m(v)m(ergence)94 2320 y(of)39 b(the)f(Gauss-Newton)g(iteration)i(solving)f(the)f(least)h (squares)e(problem.)63 b(Our)37 b(third)g(con)m(tribution)h(is)94 2433 y(the)j(form)m(ulation)g(of)g(a)g(global)g(structure-preserving)f (condition)h(n)m(um)m(b)s(er)e(measuring)i(the)f(com)m(bined)94 2546 y(sensitivit)m(y)d(of)d(all)i(ro)s(ots)f(that)g(are)g(constrained) g(in)f(a)h(m)m(ultiplicit)m(y)i(structure,)f(instead)f(of)f(an)h (isolated)94 2659 y(m)m(ultiple)43 b(ro)s(ot)f(considered)f(b)m(y)g (Kahan.)74 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FL(j)3052 4518 y FM(;)621 b FN(\(7\))94 4804 y(where)30 b(eac)m(h)i FM(g)605 4818 y FL(j)672 4804 y FN(is)f(a)f(p)s(olynomial)h(in)f FM(z)1457 4818 y Fx(1)1497 4804 y FM(;)15 b Fy(\001)g(\001)g(\001)h FM(;)f(z)1740 4818 y FL(m)1808 4804 y FN(.)40 b(W)-8 b(e)32 b(ha)m(v)m(e)f(the)g(corresp)s(ondence)525 5145 y FM(p)25 b Fy(\030)g FM(G)763 5160 y FL(`)796 5145 y FN(\()p FO(z)p FN(\))52 b Fy(\021)1085 4952 y Fu(0)1085 5098 y(B)1085 5151 y(@)1203 5009 y FM(g)1246 5023 y Fx(1)1286 5009 y FN(\()p FM(z)1363 5023 y Fx(1)1403 5009 y FM(;)15 b Fy(\001)g(\001)g(\001)i FM(;)e(z)1647 5023 y FL(m)1714 5009 y FN(\))1463 5102 y(.)1463 5136 y(.)1463 5169 y(.)1199 5282 y FM(g)1242 5296 y FL(n)1290 5282 y FN(\()p FM(z)1367 5296 y Fx(1)1407 5282 y FM(;)g Fy(\001)g(\001)g(\001)h FM(;)f(z)1650 5296 y FL(m)1718 5282 y FN(\))1794 4952 y Fu(1)1794 5098 y(C)1794 5151 y(A)1892 5145 y Fy(2)25 b FO(C)2054 5108 y FL(n)2101 5145 y FM(;)151 b FN(where)91 b FO(z)26 b FN(=)2769 4952 y Fu(0)2769 5098 y(B)2769 5151 y(@)2896 5009 y FM(z)2938 5023 y Fx(1)2925 5102 y FN(.)2925 5136 y(.)2925 5169 y(.)2883 5282 y FM(z)2925 5296 y FL(m)3033 4952 y Fu(1)3033 5098 y(C)3033 5151 y(A)3131 5145 y Fy(2)f FO(C)3293 5108 y FL(m)3359 5145 y FM(:)314 b FN(\(8\))94 5478 y(W)-8 b(e)32 b(no)m(w)e(de\014ne)g(the)h (p)s(ejorativ)m(e)g(manifold)f(rigorously)h(based)f(on)g(Kahan's)g (heuristic)h(description.)1932 5686 y(8)p eop %%Page: 9 11 9 10 bop 94 99 a FO(De\014nition)36 b(3.1)100 b FC(A)n(n)27 b(or)-5 b(der)g(e)g(d)30 b(arr)-5 b(ay)29 b(of)f(p)-5 b(ositive)28 b(inte)-5 b(gers)28 b FM(`)d FN(=)g([)p FM(`)2488 113 y Fx(1)2527 99 y FM(;)15 b Fy(\001)g(\001)g(\001)i FM(;)e(`)2767 113 y FL(m)2834 99 y FN(])27 b FC(is)h(c)-5 b(al)5 b(le)-5 b(d)28 b(a)g Fr(multiplicity)94 211 y(structur)-5 b(e)38 b FC(of)g(de)-5 b(gr)g(e)g(e)38 b FM(n)f FC(if)g FM(`)1151 225 y Fx(1)1214 211 y FN(+)24 b Fy(\001)15 b(\001)g(\001)24 b FN(+)g FM(`)1571 225 y FL(m)1671 211 y FN(=)33 b FM(n)p FC(.)56 b(F)-7 b(or)38 b(any)g(such)g(given)e (multiplicity)j(structur)-5 b(e)38 b FM(`)p FC(,)g(the)94 335 y(c)-5 b(ol)5 b(le)-5 b(ction)42 b(of)f(ve)-5 b(ctors)82 b FN(\005)1041 350 y FL(`)1114 335 y Fy(\021)1224 241 y Fu(n)1294 335 y FM(G)1365 350 y FL(`)1399 335 y FN(\()p FO(z)p FN(\))1541 237 y Fu(\014)1541 287 y(\014)1541 337 y(\014)1594 335 y FO(z)26 b Fy(2)f FO(C)1828 302 y FL(m)1919 241 y Fu(o)2014 335 y Fy(\032)39 b FO(C)2200 302 y FL(n)2287 335 y FC(is)i(c)-5 b(al)5 b(le)-5 b(d)41 b(the)g Fr(p)-5 b(ejor)g(ative)47 b(manifold)40 b FC(of)94 463 y(multiplicity)34 b(structur)-5 b(e)34 b FM(`)p FC(,)e(wher)-5 b(e)67 b FM(G)1416 478 y FL(`)1490 463 y FN(:)41 b FO(C)1632 430 y FL(m)1724 463 y Fy(\000)-15 b(!)25 b FO(C)1972 430 y FL(n)2051 463 y FC(de\014ne)-5 b(d)34 b(in)f(\(7\))h({)f(\(8\))g (is)g(c)-5 b(al)5 b(le)-5 b(d)34 b(the)g Fr(c)-5 b(o)g(e\016cient)94 576 y(op)g(er)g(ator)31 b FC(asso)-5 b(ciate)g(d)35 b(with)e(the)g (multiplicity)h(structur)-5 b(e)33 b FM(`)p FC(.)94 786 y FN(F)-8 b(or)32 b(example,)f(w)m(e)g(consider)f(p)s(olynomials)h(of)f (degree)h(3.)41 b(First,)31 b(for)f(m)m(ultiplicit)m(y)j(structure)d FM(`)25 b FN(=)g([1)p FM(;)15 b FN(2],)548 963 y(\()p FM(x)20 b Fy(\000)g FM(z)788 977 y Fx(1)828 963 y FN(\)\()p FM(x)h Fy(\000)f FM(z)1104 977 y Fx(2)1143 963 y FN(\))1178 925 y Fx(2)1243 963 y FN(=)25 b FM(x)1391 925 y Fx(3)1451 963 y FN(+)20 b(\()1577 963 y gsave currentpoint translate 0 107642 15800 300 VResolution div mul DVImag div div 3482772 15800 300 Resolution div mul DVImag div div 462465 -15800 300 VResolution div mul DVImag div div box .8 setgray fill grestore 1577 963 a 26 w Fy(\000)p FM(z)1716 977 y Fx(1)1775 963 y Fy(\000)g FN(2)p FM(z)1953 977 y Fx(2)2018 963 y FN(\))15 b FM(x)2120 925 y Fx(2)2181 963 y FN(+)k(\()2306 963 y gsave currentpoint translate 0 182309 15800 300 VResolution div mul DVImag div div 3601211 15800 300 Resolution div mul DVImag div div 598293 -15800 300 VResolution div mul DVImag div div box .8 setgray fill grestore 2306 963 a 26 w FN(2)p FM(z)2419 977 y Fx(1)2459 963 y FM(z)2501 977 y Fx(2)2561 963 y FN(+)h FM(z)2698 930 y Fx(2)2694 986 y(2)2763 963 y FN(\))15 b FM(x)21 b FN(+)f(\()3012 963 y gsave currentpoint translate 0 182309 15800 300 VResolution div mul DVImag div div 2278437 15800 300 Resolution div mul DVImag div div 598293 -15800 300 VResolution div mul DVImag div div box .8 setgray fill grestore 3012 963 a 25 w Fy(\000)p FM(z)3150 977 y Fx(1)3190 963 y FM(z)3236 930 y Fx(2)3232 986 y(2)3301 963 y FN(\))p FM(:)94 1140 y FN(A)31 b(p)s(olynomial)g (with)f(one)g(simple)h(ro)s(ot)f FM(z)1556 1154 y Fx(1)1626 1140 y FN(and)g(one)h(double)f(ro)s(ot)h FM(z)2500 1154 y Fx(2)2570 1140 y FN(corresp)s(onds)e(to)i(the)f(v)m(ector)728 1428 y FM(G)799 1446 y Fx([1)p FL(;)p Fx(2])933 1428 y FN(\()p FO(z)p FN(\))c Fy(\021)1171 1234 y Fu(0)1171 1380 y(B)1171 1433 y(@)1301 1315 y gsave currentpoint translate 0 107642 15800 300 VResolution div mul DVImag div div 3482772 15800 300 Resolution div mul DVImag div div 462465 -15800 300 VResolution div mul DVImag div div box .8 setgray fill grestore 1301 1315 a 25 w Fy(\000)p FM(z)1439 1329 y Fx(1)1498 1315 y Fy(\000)20 b FN(2)p FM(z)1676 1329 y Fx(2)1286 1428 y gsave currentpoint translate 0 182309 15800 300 VResolution div mul DVImag div div 3601211 15800 300 Resolution div mul DVImag div div 598293 -15800 300 VResolution div mul DVImag div div box .8 setgray fill grestore 1286 1428 a 25 w FN(2)p FM(z)1398 1442 y Fx(1)1438 1428 y FM(z)1480 1442 y Fx(2)1540 1428 y FN(+)g FM(z)1677 1395 y Fx(2)1673 1451 y(2)1453 1541 y gsave currentpoint translate 0 182309 15800 300 VResolution div mul DVImag div div 2278437 15800 300 Resolution div mul DVImag div div 598293 -15800 300 VResolution div mul DVImag div div box .8 setgray fill grestore 1453 1541 a 25 w Fy(\000)p FM(z)1591 1555 y Fx(1)1631 1541 y FM(z)1677 1508 y Fx(2)1673 1564 y(2)1783 1234 y Fu(1)1783 1380 y(C)1783 1433 y(A)1881 1428 y Fy(2)25 b FO(C)2043 1390 y Fx(3)2082 1428 y FM(;)91 b FN(with)g FO(z)26 b FN(=)2633 1284 y Fu( )2741 1371 y FM(z)2783 1385 y Fx(1)2741 1484 y FM(z)2783 1498 y Fx(2)2864 1284 y Fu(!)2955 1428 y Fy(2)f FO(C)3117 1390 y Fx(2)3156 1428 y FM(:)517 b FN(\(9\))94 1728 y(The)30 b(v)m(ectors)i FM(G)661 1747 y Fx([1)p FL(;)p Fx(2])795 1728 y FN(\()p FO(z)p FN(\))g(in)e(\(9\))h(for)f(all)i FO(z)25 b Fy(2)g FO(C)1694 1695 y Fx(2)1763 1728 y FN(form)30 b(the)h(p)s(ejorativ)m(e)g (manifold)g(\005)3004 1747 y Fx([1)p FL(;)p Fx(2])3137 1728 y FN(.)41 b(Similarly)-8 b(,)1224 1933 y(\005)1292 1952 y Fx([3])1396 1933 y FN(=)1492 1839 y Fu(n)1572 1933 y FN(\()p Fy(\000)p FN(3)15 b FM(z)t(;)32 b FN(3)15 b FM(z)1947 1896 y Fx(2)1987 1933 y FM(;)31 b Fy(\000)p FM(z)2160 1896 y Fx(3)2199 1933 y FN(\))2234 1896 y FE(>)2319 1836 y Fu(\014)2319 1886 y(\014)2319 1936 y(\014)2372 1933 y FM(z)e Fy(2)c FO(C)2630 1839 y Fu(o)94 2126 y FN(when)32 b FM(`)c FN(=)f([3].)47 b(\005)733 2145 y Fx([3])844 2126 y FN(is)32 b(a)h(submanifold)e(of)h(\005)1703 2145 y Fx([1)p FL(;)p Fx(2])1869 2126 y FN(that)g(con)m(tains)i(all)f (p)s(olynomials)f(with)g(a)g(single)h(triple)94 2239 y(ro)s(ot.)42 b(Figure)30 b(2)h(sho)m(ws)f(the)h(manifolds)f(\005)1585 2258 y Fx([1)p FL(;)p Fx(2])1749 2239 y FN(\(the)h(wings\))f(and)g (\005)2474 2258 y Fx([3])2583 2239 y FN(\(the)h(sharp)e(edge\))j(in)e FO(R)3450 2206 y Fx(3)3490 2239 y FN(.)379 3865 y @beginspecial 168 @llx 161 @lly 490 @urx 505 @ury 1872 @rwi 1800 @rhi @setspecial %%BeginDocument: pejo02.eps %!PS-Adobe-3.0 EPSF %%Title: Maple plot %%Creator: MapleV %%Pages: 1 %%BoundingBox: (atend) %%DocumentNeededResources: font Helvetica %%EndComments 20 dict begin gsave /drawborder false def /m {moveto} def /l {lineto} def /C {setrgbcolor} def /Y /setcmykcolor where { %%ifelse Use built-in operator /setcmykcolor get }{ %%ifelse Emulate setcmykcolor with setrgbcolor { %%def 1 sub 3 { %%repeat 3 index add neg dup 0 lt { pop 0 } if 3 1 roll } repeat setrgbcolor } bind } ifelse def /G {setgray} def /S {stroke} def /NP {newpath} def %%%%This draws a filled polygon and avoids bugs/features %%%% of some postscript interpreters %%%%GHOSTSCRIPT: has a bug in reversepath - removing %%%%the call to reversepath is a sufficient work around /P {gsave fill grestore reversepath stroke} def %%%%This function is needed for drawing text /stringbbox {gsave NP 0 0 m false charpath flattenpath pathbbox 4 2 roll pop pop 1.1 mul cvi exch 1.1 mul cvi exch grestore} def /thin 3 def /medium 7 def /thick 16 def /boundarythick 20 def % thichess of bounding box %%IncludeResource: font Helvetica 72.000000 72.000000 translate /defaultTitleFont {/Helvetica findfont 219 scalefont setfont} def /defaultFont {/Helvetica findfont 154 scalefont setfont} def 0.0936 0.0936 scale 1 setlinejoin 1 setlinecap 0.0 setgray /inch {72 mul} def /fheight 0.35 inch neg def 0 G medium setlinewidth [] 0 setdash [] 0 setdash 0.592549 G NP 2614 978 m 2623 951 l 2629 954 l 2619 978 l 2614 978 l P 0.594863 G NP 2635 979 m 2629 979 l 2623 978 l 2619 978 l 2629 954 l 2629 954 l 2640 961 l 2635 979 l P 0.742863 G NP 2638 979 m 2629 979 l 2629 979 l 2629 979 l 2638 979 l P 0.592549 G NP 2649 980 m 2638 979 l 2635 979 l 2640 961 l 2640 961 l 2653 968 l 2649 980 l P 0.594863 G NP 2652 980 m 2649 980 l 2653 968 l 2653 968 l 2655 969 l 2652 980 l P 0.592549 G NP 2657 980 m 2652 980 l 2655 969 l 2655 969 l 2658 971 l 2657 980 l P 0.642314 G NP 2669 981 m 2664 981 l 2657 980 l 2658 971 l 2658 971 l 2671 975 l 2669 981 l P 0.64 G NP 2675 981 m 2674 981 l 2669 981 l 2671 975 l 2671 975 l 2676 977 l 2675 981 l P 0.642314 G NP 2682 982 m 2675 981 l 2676 977 l 2676 977 l 2683 980 l 2682 982 l P 0.64 G NP 2688 983 m 2684 982 l 2682 982 l 2683 980 l 2683 980 l 2688 982 l 2688 983 l P 0.681647 G NP 2691 983 m 2690 983 l 2688 983 l 2688 982 l 2688 982 l 2691 983 l 2691 983 l P 0.664314 G NP 2692 983 m 2691 983 l 2691 983 l 2691 983 l 2692 983 l 2692 983 l P 0.719765 G NP 2685 1016 m 2690 983 l 2694 983 l 2685 1016 l P 0.72902 G NP 2685 1016 m 2694 983 l 2694 983 l 2685 1016 l P 0.794745 G NP 2405 1017 m 2283 996 l 2404 986 l 2404 986 l 2405 1017 l P NP 2405 1017 m 2404 986 l 2422 985 l 2422 985 l 2405 1017 l P 0.742863 G NP 2650 1049 m 2638 979 l 2664 981 l 2650 1049 l P 0.745176 G NP 2650 1049 m 2664 981 l 2674 981 l 2650 1049 l P 0.782078 G NP 2501 1061 m 2405 1017 l 2422 985 l 2505 981 l 2501 1061 l P NP 2501 1061 m 2505 981 l 2540 979 l 2501 1061 l P 0.703608 G NP 2669 1090 m 2685 1016 l 2694 983 l 2696 984 l 2669 1090 l P 0.679333 G NP 2696 984 m 2694 983 l 2694 983 l 2692 983 l 2692 983 l 2692 983 l 2696 984 l P 0.764784 G NP 2563 1125 m 2578 979 l 2623 978 l 2563 1125 l P NP 2563 1125 m 2501 1061 l 2540 979 l 2578 979 l 2563 1125 l P 0.718627 G NP 2645 1161 m 2650 1049 l 2674 981 l 2684 982 l 2645 1161 l P 0.719765 G NP 2645 1161 m 2684 982 l 2690 983 l 2685 1016 l 2645 1161 l P 0.742863 G NP 2596 1207 m 2563 1125 l 2623 978 l 2629 979 l 2596 1207 l P NP 2596 1207 m 2629 979 l 2638 979 l 2650 1049 l 2650 1049 l 2596 1207 l P 0.793529 G NP 2140 1211 m 2113 1102 l 2262 998 l 2266 998 l 2266 998 l 2140 1211 l P 0.794745 G NP 2298 1218 m 2269 998 l 2283 996 l 2283 996 l 2405 1017 l 2298 1218 l P NP 2298 1218 m 2140 1211 l 2266 998 l 2269 998 l 2269 998 l 2298 1218 l P 0.793529 G NP 1935 1227 m 2113 1102 l 2140 1211 l 1935 1227 l P 0.782078 G NP 2414 1247 m 2298 1218 l 2405 1017 l 2405 1017 l 2414 1247 l P NP 2414 1247 m 2405 1017 l 2501 1061 l 2414 1247 l P 0.689726 G NP 2623 1287 m 2645 1161 l 2685 1016 l 2623 1287 l P 0.764784 G NP 2493 1297 m 2414 1247 l 2501 1061 l 2501 1061 l 2493 1297 l P NP 2493 1297 m 2501 1061 l 2563 1125 l 2493 1297 l P 0.718627 G NP 2605 1306 m 2596 1207 l 2650 1049 l 2650 1049 l 2605 1306 l P 0.720941 G NP 2605 1306 m 2650 1049 l 2645 1161 l 2605 1306 l P 0.742863 G NP 2542 1366 m 2493 1297 l 2563 1125 l 2563 1125 l 2542 1366 l P NP 2542 1366 m 2563 1125 l 2596 1207 l 2542 1366 l P 0.687412 G NP 2595 1419 m 2605 1306 l 2645 1161 l 2595 1419 l P 0.69898 G NP 2595 1419 m 2645 1161 l 2623 1287 l 2595 1419 l P 0.794745 G NP 2191 1420 m 2140 1211 l 2298 1218 l 2191 1420 l P 0.793529 G NP 2012 1428 m 1935 1227 l 2140 1211 l 2140 1211 l 2012 1428 l P 0.794745 G NP 2012 1428 m 2140 1211 l 2191 1420 l 2012 1428 l P 0.782078 G NP 2326 1434 m 2191 1420 l 2298 1218 l 2298 1218 l 2326 1434 l P NP 2326 1434 m 2298 1218 l 2414 1247 l 2326 1434 l P 0.718627 G NP 2564 1451 m 2542 1366 l 2596 1207 l 2596 1207 l 2564 1451 l P NP 2564 1451 m 2596 1207 l 2605 1306 l 2564 1451 l P 0.793529 G NP 1784 1460 m 1935 1227 l 2012 1428 l 1784 1460 l P 0.764784 G NP 2423 1469 m 2326 1434 l 2414 1247 l 2414 1247 l 2423 1469 l P NP 2423 1469 m 2414 1247 l 2493 1297 l 2423 1469 l P 0.665451 G NP 2584 1487 m 2595 1419 l 2623 1287 l 2584 1487 l P 0.742863 G NP 2487 1524 m 2493 1297 l 2542 1366 l 2487 1524 l P NP 2487 1524 m 2423 1469 l 2493 1297 l 2493 1297 l 2487 1524 l P 0.689726 G NP 2567 1552 m 2605 1306 l 2595 1419 l 2567 1552 l P 0.687412 G NP 2567 1552 m 2564 1451 l 2605 1306 l 2605 1306 l 2567 1552 l P 0.718627 G NP 2524 1596 m 2487 1524 l 2542 1366 l 2542 1366 l 2524 1596 l P NP 2524 1596 m 2542 1366 l 2564 1451 l 2524 1596 l P 0.782078 G NP 2239 1621 m 2191 1420 l 2326 1434 l 2239 1621 l P 0.794745 G NP 2084 1621 m 2012 1428 l 2191 1420 l 2191 1420 l 2084 1621 l P 0.782078 G NP 2084 1621 m 2191 1420 l 2239 1621 l 2084 1621 l P 0.764784 G NP 2353 1641 m 2239 1621 l 2326 1434 l 2326 1434 l 2353 1641 l P NP 2353 1641 m 2326 1434 l 2423 1469 l 2353 1641 l P 0.793529 G NP 1884 1645 m 1784 1460 l 2012 1428 l 2012 1428 l 1884 1645 l P 0.794745 G NP 1884 1645 m 2012 1428 l 2084 1621 l 1884 1645 l P 0.651569 G NP 2555 1666 m 2567 1552 l 2595 1419 l 2555 1666 l P 0.742863 G NP 2433 1682 m 2353 1641 l 2423 1469 l 2423 1469 l 2433 1682 l P NP 2433 1682 m 2423 1469 l 2487 1524 l 2433 1682 l P 0.685098 G NP 2539 1684 m 2524 1596 l 2564 1451 l 2564 1451 l 2539 1684 l P 0.687412 G NP 2539 1684 m 2564 1451 l 2567 1552 l 2539 1684 l P 0.793529 G NP 1633 1693 m 1784 1460 l 1884 1645 l 1633 1693 l P 0.718627 G NP 2484 1741 m 2487 1524 l 2524 1596 l 2484 1741 l P NP 2484 1741 m 2433 1682 l 2487 1524 l 2487 1524 l 2484 1741 l P 0.649255 G NP 2537 1786 m 2539 1684 l 2567 1552 l 2537 1786 l P 0.664314 G NP 2537 1786 m 2567 1552 l 2555 1666 l 2537 1786 l P 0.782078 G NP 2151 1807 m 2084 1621 l 2239 1621 l 2239 1621 l 2151 1807 l P 0.764784 G NP 2283 1814 m 2151 1807 l 2239 1621 l 2283 1814 l P NP 2283 1814 m 2239 1621 l 2353 1641 l 2283 1814 l P 0.688588 G NP 2510 1816 m 2524 1596 l 2539 1684 l 2510 1816 l P 0.685098 G NP 2510 1816 m 2484 1741 l 2524 1596 l 2524 1596 l 2510 1816 l P 0.794745 G NP 1977 1823 m 1884 1645 l 2084 1621 l 2084 1621 l 1977 1823 l P 0.782078 G NP 1977 1823 m 2084 1621 l 2151 1807 l 1977 1823 l P 0.742863 G NP 2379 1841 m 2283 1814 l 2353 1641 l 2353 1641 l 2379 1841 l P NP 2379 1841 m 2353 1641 l 2433 1682 l 2379 1841 l P 0.62149 G NP 2530 1849 m 2537 1786 l 2555 1666 l 2530 1849 l P 0.793529 G NP 1756 1862 m 1633 1693 l 1884 1645 l 1884 1645 l 1756 1862 l P 0.794745 G NP 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2143 2158 m 2063 1994 l 2213 1986 l 2213 1986 l 2143 2158 l P 0.793529 G NP 1331 2160 m 1482 1927 l 1628 2079 l 1331 2160 l P 0.567059 G NP 2502 2176 m 2505 2118 l 2514 2009 l 2502 2176 l P 0.718627 G NP 2362 2176 m 2324 1999 l 2403 2031 l 2362 2176 l P NP 2362 2176 m 2270 2157 l 2324 1999 l 2324 1999 l 2362 2176 l P 0.764784 G NP 1976 2180 m 2063 1994 l 2143 2158 l 1976 2180 l P 0.782078 G NP 1976 2180 m 1870 2024 l 2063 1994 l 2063 1994 l 1976 2180 l P 0.686275 G NP 2426 2214 m 2403 2031 l 2454 2081 l 2426 2214 l P 0.685098 G NP 2426 2214 m 2362 2176 l 2403 2031 l 2403 2031 l 2426 2214 l P 0.782078 G NP 1763 2225 m 1870 2024 l 1976 2180 l 1763 2225 l P 0.794745 G NP 1763 2225 m 1628 2079 l 1870 2024 l 1870 2024 l 1763 2225 l P 0.604118 G NP 2495 2227 m 2501 2027 l 2505 2118 l 2495 2227 l P 0.59949 G NP 2495 2227 m 2483 2147 l 2501 2027 l 2501 2027 l 2495 2227 l P 0.646941 G NP 2465 2267 m 2454 2081 l 2483 2147 l 2465 2267 l P NP 2465 2267 m 2426 2214 l 2454 2081 l 2454 2081 l 2465 2267 l P 0.794745 G NP 1500 2296 m 1628 2079 l 1763 2225 l 1500 2296 l P 0.793529 G NP 1500 2296 m 1331 2160 l 1628 2079 l 1628 2079 l 1500 2296 l P 0.742863 G NP 2216 2316 m 2143 2158 l 2270 2157 l 2270 2157 l 2216 2316 l P 0.546196 G NP 2497 2319 m 2495 2227 l 2505 2118 l 2505 2118 l 2497 2319 l P 0.718627 G NP 2322 2321 m 2270 2157 l 2362 2176 l 2322 2321 l P NP 2322 2321 m 2216 2316 l 2270 2157 l 2270 2157 l 2322 2321 l P 0.764784 G NP 2073 2330 m 1976 2180 l 2143 2158 l 2143 2158 l 2073 2330 l P 0.742863 G NP 2073 2330 m 2143 2158 l 2216 2316 l 2073 2330 l P 0.59949 G NP 2486 2335 m 2483 2147 l 2495 2227 l 2486 2335 l P NP 2486 2335 m 2465 2267 l 2483 2147 l 2483 2147 l 2486 2335 l P 0.686275 G NP 2397 2346 m 2322 2321 l 2362 2176 l 2362 2176 l 2397 2346 l P NP 2397 2346 m 2362 2176 l 2426 2214 l 2397 2346 l P 0.782078 G NP 1888 2367 m 1763 2225 l 1976 2180 l 1976 2180 l 1888 2367 l P 0.764784 G NP 1888 2367 m 1976 2180 l 2073 2330 l 1888 2367 l P 0.646941 G NP 2447 2387 m 2397 2346 l 2426 2214 l 2426 2214 l 2447 2387 l P NP 2447 2387 m 2426 2214 l 2465 2267 l 2447 2387 l P 0.793529 G NP 1180 2393 m 1331 2160 l 1500 2296 l 1180 2393 l P 0.541569 G NP 2494 2417 m 2486 2335 l 2495 2227 l 2495 2227 l 2494 2417 l P 0.563608 G NP 2494 2417 m 2495 2227 l 2497 2319 l 2494 2417 l P 0.794745 G NP 1656 2427 m 1500 2296 l 1763 2225 l 1763 2225 l 1656 2427 l P 0.782078 G NP 1656 2427 m 1763 2225 l 1888 2367 l 1656 2427 l P 0.59949 G NP 2477 2444 m 2465 2267 l 2486 2335 l 2477 2444 l P 0.597176 G NP 2477 2444 m 2447 2387 l 2465 2267 l 2465 2267 l 2477 2444 l P 0.718627 G NP 2282 2467 m 2216 2316 l 2322 2321 l 2282 2467 l P 0.507922 G NP 2495 2470 m 2494 2417 l 2497 2319 l 2495 2470 l P 0.718627 G NP 2161 2474 m 2216 2316 l 2282 2467 l 2161 2474 l P 0.742863 G NP 2161 2474 m 2073 2330 l 2216 2316 l 2216 2316 l 2161 2474 l P 0.686275 G NP 2369 2478 m 2282 2467 l 2322 2321 l 2322 2321 l 2369 2478 l P NP 2369 2478 m 2322 2321 l 2397 2346 l 2369 2478 l P 0.764784 G NP 2003 2502 m 1888 2367 l 2073 2330 l 2073 2330 l 2003 2502 l P 0.742863 G NP 2003 2502 m 2073 2330 l 2161 2474 l 2003 2502 l P 0.646941 G NP 2429 2508 m 2397 2346 l 2447 2387 l 2429 2508 l P 0.644627 G NP 2429 2508 m 2369 2478 l 2397 2346 l 2397 2346 l 2429 2508 l P 0.793529 G NP 1372 2513 m 1180 2393 l 1500 2296 l 1500 2296 l 1372 2513 l P 0.794745 G NP 1372 2513 m 1500 2296 l 1656 2427 l 1372 2513 l P 0.539255 G NP 2491 2514 m 2477 2444 l 2486 2335 l 2486 2335 l 2491 2514 l P 0.543882 G NP 2491 2514 m 2486 2335 l 2494 2417 l 2491 2514 l P 0.597176 G NP 2467 2553 m 2447 2387 l 2477 2444 l 2467 2553 l P NP 2467 2553 m 2429 2508 l 2447 2387 l 2447 2387 l 2467 2553 l P 0.764784 G NP 1801 2553 m 1888 2367 l 2003 2502 l 1801 2553 l P 0.782078 G NP 1801 2553 m 1656 2427 l 1888 2367 l 1888 2367 l 1801 2553 l P 0.504392 G NP 2497 2597 m 2491 2514 l 2494 2417 l 2494 2417 l 2497 2597 l P 0.686275 G NP 2341 2611 m 2282 2467 l 2369 2478 l 2341 2611 l P NP 2241 2612 m 2282 2467 l 2341 2611 l 2241 2612 l P 0.718627 G NP 2241 2612 m 2161 2474 l 2282 2467 l 2282 2467 l 2241 2612 l P 0.539255 G NP 2489 2612 m 2467 2553 l 2477 2444 l 2477 2444 l 2489 2612 l P 0.541569 G NP 2489 2612 m 2477 2444 l 2491 2514 l 2489 2612 l P 0.793529 G NP 1045 2621 m 1040 2609 l 1180 2393 l 1180 2393 l 1372 2513 l 1045 2621 l P 0.644627 G NP 2411 2628 m 2341 2611 l 2369 2478 l 2369 2478 l 2411 2628 l P NP 2411 2628 m 2369 2478 l 2429 2508 l 2411 2628 l P 0.782078 G NP 1549 2628 m 1656 2427 l 1801 2553 l 1549 2628 l P 0.794745 G NP 1549 2628 m 1372 2513 l 1656 2427 l 1656 2427 l 1549 2628 l P 0.718627 G NP 2107 2632 m 2161 2474 l 2241 2612 l 2107 2632 l P 0.742863 G NP 2107 2632 m 2003 2502 l 2161 2474 l 2161 2474 l 2107 2632 l P 0.739137 G NP 2503 2648 m 2498 2626 l 2497 2621 l 2497 2621 l 2503 2648 l P NP 2503 2648 m 2499 2648 l 2498 2626 l 2498 2626 l 2503 2648 l P 0.597176 G NP 2458 2662 m 2411 2628 l 2429 2508 l 2429 2508 l 2458 2662 l P NP 2458 2662 m 2429 2508 l 2467 2553 l 2458 2662 l P 0.764784 G NP 1933 2674 m 1801 2553 l 2003 2502 l 2003 2502 l 1933 2674 l P 0.742863 G NP 1933 2674 m 2003 2502 l 2107 2632 l 1933 2674 l P 0.504392 G NP 2499 2684 m 2489 2612 l 2491 2514 l 2491 2514 l 2499 2684 l P 0.509098 G NP 2499 2684 m 2491 2514 l 2497 2597 l 2499 2684 l P 0.739137 G NP 2507 2700 m 2499 2658 l 2499 2648 l 2499 2648 l 2503 2648 l 2507 2700 l P NP 2507 2700 m 2499 2665 l 2499 2658 l 2499 2658 l 2507 2700 l P 0.539255 G NP 2486 2710 m 2467 2553 l 2489 2612 l 2486 2710 l P NP 2486 2710 m 2458 2662 l 2467 2553 l 2467 2553 l 2486 2710 l P 0.794745 G NP 1244 2730 m 1372 2513 l 1549 2628 l 1244 2730 l P 0.793529 G NP 1244 2730 m 1051 2636 l 1045 2621 l 1372 2513 l 1372 2513 l 1244 2730 l P 0.49498 G NP 2502 2734 m 2499 2684 l 2497 2597 l 2502 2734 l P 0.782078 G NP 1713 2740 m 1549 2628 l 1801 2553 l 1801 2553 l 1713 2740 l P 0.764784 G NP 1713 2740 m 1801 2553 l 1933 2674 l 1713 2740 l P 0.686275 G NP 2313 2743 m 2241 2612 l 2341 2611 l 2341 2611 l 2313 2743 l P 0.644627 G NP 2393 2748 m 2341 2611 l 2411 2628 l 2393 2748 l P NP 2393 2748 m 2313 2743 l 2341 2611 l 2341 2611 l 2393 2748 l P 0.739137 G NP 2511 2751 m 2500 2695 l 2499 2665 l 2499 2665 l 2507 2700 l 2511 2751 l P NP 2511 2751 m 2501 2705 l 2500 2695 l 2500 2695 l 2511 2751 l P NP 2511 2751 m 2505 2751 l 2502 2730 l 2501 2705 l 2501 2705 l 2511 2751 l P 0.718627 G NP 2201 2757 m 2107 2632 l 2241 2612 l 2241 2612 l 2201 2757 l P 0.686275 G NP 2201 2757 m 2241 2612 l 2313 2743 l 2201 2757 l P 0.597176 G NP 2449 2771 m 2393 2748 l 2411 2628 l 2411 2628 l 2449 2771 l P NP 2449 2771 m 2411 2628 l 2458 2662 l 2449 2771 l P 0.504392 G NP 2501 2771 m 2486 2710 l 2489 2612 l 2489 2612 l 2501 2771 l P 0.505569 G NP 2501 2771 m 2489 2612 l 2499 2684 l 2501 2771 l P 0.504392 G NP 2501 2771 m 2493 2771 l 2486 2710 l 2486 2710 l 2501 2771 l P 0.793529 G NP 1106 2779 m 1051 2636 l 1244 2730 l 1106 2779 l P 0.742863 G NP 2053 2791 m 1933 2674 l 2107 2632 l 2107 2632 l 2053 2791 l P 0.718627 G NP 2053 2791 m 2107 2632 l 2201 2757 l 2053 2791 l P 0.739137 G NP 2515 2803 m 2505 2751 l 2511 2751 l 2515 2803 l P NP 2515 2803 m 2503 2745 l 2502 2734 l 2502 2730 l 2502 2730 l 2515 2803 l P 0.539255 G NP 2483 2808 m 2449 2771 l 2458 2662 l 2458 2662 l 2483 2808 l P NP 2483 2808 m 2458 2662 l 2486 2710 l 2483 2808 l P 0.794745 G NP 1442 2830 m 1244 2730 l 1549 2628 l 1549 2628 l 1442 2830 l P 0.782078 G NP 1442 2830 m 1549 2628 l 1713 2740 l 1442 2830 l P 0.489098 G NP 2509 2844 m 2501 2771 l 2499 2684 l 2509 2844 l P 0.496157 G NP 2509 2844 m 2507 2844 l 2501 2771 l 2501 2771 l 2509 2844 l P 0.742863 G NP 1863 2846 m 1933 2674 l 2053 2791 l 1863 2846 l P 0.764784 G NP 1863 2846 m 1713 2740 l 1933 2674 l 1933 2674 l 1863 2846 l P 0.739137 G NP 2519 2854 m 2504 2771 l 2503 2745 l 2503 2745 l 2515 2803 l 2519 2854 l P NP 2519 2854 m 2505 2788 l 2504 2771 l 2504 2771 l 2519 2854 l P NP 2519 2854 m 2514 2854 l 2507 2811 l 2505 2788 l 2505 2788 l 2519 2854 l P 0.503216 G NP 2503 2859 m 2483 2808 l 2486 2710 l 2486 2710 l 2503 2859 l P 0.504392 G NP 2503 2859 m 2493 2771 l 2501 2771 l 2503 2859 l P NP 2503 2859 m 2492 2859 l 2483 2808 l 2483 2808 l 2503 2859 l P 0.783765 G NP 2570 2860 m 2507 2700 l 2503 2648 l 2503 2648 l 2570 2860 l P 0.644627 G NP 2376 2869 m 2313 2743 l 2393 2748 l 2376 2869 l P NP 2284 2876 m 2313 2743 l 2376 2869 l 2284 2876 l P 0.686275 G NP 2284 2876 m 2201 2757 l 2313 2743 l 2313 2743 l 2284 2876 l P 0.597176 G NP 2439 2879 m 2376 2869 l 2393 2748 l 2393 2748 l 2439 2879 l P NP 2439 2879 m 2393 2748 l 2449 2771 l 2439 2879 l P 0.793529 G NP 1149 2891 m 1106 2779 l 1244 2730 l 1244 2730 l 1149 2891 l P 0.718627 G NP 2161 2902 m 2053 2791 l 2201 2757 l 2201 2757 l 2161 2902 l P 0.686275 G NP 2161 2902 m 2201 2757 l 2284 2876 l 2161 2902 l P 0.783765 G NP 2570 2904 m 2507 2700 l 2570 2860 l 2570 2904 l P NP 2570 2904 m 2511 2751 l 2507 2700 l 2570 2904 l P 0.539255 G NP 2481 2906 m 2449 2771 l 2483 2808 l 2481 2906 l P 0.536941 G NP 2481 2906 m 2439 2879 l 2449 2771 l 2449 2771 l 2481 2906 l P 0.739137 G NP 2523 2906 m 2514 2854 l 2519 2854 l 2523 2906 l P NP 2523 2906 m 2508 2831 l 2507 2811 l 2507 2811 l 2523 2906 l P 0.487922 G NP 2514 2921 m 2503 2859 l 2501 2771 l 2514 2921 l P 0.496157 G NP 2514 2921 m 2507 2844 l 2509 2844 l 2514 2921 l P 0.764784 G NP 1625 2926 m 1713 2740 l 1863 2846 l 1625 2926 l P 0.782078 G NP 1625 2926 m 1442 2830 l 1713 2740 l 1713 2740 l 1625 2926 l P 0.794745 G NP 1163 2930 m 1149 2891 l 1244 2730 l 1244 2730 l 1442 2830 l 1163 2930 l P 0.503216 G NP 2506 2946 m 2481 2906 l 2483 2808 l 2483 2808 l 2506 2946 l P 0.504392 G NP 2506 2946 m 2492 2859 l 2503 2859 l 2506 2946 l P 0.503216 G NP 2506 2946 m 2489 2946 l 2481 2906 l 2481 2906 l 2506 2946 l P 0.783765 G NP 2570 2948 m 2515 2803 l 2511 2751 l 2511 2751 l 2570 2948 l P NP 2570 2948 m 2511 2751 l 2570 2904 l 2570 2948 l P 0.742863 G NP 1998 2949 m 1863 2846 l 2053 2791 l 2053 2791 l 1998 2949 l P 0.718627 G NP 1998 2949 m 2053 2791 l 2161 2902 l 1998 2949 l P 0.739137 G NP 2527 2957 m 2509 2850 l 2509 2844 l 2509 2844 l 2508 2831 l 2523 2906 l 2527 2957 l P NP 2527 2957 m 2511 2878 l 2509 2850 l 2509 2850 l 2527 2957 l P 0.47498 G NP 2519 2972 m 2514 2921 l 2509 2844 l 2519 2972 l P 0.597176 G NP 2430 2988 m 2376 2869 l 2439 2879 l 2430 2988 l P 0.644627 G NP 2358 2989 m 2284 2876 l 2376 2869 l 2376 2869 l 2358 2989 l P 0.597176 G NP 2358 2989 m 2376 2869 l 2430 2988 l 2358 2989 l P 0.783765 G NP 2570 2992 m 2519 2854 l 2515 2803 l 2570 2992 l P NP 2570 2992 m 2515 2803 l 2570 2948 l 2570 2992 l P 0.487922 G NP 2519 2999 m 2506 2946 l 2503 2859 l 2519 2999 l P 0.490275 G NP 2519 2999 m 2503 2859 l 2514 2921 l 2519 2999 l P 0.489098 G NP 2519 2999 m 2514 2999 l 2506 2946 l 2506 2946 l 2519 2999 l P 0.536941 G NP 2478 3003 m 2439 2879 l 2481 2906 l 2478 3003 l P NP 2478 3003 m 2430 2988 l 2439 2879 l 2439 2879 l 2478 3003 l P 0.644627 G NP 2256 3008 m 2284 2876 l 2358 2989 l 2256 3008 l P 0.686275 G NP 2256 3008 m 2161 2902 l 2284 2876 l 2284 2876 l 2256 3008 l P 0.739137 G NP 2531 3009 m 2515 2926 l 2513 2896 l 2513 2896 l 2531 3009 l P NP 2531 3009 m 2513 2896 l 2511 2878 l 2511 2878 l 2527 2957 l 2531 3009 l P NP 2531 3009 m 2527 3009 l 2516 2939 l 2515 2926 l 2515 2926 l 2531 3009 l P 0.764784 G NP 1793 3019 m 1625 2926 l 1863 2846 l 1863 2846 l 1793 3019 l P 0.742863 G NP 1793 3019 m 1863 2846 l 1998 2949 l 1793 3019 l P 0.794745 G NP 1335 3031 m 1178 2971 l 1163 2930 l 1442 2830 l 1442 2830 l 1335 3031 l P 0.782078 G NP 1335 3031 m 1442 2830 l 1625 2926 l 1335 3031 l P 0.503216 G NP 2508 3034 m 2489 2946 l 2506 2946 l 2508 3034 l P NP 2508 3034 m 2478 3003 l 2481 2906 l 2481 2906 l 2508 3034 l P NP 2508 3034 m 2486 3034 l 2478 3003 l 2478 3003 l 2508 3034 l P 0.783765 G NP 2570 3036 m 2519 2854 l 2570 2992 l 2570 3036 l P NP 2570 3036 m 2523 2906 l 2519 2854 l 2519 2854 l 2570 3036 l P 0.686275 G NP 2120 3047 m 2161 2902 l 2256 3008 l 2120 3047 l P 0.718627 G NP 2120 3047 m 1998 2949 l 2161 2902 l 2161 2902 l 2120 3047 l P 0.806431 G NP 2661 3056 m 2570 2904 l 2570 2860 l 2570 2860 l 2661 3056 l P 0.740314 G NP 2535 3060 m 2519 2972 l 2516 2939 l 2516 2939 l 2535 3060 l P 0.739137 G NP 2535 3060 m 2527 3009 l 2531 3009 l 2535 3060 l P NP 2535 3060 m 2531 3060 l 2520 2982 l 2519 2972 l 2519 2972 l 2519 2972 l 2535 3060 l P 0.466745 G NP 2527 3063 m 2519 2999 l 2514 2921 l 2527 3063 l P 0.794745 G NP 1218 3076 m 1178 2971 l 1335 3031 l 1218 3076 l P 0.489098 G NP 2525 3077 m 2514 2999 l 2519 2999 l 2525 3077 l P 0.487922 G NP 2525 3077 m 2508 3034 l 2506 2946 l 2525 3077 l P NP 2525 3077 m 2516 3077 l 2508 3034 l 2508 3034 l 2525 3077 l P 0.783765 G NP 2570 3080 m 2527 2957 l 2523 2906 l 2570 3080 l P NP 2570 3080 m 2523 2906 l 2570 3036 l 2570 3080 l P 0.804118 G NP 2655 3093 m 2570 2904 l 2661 3056 l 2655 3093 l P 0.806431 G NP 2655 3093 m 2570 2948 l 2570 2904 l 2570 2904 l 2655 3093 l P 0.597176 G NP 2420 3097 m 2358 2989 l 2430 2988 l 2430 2988 l 2420 3097 l P 0.536941 G NP 2475 3101 m 2430 2988 l 2478 3003 l 2475 3101 l P 0.535765 G NP 2475 3101 m 2420 3097 l 2430 2988 l 2475 3101 l P 0.71102 G NP 2532 3103 m 2527 3063 l 2526 3043 l 2532 3103 l P 0.742863 G NP 1944 3107 m 1793 3019 l 1998 2949 l 1998 2949 l 1944 3107 l P 0.718627 G NP 1944 3107 m 1998 2949 l 2120 3047 l 1944 3107 l P 0.597176 G NP 2340 3109 m 2358 2989 l 2420 3097 l 2340 3109 l P 0.644627 G NP 2340 3109 m 2256 3008 l 2358 2989 l 2358 2989 l 2340 3109 l P 0.739137 G NP 2539 3112 m 2531 3060 l 2535 3060 l 2539 3112 l P 0.740314 G NP 2539 3112 m 2524 3025 l 2520 2982 l 2520 2982 l 2539 3112 l P NP 2539 3112 m 2536 3112 l 2524 3032 l 2524 3025 l 2524 3025 l 2539 3112 l P 0.764784 G NP 1538 3113 m 1625 2926 l 1793 3019 l 1538 3113 l P 0.782078 G NP 1538 3113 m 1335 3031 l 1625 2926 l 1625 2926 l 1538 3113 l P 0.503216 G NP 2510 3121 m 2486 3034 l 2508 3034 l 2510 3121 l P NP 2510 3121 m 2475 3101 l 2478 3003 l 2478 3003 l 2510 3121 l P NP 2510 3121 m 2482 3121 l 2475 3101 l 2475 3101 l 2510 3121 l P 0.783765 G NP 2570 3124 m 2527 2957 l 2570 3080 l 2570 3124 l P NP 2570 3124 m 2531 3009 l 2527 2957 l 2570 3124 l P 0.806431 G NP 2650 3130 m 2570 2992 l 2570 2948 l 2570 2948 l 2650 3130 l P 0.804118 G NP 2650 3130 m 2570 2948 l 2655 3093 l 2650 3130 l P 0.465569 G NP 2534 3131 m 2525 3077 l 2519 2999 l 2534 3131 l P 0.476157 G NP 2534 3131 m 2519 2999 l 2527 3063 l 2534 3131 l P 0.467922 G NP 2534 3131 m 2532 3131 l 2525 3077 l 2525 3077 l 2534 3131 l P 0.686275 G NP 2228 3140 m 2120 3047 l 2256 3008 l 2256 3008 l 2228 3140 l P 0.644627 G NP 2228 3140 m 2256 3008 l 2340 3109 l 2228 3140 l P 0.487922 G NP 2530 3154 m 2516 3077 l 2525 3077 l 2530 3154 l P NP 2530 3154 m 2510 3121 l 2508 3034 l 2530 3154 l P NP 2530 3154 m 2518 3154 l 2510 3121 l 2510 3121 l 2530 3154 l P 0.740314 G NP 2543 3163 m 2529 3080 l 2526 3043 l 2524 3032 l 2524 3032 l 2543 3163 l P NP 2543 3163 m 2536 3112 l 2539 3112 l 2543 3163 l P NP 2543 3163 m 2540 3163 l 2530 3084 l 2529 3080 l 2529 3080 l 2543 3163 l P 0.804118 G NP 2644 3167 m 2570 2992 l 2650 3130 l 2644 3167 l P 0.806431 G NP 2644 3167 m 2570 3036 l 2570 2992 l 2570 2992 l 2644 3167 l P 0.784941 G NP 2570 3168 m 2535 3060 l 2531 3009 l 2531 3009 l 2570 3168 l P 0.783765 G NP 2570 3168 m 2531 3009 l 2570 3124 l 2570 3168 l P 0.794745 G NP 1256 3180 m 1218 3076 l 1335 3031 l 1335 3031 l 1256 3180 l P 0.764784 G NP 1723 3191 m 1538 3113 l 1793 3019 l 1793 3019 l 1723 3191 l P 0.742863 G NP 1723 3191 m 1793 3019 l 1944 3107 l 1723 3191 l P 0.686275 G NP 2080 3192 m 2120 3047 l 2228 3140 l 2080 3192 l P 0.718627 G NP 2080 3192 m 1944 3107 l 2120 3047 l 2120 3047 l 2080 3192 l P 0.447922 G NP 2541 3195 m 2534 3131 l 2527 3063 l 2541 3195 l P 0.702941 G NP 2541 3195 m 2541 3195 l 2534 3131 l 2534 3131 l 2541 3195 l P 0.536941 G NP 2473 3199 m 2420 3097 l 2475 3101 l 2473 3199 l P 0.464392 G NP 2541 3199 m 2530 3154 l 2525 3077 l 2541 3199 l P 0.467922 G NP 2541 3199 m 2532 3131 l 2534 3131 l 2541 3199 l P 0.465569 G NP 2541 3199 m 2537 3199 l 2530 3154 l 2530 3154 l 2541 3199 l P 0.806431 G NP 2638 3204 m 2570 3080 l 2570 3036 l 2570 3036 l 2638 3204 l P 0.804118 G NP 2638 3204 m 2570 3036 l 2644 3167 l 2638 3204 l P 0.535765 G NP 2411 3206 m 2420 3097 l 2473 3199 l 2411 3206 l P 0.597176 G NP 2411 3206 m 2340 3109 l 2420 3097 l 2420 3097 l 2411 3206 l P 0.503216 G NP 2513 3208 m 2482 3121 l 2510 3121 l 2513 3208 l P NP 2513 3208 m 2473 3199 l 2475 3101 l 2475 3101 l 2513 3208 l P NP 2513 3208 m 2477 3208 l 2473 3199 l 2473 3199 l 2513 3208 l P 0.784941 G NP 2570 3212 m 2535 3060 l 2570 3168 l 2570 3212 l P NP 2570 3212 m 2539 3112 l 2535 3060 l 2535 3060 l 2570 3212 l P 0.740314 G NP 2547 3215 m 2540 3163 l 2543 3163 l 2547 3215 l P 0.743804 G NP 2547 3215 m 2535 3137 l 2532 3103 l 2530 3084 l 2530 3084 l 2547 3215 l P 0.74149 G NP 2547 3215 m 2545 3215 l 2535 3140 l 2535 3137 l 2535 3137 l 2547 3215 l P 0.782078 G NP 1269 3217 m 1256 3180 l 1335 3031 l 1335 3031 l 1538 3113 l 1269 3217 l P 0.597176 G NP 2322 3230 m 2340 3109 l 2411 3206 l 2322 3230 l P 0.644627 G NP 2322 3230 m 2228 3140 l 2340 3109 l 2340 3109 l 2322 3230 l P 0.486745 G NP 2536 3232 m 2513 3208 l 2510 3121 l 2536 3232 l P 0.487922 G NP 2536 3232 m 2518 3154 l 2530 3154 l 2536 3232 l P NP 2536 3232 m 2519 3232 l 2513 3208 l 2513 3208 l 2536 3232 l P 0.776627 G NP 2776 3236 m 2655 3093 l 2661 3056 l 2661 3056 l 2776 3236 l P 0.804118 G NP 2632 3241 m 2570 3080 l 2638 3204 l 2632 3241 l P 0.806431 G NP 2632 3241 m 2570 3124 l 2570 3080 l 2570 3080 l 2632 3241 l P 0.446745 G NP 2547 3254 m 2541 3199 l 2534 3131 l 2547 3254 l P 0.702941 G NP 2547 3254 m 2541 3195 l 2541 3195 l 2547 3254 l P 0.449098 G NP 2547 3254 m 2547 3254 l 2541 3199 l 2541 3199 l 2547 3254 l P 0.784941 G NP 2570 3256 m 2539 3112 l 2570 3212 l 2570 3256 l P NP 2570 3256 m 2543 3163 l 2539 3112 l 2539 3112 l 2570 3256 l P 0.742863 G NP 1890 3266 m 1723 3191 l 1944 3107 l 1944 3107 l 1890 3266 l P 0.718627 G NP 1890 3266 m 1944 3107 l 2080 3192 l 1890 3266 l P 0.74149 G NP 2551 3267 m 2545 3215 l 2547 3215 l 2551 3267 l P 0.749647 G NP 2551 3267 m 2541 3195 l 2535 3140 l 2551 3267 l P 0.742667 G NP 2551 3267 m 2549 3267 l 2541 3195 l 2541 3195 l 2551 3267 l P 0.776627 G NP 2763 3267 m 2650 3130 l 2655 3093 l 2655 3093 l 2763 3267 l P NP 2763 3267 m 2655 3093 l 2776 3236 l 2763 3267 l P 0.465569 G NP 2548 3268 m 2537 3199 l 2541 3199 l 2548 3268 l P 0.464392 G NP 2548 3268 m 2536 3232 l 2530 3154 l 2530 3154 l 2548 3268 l P NP 2548 3268 m 2542 3268 l 2536 3232 l 2536 3232 l 2548 3268 l P 0.644627 G NP 2200 3273 m 2228 3140 l 2322 3230 l 2200 3273 l P 0.686275 G NP 2200 3273 m 2080 3192 l 2228 3140 l 2228 3140 l 2200 3273 l P 0.804118 G NP 2626 3278 m 2570 3168 l 2570 3124 l 2570 3124 l 2626 3278 l P NP 2626 3278 m 2570 3124 l 2632 3241 l 2626 3278 l P 0.503216 G NP 2515 3296 m 2477 3208 l 2513 3208 l 2515 3296 l P NP 2470 3297 m 2473 3199 l 2515 3296 l 2470 3297 l P 0.536941 G NP 2470 3297 m 2411 3206 l 2473 3199 l 2473 3199 l 2470 3297 l P 0.776627 G NP 2749 3297 m 2644 3167 l 2650 3130 l 2650 3130 l 2749 3297 l P NP 2749 3297 m 2650 3130 l 2763 3267 l 2749 3297 l P 0.449098 G NP 2552 3299 m 2547 3254 l 2547 3254 l 2552 3299 l P 0.764784 G NP 1450 3299 m 1538 3113 l 1723 3191 l 1450 3299 l P 0.782078 G NP 1450 3299 m 1281 3248 l 1269 3217 l 1538 3113 l 1538 3113 l 1450 3299 l P 0.784941 G NP 2570 3300 m 2543 3163 l 2570 3256 l 2570 3300 l P NP 2570 3300 m 2547 3215 l 2543 3163 l 2543 3163 l 2570 3300 l P 0.487922 G NP 2541 3309 m 2519 3232 l 2536 3232 l 2541 3309 l P 0.486745 G NP 2541 3309 m 2515 3296 l 2513 3208 l 2541 3309 l P NP 2541 3309 m 2520 3309 l 2515 3296 l 2515 3296 l 2541 3309 l P 0.446745 G NP 2553 3310 m 2548 3268 l 2541 3199 l 2552 3299 l 2553 3310 l P 0.535765 G NP 2402 3314 m 2411 3206 l 2470 3297 l 2402 3314 l P 0.597176 G NP 2402 3314 m 2322 3230 l 2411 3206 l 2411 3206 l 2402 3314 l P 0.804118 G NP 2620 3315 m 2570 3212 l 2570 3168 l 2570 3168 l 2620 3315 l P NP 2620 3315 m 2570 3168 l 2626 3278 l 2620 3315 l P 0.742667 G NP 2555 3318 m 2549 3267 l 2551 3267 l 2555 3318 l P 0.769529 G NP 2555 3318 m 2547 3254 l 2541 3195 l 2555 3318 l P 0.74502 G NP 2555 3318 m 2554 3318 l 2547 3254 l 2547 3254 l 2555 3318 l P 0.776627 G NP 2736 3328 m 2638 3204 l 2644 3167 l 2644 3167 l 2736 3328 l P NP 2736 3328 m 2644 3167 l 2749 3297 l 2736 3328 l P 0.464392 G NP 2555 3336 m 2541 3309 l 2536 3232 l 2555 3336 l P NP 2555 3336 m 2542 3268 l 2548 3268 l 2555 3336 l P NP 2555 3336 m 2547 3336 l 2541 3309 l 2541 3309 l 2555 3336 l P 0.718627 G NP 2039 3337 m 1890 3266 l 2080 3192 l 2080 3192 l 2039 3337 l P 0.686275 G NP 2039 3337 m 2080 3192 l 2200 3273 l 2039 3337 l P 0.446745 G NP 2556 3339 m 2548 3268 l 2553 3310 l 2555 3323 l 2556 3339 l P 0.786118 G NP 2570 3344 m 2551 3267 l 2547 3215 l 2570 3344 l P 0.784941 G NP 2570 3344 m 2547 3215 l 2570 3300 l 2570 3344 l P 0.644627 G NP 2304 3350 m 2200 3273 l 2322 3230 l 2322 3230 l 2304 3350 l P 0.597176 G NP 2304 3350 m 2322 3230 l 2402 3314 l 2304 3350 l P 0.804118 G NP 2614 3352 m 2570 3256 l 2570 3212 l 2570 3212 l 2614 3352 l P NP 2614 3352 m 2570 3212 l 2620 3315 l 2614 3352 l P 0.782078 G NP 1319 3353 m 1281 3248 l 1450 3299 l 1319 3353 l P 0.446745 G NP 2558 3358 m 2555 3336 l 2548 3268 l 2556 3339 l 2558 3358 l P 0.776627 G NP 2722 3358 m 2632 3241 l 2638 3204 l 2638 3204 l 2722 3358 l P NP 2722 3358 m 2638 3204 l 2736 3328 l 2722 3358 l P 0.764784 G NP 1652 3363 m 1450 3299 l 1723 3191 l 1723 3191 l 1652 3363 l P 0.742863 G NP 1652 3363 m 1723 3191 l 1890 3266 l 1652 3363 l P 0.74502 G NP 2559 3370 m 2554 3318 l 2555 3318 l 2559 3370 l P 0.760275 G NP 2559 3370 m 2559 3370 l 2555 3323 l 2555 3323 l 2559 3370 l P 0.446745 G NP 2559 3376 m 2555 3336 l 2558 3358 l 2559 3376 l P 0.503216 G NP 2517 3383 m 2470 3297 l 2515 3296 l 2515 3296 l 2517 3383 l P 0.486745 G NP 2547 3387 m 2520 3309 l 2541 3309 l 2547 3387 l P NP 2547 3387 m 2517 3383 l 2515 3296 l 2547 3387 l P NP 2547 3387 m 2519 3387 l 2517 3383 l 2517 3383 l 2547 3387 l P 0.787294 G NP 2570 3388 m 2555 3318 l 2551 3267 l 2551 3267 l 2570 3388 l P 0.784941 G NP 2570 3388 m 2551 3267 l 2570 3344 l 2570 3388 l P 0.776627 G NP 2709 3389 m 2626 3278 l 2632 3241 l 2632 3241 l 2709 3389 l P NP 2709 3389 m 2632 3241 l 2722 3358 l 2709 3389 l P 0.804118 G NP 2609 3390 m 2570 3256 l 2614 3352 l 2609 3390 l P NP 2609 3390 m 2570 3300 l 2570 3256 l 2570 3256 l 2609 3390 l P 0.536941 G NP 2468 3395 m 2402 3314 l 2470 3297 l 2470 3297 l 2468 3395 l P 0.503216 G NP 2468 3395 m 2470 3297 l 2517 3383 l 2468 3395 l P 0.714157 G NP 2915 3399 m 2763 3267 l 2776 3236 l 2776 3236 l 2915 3399 l P 0.464392 G NP 2561 3404 m 2547 3336 l 2555 3336 l 2561 3404 l P 0.463216 G NP 2561 3404 m 2547 3387 l 2541 3309 l 2561 3404 l P 0.464392 G NP 2561 3404 m 2551 3404 l 2547 3387 l 2547 3387 l 2561 3404 l P 0.686275 G NP 2171 3405 m 2039 3337 l 2200 3273 l 2200 3273 l 2171 3405 l P 0.644627 G NP 2171 3405 m 2200 3273 l 2304 3350 l 2171 3405 l P 0.446745 G NP 2563 3412 m 2561 3404 l 2555 3336 l 2555 3336 l 2559 3376 l 2563 3412 l P 0.776627 G NP 2695 3420 m 2620 3315 l 2626 3278 l 2626 3278 l 2695 3420 l P NP 2695 3420 m 2626 3278 l 2709 3389 l 2695 3420 l P 0.760275 G NP 2564 3421 m 2559 3370 l 2559 3370 l 2564 3421 l P 0.446745 G NP 2564 3421 m 2563 3421 l 2561 3404 l 2561 3404 l 2563 3412 l 2564 3421 l P 0.535765 G NP 2392 3423 m 2402 3314 l 2468 3395 l 2392 3423 l P 0.597176 G NP 2392 3423 m 2304 3350 l 2402 3314 l 2402 3314 l 2392 3423 l P 0.718627 G NP 1835 3424 m 1890 3266 l 2039 3337 l 1835 3424 l P 0.742863 G NP 1835 3424 m 1652 3363 l 1890 3266 l 1890 3266 l 1835 3424 l P 0.714157 G NP 2892 3424 m 2763 3267 l 2915 3399 l 2892 3424 l P NP 2892 3424 m 2749 3297 l 2763 3267 l 2763 3267 l 2892 3424 l P 0.804118 G NP 2603 3427 m 2570 3300 l 2609 3390 l 2603 3427 l P 0.805294 G NP 2603 3427 m 2570 3344 l 2570 3300 l 2570 3300 l 2603 3427 l P 0.786118 G NP 2570 3432 m 2555 3318 l 2570 3388 l 2570 3432 l P 0.789647 G NP 2570 3432 m 2559 3370 l 2555 3318 l 2555 3318 l 2570 3432 l P 0.446745 G NP 2565 3438 m 2561 3404 l 2564 3422 l 2565 3438 l P 0.711843 G NP 2869 3449 m 2749 3297 l 2892 3424 l 2869 3449 l P 0.714157 G NP 2869 3449 m 2736 3328 l 2749 3297 l 2749 3297 l 2869 3449 l P 0.776627 G NP 2682 3450 m 2620 3315 l 2695 3420 l 2682 3450 l P NP 2682 3450 m 2614 3352 l 2620 3315 l 2620 3315 l 2682 3450 l P 0.805294 G NP 2597 3464 m 2570 3388 l 2570 3344 l 2570 3344 l 2597 3464 l P 0.804118 G NP 2597 3464 m 2570 3344 l 2603 3427 l 2597 3464 l P 0.486745 G NP 2552 3465 m 2519 3387 l 2547 3387 l 2552 3465 l P 0.464392 G NP 2566 3466 m 2551 3404 l 2561 3404 l 2565 3438 l 2566 3466 l P 0.644627 G NP 2286 3470 m 2171 3405 l 2304 3350 l 2304 3350 l 2286 3470 l P 0.597176 G NP 2286 3470 m 2304 3350 l 2392 3423 l 2286 3470 l P 0.503216 G NP 2520 3471 m 2468 3395 l 2517 3383 l 2517 3383 l 2520 3471 l P 0.486745 G NP 2520 3471 m 2517 3383 l 2552 3465 l 2520 3471 l P 0.463216 G NP 2567 3472 m 2552 3465 l 2547 3387 l 2566 3466 l 2567 3472 l P 0.711843 G NP 2846 3473 m 2736 3328 l 2869 3449 l 2846 3473 l P 0.714157 G NP 2846 3473 m 2722 3358 l 2736 3328 l 2736 3328 l 2846 3473 l P 0.787294 G NP 2570 3476 m 2559 3370 l 2570 3432 l 2570 3476 l P 0.804902 G NP 2570 3476 m 2564 3421 l 2559 3370 l 2570 3476 l P 0.776627 G NP 2668 3481 m 2614 3352 l 2682 3450 l 2668 3481 l P NP 2668 3481 m 2609 3390 l 2614 3352 l 2614 3352 l 2668 3481 l P 0.782078 G NP 1365 3481 m 1319 3353 l 1450 3299 l 1450 3299 l 1365 3481 l P 0.686275 G NP 1999 3482 m 2039 3337 l 2171 3405 l 1999 3482 l P 0.718627 G NP 1999 3482 m 1835 3424 l 2039 3337 l 2039 3337 l 1999 3482 l P 0.764784 G NP 1366 3484 m 1365 3481 l 1450 3299 l 1652 3363 l 1366 3484 l P 0.536941 G NP 2465 3493 m 2392 3423 l 2468 3395 l 2468 3395 l 2465 3493 l P 0.503216 G NP 2465 3493 m 2468 3395 l 2520 3471 l 2465 3493 l P 0.711843 G NP 2823 3498 m 2709 3389 l 2722 3358 l 2722 3358 l 2823 3498 l P NP 2823 3498 m 2722 3358 l 2846 3473 l 2823 3498 l P 0.805294 G NP 2591 3501 m 2570 3432 l 2570 3388 l 2570 3388 l 2591 3501 l P NP 2591 3501 m 2570 3388 l 2597 3464 l 2591 3501 l P 0.776627 G NP 2655 3511 m 2609 3390 l 2668 3481 l 2655 3511 l P NP 2655 3511 m 2603 3427 l 2609 3390 l 2609 3390 l 2655 3511 l P 0.787333 G NP 2570 3520 m 2564 3421 l 2570 3476 l 2570 3520 l P 0.464392 G NP 2569 3521 m 2552 3465 l 2567 3472 l 2568 3490 l 2569 3521 l P 0.711843 G NP 2800 3523 m 2695 3420 l 2709 3389 l 2709 3389 l 2800 3523 l P NP 2800 3523 m 2709 3389 l 2823 3498 l 2800 3523 l P 0.597176 G NP 2383 3532 m 2286 3470 l 2392 3423 l 2392 3423 l 2383 3532 l P 0.535765 G NP 2383 3532 m 2392 3423 l 2465 3493 l 2383 3532 l P 0.764784 G NP 1582 3535 m 1366 3487 l 1366 3484 l 1652 3363 l 1652 3363 l 1582 3535 l P 0.742863 G NP 1582 3535 m 1652 3363 l 1835 3424 l 1582 3535 l P 0.805294 G NP 2585 3538 m 2570 3432 l 2591 3501 l 2585 3538 l P 0.806471 G NP 2585 3538 m 2570 3476 l 2570 3432 l 2570 3432 l 2585 3538 l P 0.644627 G NP 2143 3538 m 2171 3405 l 2286 3470 l 2143 3538 l P 0.686275 G NP 2143 3538 m 1999 3482 l 2171 3405 l 2171 3405 l 2143 3538 l P 0.776627 G NP 2641 3542 m 2597 3464 l 2603 3427 l 2603 3427 l 2641 3542 l P NP 2641 3542 m 2603 3427 l 2655 3511 l 2641 3542 l P 0.463216 G NP 2558 3542 m 2552 3465 l 2569 3521 l 2569 3541 l 2558 3542 l P 0.486745 G NP 2558 3542 m 2520 3471 l 2552 3465 l 2552 3465 l 2558 3542 l P 0.637804 G NP 3078 3546 m 2892 3424 l 2915 3399 l 2915 3399 l 3078 3546 l P 0.711843 G NP 2778 3548 m 2695 3420 l 2800 3523 l 2778 3548 l P NP 2778 3548 m 2682 3450 l 2695 3420 l 2695 3420 l 2778 3548 l P 0.486745 G NP 2522 3558 m 2520 3471 l 2558 3542 l 2522 3558 l P 0.503216 G NP 2522 3558 m 2465 3493 l 2520 3471 l 2520 3471 l 2522 3558 l P 0.79451 G NP 2570 3564 m 2568 3490 l 2570 3520 l 2570 3564 l P 0.463216 G NP 2570 3564 m 2565 3564 l 2558 3542 l 2569 3541 l 2569 3541 l 2570 3564 l P 0.637804 G NP 3044 3565 m 2869 3449 l 2892 3424 l 2892 3424 l 3044 3565 l P NP 3044 3565 m 2892 3424 l 3078 3546 l 3044 3565 l P 0.711843 G NP 2755 3573 m 2668 3481 l 2682 3450 l 2682 3450 l 2755 3573 l P NP 2755 3573 m 2682 3450 l 2778 3548 l 2755 3573 l P 0.776627 G NP 2628 3573 m 2591 3501 l 2597 3464 l 2597 3464 l 2628 3573 l P NP 2628 3573 m 2597 3464 l 2641 3542 l 2628 3573 l P 0.804157 G NP 2579 3575 m 2570 3476 l 2585 3538 l 2579 3575 l P 0.80651 G NP 2579 3575 m 2570 3520 l 2570 3476 l 2570 3476 l 2579 3575 l P 0.463216 G NP 2570 3576 m 2565 3564 l 2570 3564 l 2570 3564 l 2570 3576 l P 0.742863 G NP 1781 3582 m 1582 3535 l 1835 3424 l 1835 3424 l 1781 3582 l P 0.718627 G NP 1781 3582 m 1835 3424 l 1999 3482 l 1781 3582 l P 0.637804 G NP 3010 3585 m 2846 3473 l 2869 3449 l 2869 3449 l 3010 3585 l P NP 3010 3585 m 2869 3449 l 3044 3565 l 3010 3585 l P 0.535765 G NP 2462 3590 m 2383 3532 l 2465 3493 l 2465 3493 l 2462 3590 l P 0.503216 G NP 2462 3590 m 2465 3493 l 2522 3558 l 2462 3590 l P 0.597176 G NP 2268 3591 m 2286 3470 l 2383 3532 l 2268 3591 l P 0.644627 G NP 2268 3591 m 2143 3538 l 2286 3470 l 2286 3470 l 2268 3591 l P 0.711843 G NP 2732 3597 m 2668 3481 l 2755 3573 l 2732 3597 l P NP 2732 3597 m 2655 3511 l 2668 3481 l 2668 3481 l 2732 3597 l P 0.774314 G NP 2614 3603 m 2591 3501 l 2628 3573 l 2614 3603 l P NP 2614 3603 m 2585 3538 l 2591 3501 l 2591 3501 l 2614 3603 l P 0.637804 G NP 2975 3604 m 2823 3498 l 2846 3473 l 2846 3473 l 2975 3604 l P NP 2975 3604 m 2846 3473 l 3010 3585 l 2975 3604 l P 0.80902 G NP 2573 3612 m 2570 3564 l 2570 3520 l 2570 3520 l 2573 3612 l P 0.804157 G NP 2573 3612 m 2570 3520 l 2579 3575 l 2573 3612 l P 0.764784 G NP 1410 3612 m 1366 3487 l 1582 3535 l 1410 3612 l P 0.486745 G NP 2563 3620 m 2522 3558 l 2558 3542 l 2558 3542 l 2563 3620 l P 0.463216 G NP 2563 3620 m 2558 3542 l 2570 3576 l 2569 3617 l 2563 3620 l P 0.711843 G NP 2709 3622 m 2655 3511 l 2732 3597 l 2709 3622 l P NP 2709 3622 m 2641 3542 l 2655 3511 l 2655 3511 l 2709 3622 l P 0.637804 G NP 2941 3624 m 2800 3523 l 2823 3498 l 2823 3498 l 2941 3624 l P NP 2941 3624 m 2823 3498 l 2975 3604 l 2941 3624 l P 0.718627 G NP 1959 3627 m 1781 3582 l 1999 3482 l 1999 3482 l 1959 3627 l P 0.686275 G NP 1959 3627 m 1999 3482 l 2143 3538 l 1959 3627 l P 0.463216 G NP 2568 3630 m 2563 3620 l 2569 3617 l 2569 3617 l 2568 3625 l 2568 3630 l P 0.774314 G NP 2601 3634 m 2585 3538 l 2614 3603 l 2601 3634 l P NP 2601 3634 m 2579 3575 l 2585 3538 l 2585 3538 l 2601 3634 l P 0.597176 G NP 2374 3641 m 2268 3591 l 2383 3532 l 2383 3532 l 2374 3641 l P 0.535765 G NP 2374 3641 m 2383 3532 l 2462 3590 l 2374 3641 l P 0.637804 G NP 2907 3644 m 2800 3523 l 2941 3624 l 2907 3644 l P NP 2907 3644 m 2778 3548 l 2800 3523 l 2800 3523 l 2907 3644 l P 0.503216 G NP 2524 3645 m 2462 3590 l 2522 3558 l 2522 3558 l 2524 3645 l P 0.486745 G NP 2524 3645 m 2522 3558 l 2563 3620 l 2524 3645 l P 0.711843 G NP 2686 3647 m 2641 3542 l 2709 3622 l 2686 3647 l P NP 2686 3647 m 2628 3573 l 2641 3542 l 2641 3542 l 2686 3647 l P 0.805373 G NP 2568 3649 m 2570 3564 l 2573 3612 l 2568 3649 l P 0.799765 G NP 2568 3649 m 2566 3649 l 2568 3625 l 2568 3625 l 2568 3649 l P 0.463216 G NP 2565 3652 m 2563 3620 l 2568 3630 l 2565 3652 l P 0.637804 G NP 2873 3663 m 2778 3548 l 2907 3644 l 2873 3663 l P NP 2873 3663 m 2755 3573 l 2778 3548 l 2778 3548 l 2873 3663 l P 0.774314 G NP 2587 3664 m 2579 3575 l 2601 3634 l 2587 3664 l P NP 2587 3664 m 2573 3612 l 2579 3575 l 2579 3575 l 2587 3664 l P 0.644627 G NP 2115 3670 m 2143 3538 l 2268 3591 l 2115 3670 l P 0.686275 G NP 2115 3670 m 1959 3627 l 2143 3538 l 2143 3538 l 2115 3670 l P 0.711843 G NP 2663 3672 m 2628 3573 l 2686 3647 l 2663 3672 l P NP 2663 3672 m 2614 3603 l 2628 3573 l 2628 3573 l 2663 3672 l P 0.596157 G NP 3265 3676 m 3044 3565 l 3078 3546 l 3078 3546 l 3265 3676 l P 0.637804 G NP 2839 3683 m 2732 3597 l 2755 3573 l 2755 3573 l 2839 3683 l P NP 2839 3683 m 2755 3573 l 2873 3663 l 2839 3683 l P 0.799765 G NP 2562 3686 m 2566 3649 l 2568 3649 l 2568 3649 l 2562 3686 l P 0.486745 G NP 2562 3686 m 2558 3686 l 2524 3645 l 2563 3620 l 2563 3620 l 2565 3652 l 2562 3686 l P 0.503216 G NP 2460 3688 m 2462 3590 l 2524 3645 l 2460 3688 l P 0.535765 G NP 2460 3688 m 2374 3641 l 2462 3590 l 2462 3590 l 2460 3688 l P 0.486745 G NP 2561 3689 m 2558 3686 l 2562 3686 l 2562 3686 l 2561 3689 l P 0.596157 G NP 3217 3691 m 3044 3565 l 3265 3676 l 3217 3691 l P NP 3217 3691 m 3010 3585 l 3044 3565 l 3044 3565 l 3217 3691 l P 0.774314 G NP 2574 3695 m 2573 3612 l 2587 3664 l 2574 3695 l P 0.769686 G NP 2574 3695 m 2568 3649 l 2573 3612 l 2573 3612 l 2574 3695 l P 0.711843 G NP 2640 3696 m 2614 3603 l 2663 3672 l 2640 3696 l P NP 2640 3696 m 2601 3634 l 2614 3603 l 2614 3603 l 2640 3696 l P 0.637804 G NP 2804 3702 m 2732 3597 l 2839 3683 l 2804 3702 l P NP 2804 3702 m 2709 3622 l 2732 3597 l 2732 3597 l 2804 3702 l P 0.596157 G NP 3170 3706 m 3010 3585 l 3217 3691 l 3170 3706 l P NP 3170 3706 m 2975 3604 l 3010 3585 l 3010 3585 l 3170 3706 l P 0.764784 G NP 1512 3707 m 1439 3696 l 1410 3612 l 1582 3535 l 1582 3535 l 1512 3707 l P 0.742863 G NP 1512 3707 m 1582 3535 l 1781 3582 l 1512 3707 l P 0.644627 G NP 2250 3711 m 2115 3670 l 2268 3591 l 2268 3591 l 2250 3711 l P 0.597176 G NP 2250 3711 m 2268 3591 l 2374 3641 l 2250 3711 l P 0.596157 G NP 3123 3721 m 2975 3604 l 3170 3706 l 3123 3721 l P NP 3123 3721 m 2941 3624 l 2975 3604 l 2975 3604 l 3123 3721 l P 0.711843 G NP 2617 3721 m 2587 3664 l 2601 3634 l 2601 3634 l 2617 3721 l P NP 2617 3721 m 2601 3634 l 2640 3696 l 2617 3721 l P 0.637804 G NP 2770 3722 m 2686 3647 l 2709 3622 l 2709 3622 l 2770 3722 l P NP 2770 3722 m 2709 3622 l 2804 3702 l 2770 3722 l P 0.772 G NP 2560 3726 m 2568 3649 l 2574 3695 l 2560 3726 l P 0.755804 G NP 2560 3726 m 2562 3686 l 2568 3649 l 2560 3726 l P 0.486745 G NP 2526 3733 m 2524 3645 l 2561 3689 l 2557 3707 l 2526 3733 l P 0.503216 G NP 2526 3733 m 2460 3688 l 2524 3645 l 2524 3645 l 2526 3733 l P 0.764784 G NP 1453 3735 m 1439 3696 l 1512 3707 l 1453 3735 l P 0.596157 G NP 3076 3736 m 2941 3624 l 3123 3721 l 3076 3736 l P NP 3076 3736 m 2907 3644 l 2941 3624 l 2941 3624 l 3076 3736 l P 0.742863 G NP 1727 3740 m 1512 3707 l 1781 3582 l 1781 3582 l 1727 3740 l P 0.718627 G NP 1727 3740 m 1781 3582 l 1959 3627 l 1727 3740 l P 0.637804 G NP 2736 3741 m 2686 3647 l 2770 3722 l 2736 3741 l P NP 2736 3741 m 2663 3672 l 2686 3647 l 2686 3647 l 2736 3741 l P 0.709529 G NP 2594 3746 m 2587 3664 l 2617 3721 l 2594 3746 l P NP 2594 3746 m 2574 3695 l 2587 3664 l 2587 3664 l 2594 3746 l P 0.535765 G NP 2364 3749 m 2374 3641 l 2460 3688 l 2364 3749 l P 0.597176 G NP 2364 3749 m 2250 3711 l 2374 3641 l 2374 3641 l 2364 3749 l P 0.486745 G NP 2546 3750 m 2526 3733 l 2557 3707 l 2557 3707 l 2551 3737 l 2546 3750 l P 0.596157 G NP 3028 3750 m 2907 3644 l 3076 3736 l 3028 3750 l P NP 3028 3750 m 2873 3663 l 2907 3644 l 2907 3644 l 3028 3750 l P 0.767373 G NP 2547 3756 m 2562 3686 l 2560 3726 l 2547 3756 l P 0.63549 G NP 2702 3761 m 2663 3672 l 2736 3741 l 2702 3761 l P NP 2702 3761 m 2640 3696 l 2663 3672 l 2663 3672 l 2702 3761 l P 0.596157 G NP 2981 3765 m 2873 3663 l 3028 3750 l 2981 3765 l P NP 2981 3765 m 2839 3683 l 2873 3663 l 2873 3663 l 2981 3765 l P 0.709529 G NP 2571 3771 m 2574 3695 l 2594 3746 l 2571 3771 l P 0.707216 G NP 2571 3771 m 2560 3726 l 2574 3695 l 2574 3695 l 2571 3771 l P 0.686275 G NP 1918 3772 m 1959 3627 l 2115 3670 l 1918 3772 l P 0.718627 G NP 1918 3772 m 1727 3740 l 1959 3627 l 1959 3627 l 1918 3772 l P 0.63549 G NP 2667 3780 m 2640 3696 l 2702 3761 l 2667 3780 l P NP 2667 3780 m 2617 3721 l 2640 3696 l 2640 3696 l 2667 3780 l P 0.596157 G NP 2934 3780 m 2839 3683 l 2981 3765 l 2934 3780 l P NP 2934 3780 m 2804 3702 l 2839 3683 l 2839 3683 l 2934 3780 l P 0.535765 G NP 2457 3786 m 2364 3749 l 2460 3688 l 2460 3688 l 2457 3786 l P 0.503216 G NP 2457 3786 m 2460 3688 l 2526 3733 l 2457 3786 l P 0.746549 G NP 2533 3787 m 2551 3737 l 2547 3756 l 2533 3787 l P 0.486745 G NP 2533 3787 m 2528 3787 l 2526 3733 l 2526 3733 l 2546 3750 l 2533 3787 l P 0.596157 G NP 3475 3790 m 3217 3691 l 3265 3676 l 3265 3676 l 3475 3790 l P NP 2886 3795 m 2770 3722 l 2804 3702 l 2804 3702 l 2886 3795 l P NP 2886 3795 m 2804 3702 l 2934 3780 l 2886 3795 l P 0.704902 G NP 2548 3796 m 2547 3756 l 2560 3726 l 2560 3726 l 2548 3796 l P 0.709529 G NP 2548 3796 m 2560 3726 l 2571 3771 l 2548 3796 l P 0.486745 G NP 2528 3796 m 2528 3787 l 2533 3787 l 2533 3787 l 2528 3796 l P 0.764784 G NP 1475 3799 m 1453 3735 l 1512 3707 l 1512 3707 l 1475 3799 l P 0.63549 G NP 2633 3800 m 2594 3746 l 2617 3721 l 2617 3721 l 2633 3800 l P NP 2633 3800 m 2617 3721 l 2667 3780 l 2633 3800 l P 0.596157 G NP 3413 3801 m 3217 3691 l 3475 3790 l 3413 3801 l P NP 3413 3801 m 3170 3706 l 3217 3691 l 3217 3691 l 3413 3801 l P 0.644627 G NP 2087 3802 m 2115 3670 l 2250 3711 l 2087 3802 l P 0.686275 G NP 2087 3802 m 1918 3772 l 2115 3670 l 2115 3670 l 2087 3802 l P 0.596157 G NP 2839 3810 m 2736 3741 l 2770 3722 l 2770 3722 l 2839 3810 l P NP 2839 3810 m 2770 3722 l 2886 3795 l 2839 3810 l P NP 3351 3812 m 3170 3706 l 3413 3801 l 3351 3812 l P NP 3351 3812 m 3123 3721 l 3170 3706 l 3170 3706 l 3351 3812 l P 0.503216 G NP 2518 3815 m 2457 3786 l 2526 3733 l 2526 3733 l 2528 3796 l 2518 3815 l P 0.63549 G NP 2599 3819 m 2571 3771 l 2594 3746 l 2594 3746 l 2599 3819 l P NP 2599 3819 m 2594 3746 l 2633 3800 l 2599 3819 l P 0.684078 G NP 2525 3820 m 2533 3787 l 2547 3756 l 2547 3756 l 2525 3820 l P 0.707216 G NP 2525 3820 m 2547 3756 l 2548 3796 l 2525 3820 l P 0.596157 G NP 3289 3823 m 3123 3721 l 3351 3812 l 3289 3823 l P NP 3289 3823 m 3076 3736 l 3123 3721 l 3123 3721 l 3289 3823 l P NP 2792 3825 m 2736 3741 l 2839 3810 l 2792 3825 l P NP 2792 3825 m 2702 3761 l 2736 3741 l 2736 3741 l 2792 3825 l P 0.644627 G NP 2232 3831 m 2087 3802 l 2250 3711 l 2250 3711 l 2232 3831 l P 0.597176 G NP 2232 3831 m 2250 3711 l 2364 3749 l 2232 3831 l P 0.596157 G NP 3227 3833 m 3076 3736 l 3289 3823 l 3227 3833 l P NP 3227 3833 m 3028 3750 l 3076 3736 l 3076 3736 l 3227 3833 l P 0.63549 G NP 2565 3839 m 2571 3771 l 2599 3819 l 2565 3839 l P 0.633177 G NP 2565 3839 m 2548 3796 l 2571 3771 l 2571 3771 l 2565 3839 l P 0.596157 G NP 2745 3839 m 2702 3761 l 2792 3825 l 2745 3839 l P NP 2745 3839 m 2667 3780 l 2702 3761 l 2702 3761 l 2745 3839 l P NP 3164 3844 m 3028 3750 l 3227 3833 l 3164 3844 l P NP 3164 3844 m 2981 3765 l 3028 3750 l 3028 3750 l 3164 3844 l P 0.700275 G NP 2502 3845 m 2533 3787 l 2525 3820 l 2502 3845 l P 0.596157 G NP 2697 3854 m 2667 3780 l 2745 3839 l 2697 3854 l P NP 2697 3854 m 2633 3800 l 2667 3780 l 2667 3780 l 2697 3854 l P 0.742863 G NP 1493 3854 m 1475 3799 l 1512 3707 l 1512 3707 l 1727 3740 l 1493 3854 l P 0.596157 G NP 3102 3855 m 2934 3780 l 2981 3765 l 2981 3765 l 3102 3855 l P NP 3102 3855 m 2981 3765 l 3164 3844 l 3102 3855 l P 0.630863 G NP 2530 3858 m 2525 3820 l 2548 3796 l 2548 3796 l 2530 3858 l P 0.633177 G NP 2530 3858 m 2548 3796 l 2565 3839 l 2530 3858 l P 0.597176 G NP 2355 3858 m 2232 3831 l 2364 3749 l 2364 3749 l 2355 3858 l P 0.535765 G NP 2355 3858 m 2364 3749 l 2457 3786 l 2355 3858 l P 0.596157 G NP 3040 3865 m 2934 3780 l 3102 3855 l 3040 3865 l P NP 3040 3865 m 2886 3795 l 2934 3780 l 2934 3780 l 3040 3865 l P NP 2650 3869 m 2599 3819 l 2633 3800 l 2633 3800 l 2650 3869 l P NP 2650 3869 m 2633 3800 l 2697 3854 l 2650 3869 l P 0.674824 G NP 2479 3870 m 2510 3830 l 2502 3845 l 2479 3870 l P 0.503216 G NP 2479 3870 m 2471 3870 l 2495 3849 l 2495 3849 l 2479 3870 l P 0.596157 G NP 2978 3876 m 2886 3795 l 3040 3865 l 2978 3876 l P NP 2978 3876 m 2839 3810 l 2886 3795 l 2886 3795 l 2978 3876 l P 0.628549 G NP 2496 3878 m 2502 3845 l 2525 3820 l 2496 3878 l P 0.633177 G NP 2496 3878 m 2525 3820 l 2530 3858 l 2496 3878 l P 0.535765 G NP 2454 3884 m 2355 3858 l 2457 3786 l 2457 3786 l 2454 3884 l P 0.503216 G NP 2454 3884 m 2457 3786 l 2518 3815 l 2510 3830 l 2495 3849 l 2454 3884 l P 0.596157 G NP 2603 3884 m 2565 3839 l 2599 3819 l 2599 3819 l 2603 3884 l P NP 2603 3884 m 2599 3819 l 2650 3869 l 2603 3884 l P 0.503216 G NP 2461 3886 m 2454 3884 l 2471 3870 l 2479 3870 l 2479 3870 l 2461 3886 l P 0.596157 G NP 2916 3887 m 2839 3810 l 2978 3876 l 2916 3887 l P NP 2916 3887 m 2792 3825 l 2839 3810 l 2839 3810 l 2916 3887 l P NP 3710 3888 m 3413 3801 l 3475 3790 l 3475 3790 l 3710 3888 l P 0.503216 G NP 2454 3892 m 2454 3884 l 2461 3886 l 2454 3892 l P 0.596157 G NP 3631 3895 m 3351 3812 l 3413 3801 l 3413 3801 l 3631 3895 l P NP 3631 3895 m 3413 3801 l 3710 3888 l 3631 3895 l P 0.605412 G NP 2462 3897 m 2479 3870 l 2502 3845 l 2462 3897 l P 0.630863 G NP 2462 3897 m 2502 3845 l 2496 3878 l 2462 3897 l P 0.596157 G NP 2854 3897 m 2792 3825 l 2916 3887 l 2854 3897 l P NP 2854 3897 m 2745 3839 l 2792 3825 l 2792 3825 l 2854 3897 l P NP 2555 3899 m 2530 3858 l 2565 3839 l 2565 3839 l 2555 3899 l P NP 2555 3899 m 2565 3839 l 2603 3884 l 2555 3899 l P 0.718627 G NP 1672 3899 m 1727 3740 l 1918 3772 l 1672 3899 l P 0.742863 G NP 1672 3899 m 1503 3884 l 1493 3854 l 1727 3740 l 1727 3740 l 1672 3899 l P 0.596157 G NP 3552 3902 m 3351 3812 l 3631 3895 l 3552 3902 l P NP 3552 3902 m 3289 3823 l 3351 3812 l 3351 3812 l 3552 3902 l P NP 2792 3908 m 2697 3854 l 2745 3839 l 2745 3839 l 2792 3908 l P NP 2792 3908 m 2745 3839 l 2854 3897 l 2792 3908 l P NP 3473 3910 m 3289 3823 l 3552 3902 l 3473 3910 l P NP 3473 3910 m 3227 3833 l 3289 3823 l 3289 3823 l 3473 3910 l P NP 2508 3913 m 2530 3858 l 2555 3899 l 2508 3913 l P NP 2508 3913 m 2496 3878 l 2530 3858 l 2530 3858 l 2508 3913 l P 0.623922 G NP 2428 3917 m 2479 3870 l 2462 3897 l 2428 3917 l P 0.596157 G NP 3394 3917 m 3227 3833 l 3473 3910 l 3394 3917 l P NP 3394 3917 m 3164 3844 l 3227 3833 l 3227 3833 l 3394 3917 l P 0.686275 G NP 1878 3917 m 1918 3772 l 2087 3802 l 1878 3917 l P 0.718627 G NP 1878 3917 m 1672 3899 l 1918 3772 l 1918 3772 l 1878 3917 l P 0.596157 G NP 2729 3919 m 2650 3869 l 2697 3854 l 2697 3854 l 2729 3919 l P NP 2729 3919 m 2697 3854 l 2792 3908 l 2729 3919 l P NP 3315 3924 m 3102 3855 l 3164 3844 l 3164 3844 l 3315 3924 l P NP 3315 3924 m 3164 3844 l 3394 3917 l 3315 3924 l P NP 2461 3928 m 2462 3897 l 2496 3878 l 2461 3928 l P NP 2461 3928 m 2496 3878 l 2508 3913 l 2461 3928 l P NP 2667 3929 m 2650 3869 l 2729 3919 l 2667 3929 l P NP 2667 3929 m 2603 3884 l 2650 3869 l 2650 3869 l 2667 3929 l P NP 3237 3931 m 3102 3855 l 3315 3924 l 3237 3931 l P NP 3237 3931 m 3040 3865 l 3102 3855 l 3102 3855 l 3237 3931 l P 0.644627 G NP 2058 3935 m 2087 3802 l 2232 3831 l 2058 3935 l P 0.686275 G NP 2058 3935 m 1878 3917 l 2087 3802 l 2087 3802 l 2058 3935 l P 0.598471 G NP 2393 3936 m 2441 3905 l 2428 3917 l 2393 3936 l P 0.596157 G NP 3158 3938 m 3040 3865 l 3237 3931 l 3158 3938 l P NP 3158 3938 m 2978 3876 l 3040 3865 l 3040 3865 l 3158 3938 l P NP 2605 3940 m 2603 3884 l 2667 3929 l 2605 3940 l P NP 2605 3940 m 2555 3899 l 2603 3884 l 2603 3884 l 2605 3940 l P NP 2413 3943 m 2428 3917 l 2462 3897 l 2413 3943 l P NP 2413 3943 m 2462 3897 l 2461 3928 l 2413 3943 l P NP 3079 3945 m 2978 3876 l 3158 3938 l 3079 3945 l P NP 3079 3945 m 2916 3887 l 2978 3876 l 2978 3876 l 3079 3945 l P 0.535765 G NP 2372 3946 m 2454 3884 l 2454 3892 l 2441 3905 l 2393 3936 l 2393 3936 l 2372 3946 l P 0.596157 G NP 2543 3951 m 2555 3899 l 2605 3940 l 2543 3951 l P NP 2543 3951 m 2508 3913 l 2555 3899 l 2555 3899 l 2543 3951 l P 0.644627 G NP 2214 3951 m 2058 3935 l 2232 3831 l 2232 3831 l 2214 3951 l P 0.597176 G NP 2214 3951 m 2232 3831 l 2355 3858 l 2214 3951 l P 0.596157 G NP 3000 3952 m 2916 3887 l 3079 3945 l 3000 3952 l P NP 3000 3952 m 2854 3897 l 2916 3887 l 2916 3887 l 3000 3952 l P NP 2366 3958 m 2428 3917 l 2413 3943 l 2366 3958 l P NP 2366 3958 m 2393 3936 l 2428 3917 l 2428 3917 l 2366 3958 l P 0.535765 G NP 2346 3959 m 2355 3858 l 2454 3884 l 2372 3946 l 2346 3959 l P 0.596157 G NP 2921 3959 m 2792 3908 l 2854 3897 l 2854 3897 l 2921 3959 l P NP 2921 3959 m 2854 3897 l 3000 3952 l 2921 3959 l P NP 2481 3961 m 2508 3913 l 2543 3951 l 2481 3961 l P NP 2481 3961 m 2461 3928 l 2508 3913 l 2508 3913 l 2481 3961 l P 0.597176 G NP 2332 3965 m 2214 3951 l 2355 3858 l 2355 3858 l 2346 3959 l 2338 3963 l 2332 3965 l P 0.596157 G NP 2843 3967 m 2792 3908 l 2921 3959 l 2843 3967 l P NP 2843 3967 m 2729 3919 l 2792 3908 l 2792 3908 l 2843 3967 l P NP 3968 3969 m 3631 3895 l 3710 3888 l 3710 3888 l 3968 3969 l P 0.742863 G NP 1532 3970 m 1503 3884 l 1672 3899 l 1532 3970 l P 0.596157 G NP 2419 3972 m 2413 3943 l 2461 3928 l 2461 3928 l 2419 3972 l P NP 2419 3972 m 2461 3928 l 2481 3961 l 2419 3972 l P NP 2319 3973 m 2393 3936 l 2366 3958 l 2319 3973 l P NP 2319 3973 m 2312 3973 l 2338 3963 l 2338 3963 l 2319 3973 l P NP 3870 3974 m 3552 3902 l 3631 3895 l 3631 3895 l 3870 3974 l P NP 3870 3974 m 3631 3895 l 3968 3969 l 3870 3974 l P NP 2764 3974 m 2729 3919 l 2843 3967 l 2764 3974 l P NP 2764 3974 m 2667 3929 l 2729 3919 l 2729 3919 l 2764 3974 l P NP 3773 3978 m 3552 3902 l 3870 3974 l 3773 3978 l P NP 3773 3978 m 3473 3910 l 3552 3902 l 3552 3902 l 3773 3978 l P NP 2685 3981 m 2667 3929 l 2764 3974 l 2685 3981 l P NP 2685 3981 m 2605 3940 l 2667 3929 l 2667 3929 l 2685 3981 l P NP 3676 3982 m 3473 3910 l 3773 3978 l 3676 3982 l P NP 3676 3982 m 3394 3917 l 3473 3910 l 3473 3910 l 3676 3982 l P NP 2356 3983 m 2366 3958 l 2413 3943 l 2356 3983 l P NP 2356 3983 m 2413 3943 l 2419 3972 l 2356 3983 l P NP 3579 3986 m 3394 3917 l 3676 3982 l 3579 3986 l P NP 3579 3986 m 3315 3924 l 3394 3917 l 3394 3917 l 3579 3986 l P NP 2272 3987 m 2312 3973 l 2319 3973 l 2319 3973 l 2272 3987 l P 0.597176 G NP 2272 3987 m 2209 3987 l 2214 3951 l 2214 3951 l 2332 3965 l 2272 3987 l P 0.596157 G NP 2606 3988 m 2543 3951 l 2605 3940 l 2605 3940 l 2606 3988 l P NP 2606 3988 m 2605 3940 l 2685 3981 l 2606 3988 l P NP 3481 3990 m 3315 3924 l 3579 3986 l 3481 3990 l P NP 3481 3990 m 3237 3931 l 3315 3924 l 3315 3924 l 3481 3990 l P NP 2294 3994 m 2366 3958 l 2356 3983 l 2294 3994 l P NP 2294 3994 m 2319 3973 l 2366 3958 l 2294 3994 l P NP 3384 3994 m 3237 3931 l 3481 3990 l 3384 3994 l P NP 3384 3994 m 3158 3938 l 3237 3931 l 3237 3931 l 3384 3994 l P NP 2527 3995 m 2543 3951 l 2606 3988 l 2527 3995 l P NP 2527 3995 m 2481 3961 l 2543 3951 l 2543 3951 l 2527 3995 l P NP 3287 3998 m 3158 3938 l 3384 3994 l 3287 3998 l P NP 3287 3998 m 3079 3945 l 3158 3938 l 3158 3938 l 3287 3998 l P NP 2449 4002 m 2419 3972 l 2481 3961 l 2481 3961 l 2449 4002 l P NP 2449 4002 m 2481 3961 l 2527 3995 l 2449 4002 l P NP 3189 4003 m 3079 3945 l 3287 3998 l 3189 4003 l P NP 3189 4003 m 3000 3952 l 3079 3945 l 3079 3945 l 3189 4003 l P NP 2232 4004 m 2319 3973 l 2294 3994 l 2232 4004 l P NP 2232 4004 m 2272 3987 l 2319 3973 l 2319 3973 l 2232 4004 l P 0.597176 G NP 2206 4005 m 2209 3987 l 2272 3987 l 2272 3987 l 2206 4005 l P 0.596157 G NP 3092 4007 m 2921 3959 l 3000 3952 l 3000 3952 l 3092 4007 l P NP 3092 4007 m 3000 3952 l 3189 4003 l 3092 4007 l P NP 2370 4009 m 2356 3983 l 2419 3972 l 2419 3972 l 2370 4009 l P NP 2370 4009 m 2419 3972 l 2449 4002 l 2370 4009 l P NP 2995 4011 m 2843 3967 l 2921 3959 l 2921 3959 l 2995 4011 l P NP 2995 4011 m 2921 3959 l 3092 4007 l 2995 4011 l P NP 2170 4015 m 2272 3987 l 2232 4004 l 2170 4015 l P NP 2170 4015 m 2161 4015 l 2195 4008 l 2195 4008 l 2170 4015 l P NP 2898 4015 m 2843 3967 l 2995 4011 l 2898 4015 l P NP 2898 4015 m 2764 3974 l 2843 3967 l 2843 3967 l 2898 4015 l P NP 2291 4017 m 2356 3983 l 2370 4009 l 2291 4017 l P NP 2291 4017 m 2294 3994 l 2356 3983 l 2291 4017 l P NP 2800 4019 m 2685 3981 l 2764 3974 l 2764 3974 l 2800 4019 l P NP 2800 4019 m 2764 3974 l 2898 4015 l 2800 4019 l P NP 2703 4023 m 2606 3988 l 2685 3981 l 2685 3981 l 2703 4023 l P NP 2703 4023 m 2685 3981 l 2800 4019 l 2703 4023 l P NP 2212 4024 m 2294 3994 l 2291 4017 l 2212 4024 l P NP 2212 4024 m 2232 4004 l 2294 3994 l 2212 4024 l P NP 2108 4026 m 2161 4015 l 2170 4015 l 2170 4015 l 2108 4026 l P NP 2606 4027 m 2606 3988 l 2703 4023 l 2606 4027 l P NP 2606 4027 m 2527 3995 l 2606 3988 l 2606 3988 l 2606 4027 l P 0.644627 G NP 2093 4028 m 2214 3951 l 2206 4005 l 2195 4008 l 2108 4026 l 2108 4026 l 2093 4028 l P 0.596157 G NP 2133 4031 m 2170 4015 l 2232 4004 l 2133 4031 l P NP 2133 4031 m 2232 4004 l 2212 4024 l 2133 4031 l P NP 2509 4032 m 2527 3995 l 2606 4027 l 2509 4032 l P NP 2509 4032 m 2449 4002 l 2527 3995 l 2527 3995 l 2509 4032 l P NP 4250 4034 m 3870 3974 l 3968 3969 l 3968 3969 l 4250 4034 l P NP 2411 4036 m 2449 4002 l 2509 4032 l 2411 4036 l P NP 2411 4036 m 2370 4009 l 2449 4002 l 2449 4002 l 2411 4036 l P 0.644627 G NP 2037 4036 m 2058 3935 l 2214 3951 l 2093 4028 l 2037 4036 l P 0.596157 G NP 4132 4036 m 3870 3974 l 4250 4034 l 4132 4036 l P NP 4132 4036 m 3773 3978 l 3870 3974 l 3870 3974 l 4132 4036 l P NP 4015 4038 m 3773 3978 l 4132 4036 l 4015 4038 l P NP 4015 4038 m 3676 3982 l 3773 3978 l 3773 3978 l 4015 4038 l P NP 2054 4038 m 2170 4015 l 2133 4031 l 2054 4038 l P NP 2054 4038 m 2108 4026 l 2170 4015 l 2054 4038 l P NP 3761 4038 m 3481 3990 l 3579 3986 l 3579 3986 l 3768 4038 l 3761 4038 l P NP 3646 4038 m 3481 3990 l 3761 4038 l 3646 4038 l P NP 3636 4038 m 3384 3994 l 3481 3990 l 3481 3990 l 3646 4038 l 3636 4038 l P NP 3525 4039 m 3384 3994 l 3636 4038 l 3525 4039 l P NP 3511 4039 m 3287 3998 l 3384 3994 l 3384 3994 l 3525 4039 l 3511 4039 l P NP 3405 4039 m 3287 3998 l 3511 4039 l 3484 4039 l 3405 4039 l P NP 3387 4039 m 3189 4003 l 3287 3998 l 3287 3998 l 3405 4039 l 3387 4039 l P NP 3286 4039 m 3189 4003 l 3387 4039 l 3286 4039 l P NP 3263 4039 m 3092 4007 l 3189 4003 l 3189 4003 l 3286 4039 l 3263 4039 l P NP 3168 4040 m 3092 4007 l 3263 4039 l 3174 4040 l 3168 4040 l P NP 3897 4040 m 3676 3982 l 4015 4038 l 3897 4040 l P NP 3897 4040 m 3579 3986 l 3676 3982 l 3676 3982 l 3897 4040 l P NP 3140 4040 m 2995 4011 l 3092 4007 l 3092 4007 l 3168 4040 l 3140 4040 l P NP 3051 4040 m 2995 4011 l 3140 4040 l 3051 4040 l P NP 2314 4040 m 2370 4009 l 2411 4036 l 2314 4040 l P NP 2314 4040 m 2291 4017 l 2370 4009 l 2370 4009 l 2314 4040 l P NP 3018 4040 m 2898 4015 l 2995 4011 l 2995 4011 l 3051 4040 l 3018 4040 l P NP 2936 4040 m 2898 4015 l 3018 4040 l 2936 4040 l P NP 2897 4040 m 2800 4019 l 2898 4015 l 2898 4015 l 2936 4040 l 2897 4040 l P NP 2823 4040 m 2800 4019 l 2897 4040 l 2864 4040 l 2823 4040 l P NP 2777 4041 m 2703 4023 l 2800 4019 l 2800 4019 l 2823 4040 l 2777 4041 l P NP 2713 4041 m 2703 4023 l 2777 4041 l 2713 4041 l P NP 2659 4041 m 2606 4027 l 2703 4023 l 2703 4023 l 2713 4041 l 2659 4041 l P NP 2605 4041 m 2606 4027 l 2659 4041 l 2605 4041 l P NP 2501 4041 m 2509 4032 l 2542 4041 l 2501 4041 l P NP 3781 4041 m 3771 4038 l 3768 4038 l 3579 3986 l 3579 3986 l 3897 4040 l 3781 4041 l P NP 2429 4041 m 2411 4036 l 2509 4032 l 2509 4032 l 2501 4041 l 2429 4041 l P NP 2402 4041 m 2411 4036 l 2429 4041 l 2402 4041 l P NP 2318 4042 m 2314 4040 l 2411 4036 l 2411 4036 l 2402 4041 l 2318 4042 l P NP 2309 4042 m 2314 4040 l 2318 4042 l 2309 4042 l P NP 2269 4042 m 2314 4040 l 2309 4042 l 2269 4042 l P NP 2223 4042 m 2223 4042 l 2245 4042 l 2223 4042 l P NP 2216 4042 m 2212 4024 l 2291 4017 l 2291 4017 l 2223 4042 l 2216 4042 l P NP 2142 4042 m 2212 4024 l 2216 4042 l 2142 4042 l P NP 2124 4042 m 2133 4031 l 2212 4024 l 2142 4042 l 2124 4042 l P NP 2074 4042 m 2133 4031 l 2124 4042 l 2074 4042 l P NP 2044 4042 m 2054 4038 l 2133 4031 l 2074 4042 l 2044 4042 l P NP 2023 4042 m 2054 4038 l 2044 4042 l 2023 4042 l P NP 2004 4042 m 2054 4038 l 2023 4042 l 2004 4042 l P NP 1993 4042 m 2108 4026 l 2054 4038 l 2004 4042 l 1993 4042 l P NP 1987 4043 m 2008 4040 l 1993 4042 l 1987 4043 l P NP 2245 4042 m 2223 4042 l 2291 4017 l 2291 4017 l 2314 4040 l 2269 4042 l 2245 4042 l P NP 2555 4041 m 2542 4041 l 2509 4032 l 2606 4027 l 2606 4027 l 2605 4041 l 2555 4041 l P 0.686275 G NP 1866 4046 m 1872 4043 l 1935 4043 l 1866 4046 l P NP 1935 4043 m 1872 4043 l 2058 3935 l 2058 3935 l 2037 4036 l 2008 4040 l 1987 4043 l 1935 4043 l P NP 1842 4047 m 1878 3917 l 2058 3935 l 1866 4046 l 1842 4047 l P 0.742863 G NP 1618 4057 m 1560 4056 l 1532 3970 l 1672 3899 l 1672 3899 l 1618 4057 l P 0.718627 G NP 1618 4057 m 1672 3899 l 1878 3917 l 1618 4057 l P NP 1628 4057 m 1618 4057 l 1622 4055 l 1622 4055 l 1680 4055 l 1628 4057 l P NP 1618 4058 m 1618 4057 l 1628 4057 l 1618 4058 l P 0.742863 G NP 1617 4058 m 1618 4057 l 1618 4058 l 1617 4058 l P 0.718627 G NP 1680 4055 m 1622 4055 l 1878 3917 l 1878 3917 l 1842 4047 l 1680 4055 l P 0.742863 G NP 1562 4061 m 1560 4056 l 1618 4057 l 1617 4058 l 1562 4061 l P 0.740588 G NP 1735 4066 m 1680 4055 l 1935 4043 l 1935 4043 l 1735 4066 l P NP 1735 4066 m 1463 4066 l 1467 4065 l 1680 4055 l 1680 4055 l 1735 4066 l P NP 2012 4067 m 1935 4043 l 2245 4042 l 2245 4042 l 2012 4067 l P NP 2012 4067 m 1735 4066 l 1935 4043 l 1935 4043 l 2012 4067 l P NP 2289 4068 m 2245 4042 l 2555 4041 l 2555 4041 l 2289 4068 l P NP 2289 4068 m 2012 4067 l 2245 4042 l 2245 4042 l 2289 4068 l P NP 2566 4069 m 2555 4041 l 2864 4040 l 2864 4040 l 2566 4069 l P NP 2566 4069 m 2289 4068 l 2555 4041 l 2555 4041 l 2566 4069 l P NP 2843 4070 m 2864 4040 l 3174 4040 l 2843 4070 l P NP 2843 4070 m 2566 4069 l 2864 4040 l 2864 4040 l 2843 4070 l P NP 3120 4070 m 3174 4040 l 3484 4039 l 3120 4070 l P NP 3120 4070 m 2843 4070 l 3174 4040 l 3174 4040 l 3120 4070 l P NP 3396 4071 m 3120 4070 l 3484 4039 l 3484 4039 l 3396 4071 l P NP 3396 4071 m 3484 4039 l 3771 4038 l 3775 4040 l 3396 4071 l P NP 3673 4072 m 3396 4071 l 3775 4040 l 3775 4040 l 3781 4042 l 3673 4072 l P NP 3673 4072 m 3781 4042 l 3836 4059 l 3673 4072 l P NP 3879 4073 m 3673 4072 l 3836 4059 l 3836 4059 l 3879 4073 l P 0.596157 G NP 4464 4073 m 4132 4036 l 4250 4034 l 4250 4034 l 4472 4070 l 4464 4073 l P NP 1311 4078 m 1325 4077 l 1324 4077 l 1311 4078 l P 0.740588 G NP 1347 4079 m 1326 4077 l 1446 4067 l 1446 4067 l 1347 4079 l P NP 1347 4079 m 1307 4079 l 1326 4077 l 1326 4077 l 1347 4079 l P 0.596157 G NP 1307 4079 m 1275 4079 l 1317 4078 l 1317 4078 l 1307 4079 l P NP 1066 4080 m 1035 4080 l 1033 4080 l 1118 4079 l 1118 4079 l 1066 4080 l P NP 1035 4080 m 1035 4080 l 1066 4080 l 1035 4080 l P NP 1205 4080 m 1311 4078 l 1324 4077 l 1317 4078 l 1205 4080 l P NP 1205 4080 m 1066 4080 l 1311 4078 l 1311 4078 l 1205 4080 l P NP 1279 4080 m 1205 4080 l 1275 4079 l 1307 4079 l 1279 4080 l P NP 1039 4080 m 1035 4080 l 1066 4080 l 1066 4080 l 1039 4080 l P NP 1042 4081 m 1037 4081 l 1034 4080 l 1035 4080 l 1035 4080 l 1039 4080 l 1042 4081 l P 0.740588 G NP 1592 4081 m 1461 4066 l 1463 4066 l 1463 4066 l 1735 4066 l 1592 4081 l P NP 1592 4081 m 1347 4079 l 1446 4067 l 1461 4066 l 1461 4066 l 1592 4081 l P 0.596157 G NP 1043 4081 m 1038 4081 l 1037 4081 l 1037 4081 l 1042 4081 l 1043 4081 l P NP 1045 4081 m 1039 4080 l 1066 4080 l 1205 4080 l 1045 4081 l P NP 1045 4081 m 1039 4081 l 1038 4081 l 1038 4081 l 1043 4081 l 1045 4081 l P NP 1041 4082 m 1039 4081 l 1045 4081 l 1045 4081 l 1048 4081 l 1041 4082 l P NP 3997 4083 m 3920 4082 l 3909 4080 l 3909 4080 l 3893 4076 l 3879 4073 l 3859 4066 l 3997 4083 l P NP 3997 4083 m 3859 4066 l 3836 4059 l 3836 4059 l 3782 4042 l 3997 4083 l P NP 3997 4083 m 3920 4083 l 3920 4082 l 3920 4082 l 3997 4083 l P NP 4137 4083 m 3997 4083 l 3782 4042 l 3781 4042 l 3781 4042 l 4137 4083 l P NP 4137 4083 m 3781 4042 l 3781 4042 l 3781 4041 l 3897 4040 l 4137 4083 l P NP 4137 4083 m 3999 4083 l 3997 4083 l 3997 4083 l 4137 4083 l P 0.740588 G NP 1838 4083 m 1735 4066 l 2012 4067 l 1838 4083 l P NP 1838 4083 m 1592 4081 l 1735 4066 l 1735 4066 l 1838 4083 l P 0.596157 G NP 4276 4083 m 4137 4083 l 3897 4040 l 4276 4083 l P NP 4276 4083 m 3897 4040 l 4015 4038 l 4015 4038 l 4276 4083 l P NP 4276 4083 m 4138 4083 l 4137 4083 l 4137 4083 l 4276 4083 l P NP 4416 4083 m 4276 4083 l 4015 4038 l 4015 4038 l 4416 4083 l P NP 4416 4083 m 4015 4038 l 4132 4036 l 4132 4036 l 4416 4083 l P NP 4440 4083 m 4416 4083 l 4132 4036 l 4132 4036 l 4464 4073 l 4440 4083 l P NP 1126 4083 m 1065 4083 l 1056 4082 l 1056 4082 l 1126 4083 l P NP 1065 4083 m 1044 4083 l 1043 4082 l 1043 4082 l 1056 4082 l 1065 4083 l P NP 1126 4083 m 1056 4082 l 1045 4081 l 1205 4080 l 1205 4080 l 1126 4083 l P NP 1056 4082 m 1043 4082 l 1041 4082 l 1048 4081 l 1048 4081 l 1056 4082 l P NP 1126 4083 m 1205 4080 l 1279 4080 l 1254 4081 l 1126 4083 l P NP 1207 4084 m 1126 4083 l 1140 4083 l 1140 4083 l 1225 4083 l 1207 4084 l P NP 1087 4084 m 1046 4084 l 1044 4083 l 1044 4083 l 1065 4083 l 1087 4084 l P NP 1225 4083 m 1140 4083 l 1254 4081 l 1254 4081 l 1225 4083 l P NP 4434 4084 m 4416 4083 l 4440 4083 l 4434 4084 l P NP 1091 4084 m 1065 4083 l 1126 4083 l 1091 4084 l P NP 1091 4084 m 1048 4084 l 1046 4084 l 1046 4084 l 1087 4084 l 1091 4084 l P 0.740588 G NP 2084 4084 m 2012 4067 l 2289 4068 l 2084 4084 l P NP 2084 4084 m 1838 4083 l 2012 4067 l 2012 4067 l 2084 4084 l P 0.596157 G NP 4432 4085 m 4416 4083 l 4434 4084 l 4432 4085 l P NP 1100 4085 m 1091 4084 l 1126 4083 l 1126 4083 l 1207 4084 l 1190 4085 l 1100 4085 l P NP 1100 4085 m 1049 4085 l 1048 4084 l 1091 4084 l 1091 4084 l 1100 4085 l P NP 1189 4085 m 1163 4085 l 1190 4085 l 1190 4085 l 1189 4085 l P 0.740588 G NP 1285 4086 m 1189 4085 l 1225 4083 l 1225 4083 l 1285 4086 l P NP 1285 4086 m 1225 4083 l 1307 4079 l 1307 4079 l 1347 4079 l 1285 4086 l P NP 2330 4086 m 2289 4068 l 2566 4069 l 2330 4086 l P NP 2330 4086 m 2084 4084 l 2289 4068 l 2289 4068 l 2330 4086 l P NP 1267 4087 m 1151 4085 l 1189 4085 l 1189 4085 l 1285 4086 l 1267 4087 l P NP 1267 4087 m 1106 4085 l 1151 4085 l 1151 4085 l 1267 4087 l P 0.596157 G NP 1106 4085 m 1102 4085 l 1151 4085 l 1106 4085 l P NP 1151 4085 m 1102 4085 l 1100 4085 l 1163 4085 l 1163 4085 l 1189 4085 l 1151 4085 l P NP 1267 4087 m 1214 4087 l 1225 4086 l 1225 4086 l 1267 4087 l P 0.740588 G NP 1501 4088 m 1285 4086 l 1347 4079 l 1347 4079 l 1501 4088 l P NP 1501 4088 m 1347 4079 l 1592 4081 l 1501 4088 l P NP 2576 4088 m 2330 4086 l 2566 4069 l 2566 4069 l 2576 4088 l P NP 2576 4088 m 2566 4069 l 2843 4070 l 2576 4088 l P NP 1457 4089 m 1267 4087 l 1285 4086 l 1285 4086 l 1457 4089 l P NP 1457 4089 m 1285 4086 l 1501 4088 l 1457 4089 l P 0.596157 G NP 1457 4089 m 1308 4089 l 1269 4087 l 1269 4087 l 1457 4089 l P NP 1147 4090 m 1058 4088 l 1053 4086 l 1053 4086 l 1147 4090 l P NP 1147 4090 m 1053 4086 l 1049 4085 l 1100 4085 l 1100 4085 l 1106 4085 l 1225 4086 l 1147 4090 l P 0.740588 G NP 2821 4090 m 2576 4088 l 2843 4070 l 2843 4070 l 2821 4090 l P NP 2821 4090 m 2843 4070 l 3120 4070 l 2821 4090 l P NP 1718 4090 m 1592 4081 l 1838 4083 l 1718 4090 l P NP 1718 4090 m 1501 4088 l 1592 4081 l 1592 4081 l 1718 4090 l P NP 1646 4091 m 1457 4089 l 1501 4088 l 1501 4088 l 1646 4091 l P NP 1646 4091 m 1501 4088 l 1718 4090 l 1646 4091 l P 0.596157 G NP 1646 4091 m 1546 4091 l 1528 4090 l 1528 4090 l 1646 4091 l P NP 4402 4091 m 4276 4083 l 4416 4083 l 4432 4085 l 4402 4091 l P 0.740588 G NP 3067 4092 m 3120 4070 l 3396 4071 l 3067 4092 l P NP 3067 4092 m 2821 4090 l 3120 4070 l 3120 4070 l 3067 4092 l P NP 1934 4092 m 1838 4083 l 2084 4084 l 1934 4092 l P NP 1934 4092 m 1718 4090 l 1838 4083 l 1838 4083 l 1934 4092 l P 0.596157 G NP 1071 4093 m 1058 4088 l 1147 4090 l 1071 4093 l P 0.740588 G NP 1835 4093 m 1718 4090 l 1934 4092 l 1835 4093 l P NP 1835 4093 m 1646 4091 l 1718 4090 l 1718 4090 l 1835 4093 l P 0.596157 G NP 1835 4093 m 1768 4093 l 1760 4092 l 1760 4092 l 1835 4093 l P 0.740588 G NP 3313 4094 m 3067 4092 l 3396 4071 l 3396 4071 l 3313 4094 l P NP 3313 4094 m 3396 4071 l 3673 4072 l 3313 4094 l P NP 2151 4095 m 1934 4092 l 2084 4084 l 2084 4084 l 2151 4095 l P NP 2151 4095 m 2084 4084 l 2330 4086 l 2151 4095 l P 0.596157 G NP 4388 4095 m 4276 4083 l 4402 4091 l 4388 4095 l P NP 1424 4095 m 1308 4089 l 1457 4089 l 1457 4089 l 1487 4089 l 1424 4095 l P NP 1424 4095 m 1147 4090 l 1214 4087 l 1267 4087 l 1267 4087 l 1269 4087 l 1424 4095 l P 0.740588 G NP 2024 4095 m 1835 4093 l 1934 4092 l 1934 4092 l 2024 4095 l P NP 2024 4095 m 1934 4092 l 2151 4095 l 2024 4095 l P 0.596157 G NP 2024 4095 m 1969 4095 l 1964 4095 l 1964 4095 l 2024 4095 l P 0.740588 G NP 3559 4095 m 3673 4072 l 3879 4073 l 3893 4076 l 3559 4095 l P NP 3559 4095 m 3313 4094 l 3673 4072 l 3673 4072 l 3559 4095 l P NP 2367 4097 m 2330 4086 l 2576 4088 l 2367 4097 l P NP 2367 4097 m 2151 4095 l 2330 4086 l 2330 4086 l 2367 4097 l P NP 3805 4097 m 3909 4080 l 3948 4089 l 3805 4097 l P NP 3805 4097 m 3559 4095 l 3893 4076 l 3893 4076 l 3909 4080 l 3805 4097 l P NP 2213 4097 m 2151 4095 l 2367 4097 l 2213 4097 l P NP 2213 4097 m 2024 4095 l 2151 4095 l 2151 4095 l 2213 4097 l P 0.640118 G NP 2213 4097 m 2145 4097 l 2141 4096 l 2141 4096 l 2213 4097 l P 0.740588 G NP 3987 4099 m 3805 4097 l 3948 4089 l 3948 4089 l 3987 4099 l P NP 2584 4099 m 2367 4097 l 2576 4088 l 2576 4088 l 2584 4099 l P NP 2584 4099 m 2576 4088 l 2821 4090 l 2584 4099 l P 0.596157 G NP 4369 4099 m 4138 4083 l 4276 4083 l 4388 4095 l 4369 4099 l P 0.740588 G NP 2402 4099 m 2213 4097 l 2367 4097 l 2367 4097 l 2402 4099 l P NP 2402 4099 m 2367 4097 l 2584 4099 l 2402 4099 l P 0.778941 G NP 2402 4099 m 2395 4099 l 2395 4099 l 2395 4099 l 2402 4099 l P 0.596157 G NP 1673 4100 m 1546 4091 l 1646 4091 l 1646 4091 l 1722 4092 l 1673 4100 l P NP 1673 4100 m 1424 4095 l 1487 4089 l 1487 4089 l 1528 4090 l 1673 4100 l P 0.740588 G NP 2801 4101 m 2821 4090 l 3067 4092 l 2801 4101 l P NP 2801 4101 m 2584 4099 l 2821 4090 l 2821 4090 l 2801 4101 l P NP 2591 4101 m 2584 4099 l 2801 4101 l 2591 4101 l P NP 2591 4101 m 2402 4099 l 2584 4099 l 2584 4099 l 2591 4101 l P 0.782588 G NP 2591 4101 m 2579 4101 l 2579 4101 l 2579 4101 l 2591 4101 l P 0.596157 G NP 1345 4102 m 1147 4090 l 1424 4095 l 1345 4102 l P NP 1345 4102 m 1074 4094 l 1071 4093 l 1147 4090 l 1147 4090 l 1345 4102 l P 0.740588 G NP 3017 4103 m 2801 4101 l 3067 4092 l 3067 4092 l 3017 4103 l P NP 3017 4103 m 3067 4092 l 3313 4094 l 3017 4103 l P NP 2780 4104 m 2591 4101 l 2801 4101 l 2801 4101 l 2780 4104 l P NP 2780 4104 m 2801 4101 l 3017 4103 l 2780 4104 l P 0.596157 G NP 4342 4105 m 4137 4083 l 4369 4099 l 4342 4105 l P 0.740588 G NP 2969 4106 m 2780 4104 l 3017 4103 l 3017 4103 l 2969 4106 l P NP 3234 4106 m 3313 4094 l 3559 4095 l 3234 4106 l P NP 3234 4106 m 2969 4106 l 3017 4103 l 3017 4103 l 3234 4106 l P NP 3234 4106 m 3017 4103 l 3313 4094 l 3313 4094 l 3234 4106 l P 0.596157 G NP 1893 4106 m 1673 4100 l 1722 4092 l 1722 4092 l 1760 4092 l 1893 4106 l P NP 1893 4106 m 1768 4093 l 1835 4093 l 1835 4093 l 1931 4094 l 1893 4106 l P NP 4335 4107 m 3999 4083 l 4137 4083 l 4342 4105 l 4335 4107 l P 0.740588 G NP 3158 4108 m 2969 4106 l 3234 4106 l 3158 4108 l P NP 3450 4108 m 3158 4108 l 3234 4106 l 3234 4106 l 3450 4108 l P NP 3450 4108 m 3234 4106 l 3559 4095 l 3559 4095 l 3450 4108 l P NP 3450 4108 m 3559 4095 l 3805 4097 l 3450 4108 l P 0.596157 G NP 4068 4108 m 4035 4106 l 4011 4102 l 4011 4102 l 4000 4100 l 3987 4099 l 3984 4098 l 4068 4108 l P NP 4068 4108 m 4050 4108 l 4049 4108 l 4035 4106 l 4068 4108 l P NP 1266 4109 m 1089 4103 l 1076 4096 l 1076 4096 l 1266 4109 l P NP 1266 4109 m 1076 4096 l 1074 4094 l 1074 4094 l 1345 4102 l 1266 4109 l P NP 4231 4110 m 3946 4089 l 3920 4083 l 3920 4083 l 3997 4083 l 4231 4110 l P NP 4231 4110 m 4068 4108 l 3984 4098 l 3948 4089 l 3948 4089 l 3946 4089 l 4231 4110 l P 0.740588 G NP 3348 4110 m 3158 4108 l 3450 4108 l 3348 4110 l P NP 3667 4110 m 3348 4110 l 3450 4108 l 3450 4108 l 3667 4110 l P NP 3667 4110 m 3805 4097 l 3987 4099 l 4000 4100 l 3667 4110 l P NP 3667 4110 m 3450 4108 l 3805 4097 l 3805 4097 l 3667 4110 l P 0.596157 G NP 4319 4110 m 4231 4110 l 3997 4083 l 3997 4083 l 4335 4107 l 4319 4110 l P NP 1611 4111 m 1424 4095 l 1673 4100 l 1611 4111 l P NP 1611 4111 m 1345 4102 l 1424 4095 l 1424 4095 l 1611 4111 l P NP 2085 4112 m 1893 4106 l 1931 4094 l 1931 4094 l 1964 4095 l 2085 4112 l P NP 2085 4112 m 1969 4095 l 2024 4095 l 2024 4095 l 2113 4096 l 2085 4112 l P NP 4299 4112 m 4231 4110 l 4319 4110 l 4299 4112 l P 0.740588 G NP 3537 4112 m 3348 4110 l 3667 4110 l 3537 4112 l P NP 3883 4112 m 3537 4112 l 3667 4110 l 3667 4110 l 3883 4112 l P NP 3883 4112 m 4011 4102 l 4049 4108 l 3883 4112 l P NP 3883 4112 m 3667 4110 l 4000 4100 l 4000 4100 l 4011 4102 l 3883 4112 l P 0.596157 G NP 4290 4113 m 4231 4110 l 4299 4112 l 4290 4113 l P 0.740588 G NP 3726 4114 m 3537 4112 l 3883 4112 l 3726 4114 l P NP 4095 4115 m 3883 4112 l 4049 4108 l 4049 4108 l 4095 4115 l P NP 4096 4115 m 3726 4114 l 3883 4112 l 3883 4112 l 4095 4115 l 4096 4115 l P 0.596157 G NP 4262 4115 m 4068 4108 l 4231 4110 l 4290 4113 l 4262 4115 l P 0.740588 G NP 3915 4116 m 3726 4114 l 4096 4115 l 4097 4115 l 3915 4116 l P 0.596157 G NP 1188 4116 m 1107 4113 l 1101 4110 l 1101 4110 l 1188 4116 l P NP 1188 4116 m 1101 4110 l 1089 4103 l 1089 4103 l 1266 4109 l 1188 4116 l P 0.740588 G NP 4140 4117 m 3915 4116 l 4097 4115 l 4097 4115 l 4140 4117 l P 0.596157 G NP 4233 4117 m 4068 4108 l 4262 4115 l 4233 4117 l P 0.642431 G NP 2249 4118 m 2085 4112 l 2113 4096 l 2113 4096 l 2141 4096 l 2249 4118 l P 0.640118 G NP 2249 4118 m 2145 4097 l 2213 4097 l 2213 4097 l 2267 4098 l 2249 4118 l P 0.596157 G NP 4223 4118 m 4085 4113 l 4050 4108 l 4050 4108 l 4068 4108 l 4233 4117 l 4223 4118 l P 0.740588 G NP 4104 4118 m 3915 4116 l 4140 4117 l 4169 4118 l 4104 4118 l P NP 4183 4119 m 4104 4118 l 4169 4118 l 4169 4118 l 4183 4119 l P NP 4205 4119 m 4104 4118 l 4183 4119 l 4205 4119 l P 0.596157 G NP 4205 4119 m 4183 4119 l 4169 4118 l 4140 4117 l 4095 4115 l 4085 4113 l 4223 4118 l 4205 4119 l P NP 1846 4121 m 1611 4111 l 1673 4100 l 1673 4100 l 1846 4121 l P NP 1846 4121 m 1673 4100 l 1893 4106 l 1846 4121 l P NP 1548 4122 m 1266 4109 l 1345 4102 l 1345 4102 l 1548 4122 l P NP 1548 4122 m 1345 4102 l 1611 4111 l 1548 4122 l P 0.716471 G NP 2384 4124 m 2249 4118 l 2267 4098 l 2267 4098 l 2291 4098 l 2384 4124 l P NP 2384 4124 m 2291 4098 l 2395 4099 l 2384 4124 l P 0.778941 G NP 2491 4130 m 2384 4124 l 2395 4099 l 2402 4099 l 2402 4099 l 2414 4100 l 2491 4130 l P NP 2491 4130 m 2414 4100 l 2496 4100 l 2491 4130 l P 0.596157 G NP 2051 4131 m 1893 4106 l 2085 4112 l 2051 4131 l P NP 2051 4131 m 1846 4121 l 1893 4106 l 1893 4106 l 2051 4131 l P NP 1486 4132 m 1188 4116 l 1266 4109 l 1266 4109 l 1486 4132 l P NP 1486 4132 m 1266 4109 l 1548 4122 l 1486 4132 l P NP 1799 4136 m 1548 4122 l 1611 4111 l 1611 4111 l 1799 4136 l P NP 1799 4136 m 1611 4111 l 1846 4121 l 1799 4136 l P 0.804078 G NP 2570 4136 m 2510 4101 l 2570 4101 l 2570 4136 l P NP 2570 4136 m 2491 4130 l 2496 4100 l 2496 4100 l 2510 4101 l 2570 4136 l P 0.742863 G NP 1589 4142 m 1588 4139 l 1590 4139 l 1589 4142 l P 0.640118 G NP 2226 4142 m 2085 4112 l 2249 4118 l 2226 4142 l P NP 2226 4142 m 2051 4131 l 2085 4112 l 2085 4112 l 2226 4142 l P 0.782588 G NP 2620 4143 m 2579 4101 l 2591 4101 l 2591 4101 l 2617 4102 l 2620 4143 l P NP 2620 4143 m 2570 4136 l 2570 4101 l 2570 4101 l 2579 4101 l 2620 4143 l P 0.735647 G NP 2642 4149 m 2620 4143 l 2617 4102 l 2621 4102 l 2642 4149 l P NP 2642 4149 m 2621 4102 l 2637 4102 l 2642 4149 l P 0.596157 G NP 1751 4150 m 1486 4132 l 1548 4122 l 1548 4122 l 1751 4150 l P 0.718627 G NP 1751 4150 m 1680 4150 l 1688 4146 l 1688 4146 l 1751 4150 l P 0.596157 G NP 1751 4150 m 1548 4122 l 1799 4136 l 1751 4150 l P NP 2017 4151 m 1846 4121 l 2051 4131 l 2017 4151 l P NP 2017 4151 m 1799 4136 l 1846 4121 l 1846 4121 l 2017 4151 l P 0.716471 G NP 2371 4154 m 2226 4142 l 2249 4118 l 2249 4118 l 2371 4154 l P NP 2371 4154 m 2249 4118 l 2384 4124 l 2371 4154 l P 0.778941 G NP 2486 4167 m 2371 4154 l 2384 4124 l 2384 4124 l 2486 4167 l P NP 2486 4167 m 2384 4124 l 2491 4130 l 2486 4167 l P 0.640118 G NP 2203 4167 m 2017 4151 l 2051 4131 l 2051 4131 l 2203 4167 l P NP 2203 4167 m 2051 4131 l 2226 4142 l 2203 4167 l P 0.596157 G NP 1983 4170 m 1751 4150 l 1799 4136 l 1799 4136 l 1983 4170 l P 0.686275 G NP 1983 4170 m 1859 4170 l 1874 4161 l 1874 4161 l 1983 4170 l P 0.596157 G NP 1983 4170 m 1799 4136 l 2017 4151 l 1983 4170 l P 0.804078 G NP 2570 4180 m 2491 4130 l 2570 4136 l 2570 4180 l P NP 2570 4180 m 2486 4167 l 2491 4130 l 2491 4130 l 2570 4180 l P 0.716471 G NP 2357 4185 m 2203 4167 l 2226 4142 l 2226 4142 l 2357 4185 l P NP 2357 4185 m 2226 4142 l 2371 4154 l 2357 4185 l P 0.640118 G NP 2180 4192 m 1983 4170 l 2017 4151 l 2017 4151 l 2180 4192 l P NP 2180 4192 m 2017 4151 l 2203 4167 l 2180 4192 l P 0.718627 G NP 1605 4193 m 1589 4142 l 1590 4139 l 1688 4146 l 1605 4193 l P 0.782588 G NP 2624 4194 m 2570 4180 l 2570 4136 l 2570 4136 l 2624 4194 l P NP 2624 4194 m 2570 4136 l 2620 4143 l 2624 4194 l P 0.644627 G NP 2002 4200 m 2008 4173 l 2038 4176 l 2002 4200 l P NP 2002 4200 m 2038 4176 l 2179 4192 l 2178 4192 l 2002 4200 l P 0.778941 G NP 2480 4204 m 2357 4185 l 2371 4154 l 2371 4154 l 2480 4204 l P NP 2480 4204 m 2371 4154 l 2486 4167 l 2480 4204 l P 0.686275 G NP 1797 4208 m 1812 4156 l 1874 4161 l 1797 4208 l P NP 1797 4208 m 1859 4170 l 1983 4170 l 1983 4170 l 2008 4173 l 2002 4200 l 1797 4208 l P 0.735647 G NP 2649 4209 m 2624 4194 l 2620 4143 l 2649 4209 l P NP 2649 4209 m 2620 4143 l 2642 4149 l 2649 4209 l P 0.718627 G NP 1612 4214 m 1605 4193 l 1680 4150 l 1751 4150 l 1751 4150 l 1812 4156 l 1797 4208 l 1612 4214 l P 0.716471 G NP 2344 4215 m 2180 4192 l 2203 4167 l 2203 4167 l 2344 4215 l P 0.535765 G NP 2344 4215 m 2324 4215 l 2324 4213 l 2324 4213 l 2344 4215 l P 0.714157 G NP 2344 4215 m 2203 4167 l 2357 4185 l 2344 4215 l P 0.804078 G NP 2570 4224 m 2486 4167 l 2570 4180 l 2570 4224 l P NP 2570 4224 m 2480 4204 l 2486 4167 l 2486 4167 l 2570 4224 l P 0.778941 G NP 2474 4241 m 2344 4215 l 2357 4185 l 2357 4185 l 2474 4241 l P 0.503216 G NP 2474 4241 m 2445 4241 l 2445 4235 l 2445 4235 l 2474 4241 l P 0.778941 G NP 2474 4241 m 2357 4185 l 2480 4204 l 2474 4241 l P 0.782588 G NP 2628 4246 m 2570 4224 l 2570 4180 l 2570 4180 l 2628 4246 l P NP 2628 4246 m 2570 4180 l 2624 4194 l 2628 4246 l P NP 2628 4246 m 2588 4246 l 2570 4224 l 2570 4224 l 2628 4246 l P 0.804078 G NP 2570 4268 m 2474 4241 l 2480 4204 l 2480 4204 l 2570 4268 l P 0.486745 G NP 2570 4268 m 2541 4268 l 2540 4260 l 2540 4260 l 2570 4268 l P 0.805255 G NP 2570 4268 m 2480 4204 l 2570 4224 l 2570 4268 l P 0.735647 G NP 2655 4269 m 2628 4246 l 2624 4194 l 2655 4269 l P NP 2655 4269 m 2624 4194 l 2649 4209 l 2655 4269 l P 0.503216 G NP 2444 4275 m 2474 4241 l 2535 4258 l 2444 4275 l P NP 2444 4275 m 2445 4241 l 2474 4241 l 2474 4241 l 2474 4241 l 2444 4275 l P 0.535765 G NP 2317 4293 m 2324 4215 l 2344 4215 l 2344 4215 l 2393 4225 l 2317 4293 l P NP 2317 4293 m 2393 4225 l 2445 4235 l 2444 4275 l 2317 4293 l P 0.782588 G NP 2633 4297 m 2570 4268 l 2570 4224 l 2570 4224 l 2633 4297 l P 0.463216 G NP 2633 4297 m 2619 4297 l 2621 4292 l 2621 4292 l 2633 4297 l P 0.782588 G NP 2633 4297 m 2588 4246 l 2628 4246 l 2633 4297 l P 0.718627 G NP 1639 4299 m 1612 4214 l 1797 4208 l 1797 4208 l 1639 4299 l P 0.597176 G NP 2161 4312 m 2178 4192 l 2180 4192 l 2296 4209 l 2161 4312 l P 0.594863 G NP 2161 4312 m 2296 4209 l 2324 4213 l 2317 4293 l 2161 4312 l P 0.463216 G NP 2612 4318 m 2610 4287 l 2621 4292 l 2612 4318 l P NP 2612 4318 m 2619 4297 l 2633 4297 l 2633 4297 l 2636 4301 l 2612 4318 l P 0.735647 G NP 2661 4328 m 2628 4246 l 2655 4269 l 2661 4328 l P NP 2661 4328 m 2633 4297 l 2628 4246 l 2628 4246 l 2661 4328 l P 0.446745 G NP 2661 4328 m 2656 4328 l 2656 4322 l 2656 4322 l 2661 4328 l P 0.644627 G NP 1974 4332 m 2178 4192 l 2161 4312 l 1974 4332 l P NP 1974 4332 m 2002 4200 l 2178 4192 l 1974 4332 l P 0.486745 G NP 2543 4345 m 2541 4268 l 2570 4268 l 2570 4268 l 2585 4275 l 2543 4345 l P NP 2543 4345 m 2585 4275 l 2610 4287 l 2612 4318 l 2543 4345 l P 0.686275 G NP 1757 4353 m 1797 4208 l 2002 4200 l 1757 4353 l P NP 1757 4353 m 2002 4200 l 1974 4332 l 1757 4353 l P 0.446745 G NP 2656 4361 m 2656 4328 l 2661 4328 l 2661 4328 l 2656 4361 l P NP 2656 4361 m 2652 4318 l 2656 4322 l 2656 4361 l P 0.718627 G NP 1658 4361 m 1639 4299 l 1797 4208 l 1797 4208 l 1757 4353 l 1658 4361 l P 0.503216 G NP 2441 4373 m 2444 4275 l 2535 4258 l 2539 4259 l 2441 4373 l P NP 2441 4373 m 2539 4259 l 2540 4260 l 2543 4345 l 2441 4373 l P 0.463216 G NP 2617 4396 m 2644 4310 l 2652 4318 l 2656 4361 l 2617 4396 l P NP 2617 4396 m 2612 4318 l 2636 4301 l 2636 4301 l 2644 4310 l 2617 4396 l P 0.535765 G NP 2308 4402 m 2444 4275 l 2441 4373 l 2308 4402 l P NP 2308 4402 m 2317 4293 l 2444 4275 l 2308 4402 l P 0.718627 G NP 1672 4403 m 1658 4361 l 1757 4353 l 1757 4353 l 1672 4403 l P 0.486745 G NP 2545 4432 m 2612 4318 l 2617 4396 l 2545 4432 l P NP 2545 4432 m 2543 4345 l 2612 4318 l 2612 4318 l 2545 4432 l P 0.594863 G NP 2143 4433 m 2317 4293 l 2308 4402 l 2143 4433 l P 0.597176 G NP 2143 4433 m 2161 4312 l 2317 4293 l 2143 4433 l P 0.644627 G NP 1945 4465 m 2161 4312 l 2143 4433 l 1945 4465 l P NP 1945 4465 m 1974 4332 l 2161 4312 l 1945 4465 l P 0.503216 G NP 2439 4471 m 2441 4373 l 2543 4345 l 2439 4471 l P NP 2439 4471 m 2543 4345 l 2545 4432 l 2439 4471 l P 0.686275 G NP 1716 4498 m 1757 4353 l 1974 4332 l 1716 4498 l P NP 1716 4498 m 1974 4332 l 1945 4465 l 1716 4498 l P 0.716314 G NP 1702 4500 m 1672 4403 l 1757 4353 l 1757 4353 l 1716 4498 l 1702 4500 l P 0.718627 G NP 1704 4505 m 1702 4500 l 1716 4498 l 1716 4498 l 1704 4505 l P 0.535765 G NP 2299 4511 m 2308 4402 l 2441 4373 l 2299 4511 l P NP 2299 4511 m 2441 4373 l 2439 4471 l 2299 4511 l P 0.716314 G NP 1709 4522 m 1704 4505 l 1716 4498 l 1716 4498 l 1709 4522 l P 0.594863 G NP 2125 4553 m 2308 4402 l 2299 4511 l 2125 4553 l P 0.597176 G NP 2125 4553 m 2143 4433 l 2308 4402 l 2125 4553 l P 0.644627 G NP 1917 4597 m 1945 4465 l 2143 4433 l 1917 4597 l P NP 1917 4597 m 2143 4433 l 2125 4553 l 1917 4597 l P 0.686275 G NP 1734 4604 m 1709 4522 l 1716 4498 l 1945 4465 l 1734 4604 l P NP 1742 4630 m 1734 4604 l 1945 4465 l 1945 4465 l 1917 4597 l 1742 4630 l P defaultFont 0.3 G 30 setlinewidth [] 0 setdash NP 2697 984 m 2668 1094 l S NP 2664 1111 m 2608 1359 l 2571 1568 l S NP 2564 1606 m 2532 1838 l 2512 2036 l S NP 2510 2055 m 2500 2231 l S NP 2499 2258 m 2495 2447 l 2497 2612 l S NP 2497 2623 m 2504 2769 l S NP 2505 2786 m 2516 2937 l 2528 3070 l S NP 2529 3075 m 2542 3202 l 2554 3319 l 2564 3423 l S NP 2564 3429 m 2569 3520 l 2569 3592 l S NP 2569 3613 m 2562 3683 l S NP 2562 3684 m 2546 3752 l 2525 3803 l S NP 2516 3819 m 2483 3865 l S NP 2481 3867 m 2432 3911 l 2377 3944 l S NP 2355 3955 m 2286 3983 l S NP 2279 3985 m 2187 4010 l 2083 4029 l S NP 2045 4035 m 1983 4043 l S NP 1327 4077 m 1175 4079 l S NP 1129 4079 m 1032 4080 l S 0 G boundarythick setlinewidth % The following draws a box around the plot, % if the variable drawborder is true drawborder { /bd boundarythick 2 idiv def [] 0 setdash NP bd bd m bd 6923 bd sub l 5000 bd sub 6923 bd sub l 5000 bd sub bd l bd bd l S } if % end of if to draw the border showpage grestore end %%Trailer %%BoundingBox: 168 161 490 505 %%EOF %%EndDocument @endspecial 1591 w @beginspecial 152 @llx 220 @lly 488 @urx 571 @ury 1872 @rwi 1800 @rhi @setspecial %%BeginDocument: pejo03.eps %!PS-Adobe-3.0 EPSF %%Title: Maple plot %%Creator: MapleV %%Pages: 1 %%BoundingBox: (atend) %%DocumentNeededResources: font Helvetica %%EndComments 20 dict begin gsave /drawborder false def /m {moveto} def /l {lineto} def /C {setrgbcolor} def /Y /setcmykcolor where { %%ifelse Use built-in operator /setcmykcolor get }{ %%ifelse Emulate setcmykcolor with setrgbcolor { %%def 1 sub 3 { %%repeat 3 index add neg dup 0 lt { pop 0 } if 3 1 roll } repeat setrgbcolor } bind } ifelse def /G {setgray} def /S {stroke} def /NP {newpath} def %%%%This draws a filled polygon and avoids bugs/features %%%% of some postscript interpreters %%%%GHOSTSCRIPT: has a bug in reversepath - removing %%%%the call to reversepath is a sufficient work around /P {gsave fill grestore reversepath stroke} def %%%%This function is needed for drawing text /stringbbox {gsave NP 0 0 m false charpath flattenpath pathbbox 4 2 roll pop pop 1.1 mul cvi exch 1.1 mul cvi exch grestore} def /thin 3 def /medium 7 def /thick 16 def /boundarythick 20 def % thichess of bounding box %%IncludeResource: font Helvetica 72.000000 72.000000 translate /defaultTitleFont {/Helvetica findfont 219 scalefont setfont} def /defaultFont {/Helvetica findfont 154 scalefont setfont} def 0.0936 0.0936 scale 1 setlinejoin 1 setlinecap 0.0 setgray /inch {72 mul} def /fheight 0.35 inch neg def 0 G medium setlinewidth [] 0 setdash [] 0 setdash 0.839098 G NP 3257 1823 m 3380 1722 l 3452 1812 l 3257 1823 l P NP 3257 1823 m 3288 1607 l 3380 1722 l 3257 1823 l P NP 2994 1838 m 3278 1594 l 3288 1607 l 3257 1823 l 2994 1838 l P NP 2994 1838 m 3001 1610 l 3270 1584 l 3278 1594 l 2994 1838 l P NP 2730 1853 m 3001 1610 l 2994 1838 l 2730 1853 l P NP 2730 1853 m 2710 1638 l 3001 1610 l 3001 1610 l 2730 1853 l P NP 2467 1868 m 2710 1638 l 2730 1853 l 2467 1868 l P NP 2467 1868 m 2420 1666 l 2710 1638 l 2710 1638 l 2467 1868 l P NP 2204 1883 m 2129 1695 l 2420 1666 l 2420 1666 l 2204 1883 l P NP 2204 1883 m 2420 1666 l 2467 1868 l 2204 1883 l P NP 1941 1898 m 2129 1695 l 2204 1883 l 1941 1898 l P NP 1941 1898 m 1838 1723 l 2129 1695 l 2129 1695 l 1941 1898 l P NP 1677 1913 m 1838 1723 l 1941 1898 l 1677 1913 l P NP 1677 1913 m 1548 1751 l 1838 1723 l 1838 1723 l 1677 1913 l P NP 1414 1927 m 1548 1751 l 1677 1913 l 1414 1927 l P NP 1414 1927 m 1335 1853 l 1548 1751 l 1548 1751 l 1414 1927 l P NP 1156 1942 m 1158 1939 l 1335 1853 l 1335 1853 l 1414 1927 l 1156 1942 l P 0.85298 G NP 3459 2047 m 3569 1961 l 3634 2045 l 3459 2047 l P NP 3459 2047 m 3503 1876 l 3569 1961 l 3459 2047 l P NP 3222 2050 m 3477 1843 l 3503 1876 l 3459 2047 l 3222 2050 l P NP 3222 2050 m 3257 1823 l 3452 1812 l 3477 1843 l 3222 2050 l P NP 2984 2053 m 2994 1838 l 3257 1823 l 2984 2053 l P NP 2984 2053 m 3257 1823 l 3222 2050 l 2984 2053 l P NP 2747 2056 m 2730 1853 l 2994 1838 l 2994 1838 l 2747 2056 l P NP 2747 2056 m 2994 1838 l 2984 2053 l 2747 2056 l P NP 2510 2059 m 2730 1853 l 2747 2056 l 2510 2059 l P NP 2510 2059 m 2467 1868 l 2730 1853 l 2730 1853 l 2510 2059 l P NP 2273 2062 m 2204 1883 l 2467 1868 l 2467 1868 l 2273 2062 l P NP 2273 2062 m 2467 1868 l 2510 2059 l 2273 2062 l P NP 2036 2065 m 1941 1898 l 2204 1883 l 2204 1883 l 2036 2065 l P NP 2036 2065 m 2204 1883 l 2273 2062 l 2036 2065 l P NP 1799 2068 m 1677 1913 l 1941 1898 l 1941 1898 l 1799 2068 l P NP 1799 2068 m 1941 1898 l 2036 2065 l 1799 2068 l P NP 1561 2071 m 1414 1927 l 1677 1913 l 1677 1913 l 1561 2071 l P NP 1561 2071 m 1677 1913 l 1799 2068 l 1561 2071 l P NP 1324 2074 m 1154 1945 l 1156 1942 l 1414 1927 l 1414 1927 l 1324 2074 l P NP 1324 2074 m 1414 1927 l 1561 2071 l 1324 2074 l P NP 1087 2077 m 1144 1957 l 1154 1945 l 1324 2074 l 1087 2077 l P NP 1087 2077 m 1062 2062 l 1144 1957 l 1144 1957 l 1087 2077 l P NP 1049 2077 m 1062 2062 l 1087 2077 l 1049 2077 l P 0.869176 G NP 1062 2186 m 993 2151 l 1049 2077 l 1087 2077 l 1087 2077 l 1062 2186 l P NP 1062 2186 m 969 2182 l 993 2151 l 993 2151 l 1062 2186 l P NP 1275 2193 m 1087 2077 l 1324 2074 l 1324 2074 l 1275 2193 l P NP 1275 2193 m 1062 2186 l 1087 2077 l 1087 2077 l 1275 2193 l P NP 1487 2201 m 1324 2074 l 1561 2071 l 1561 2071 l 1487 2201 l P NP 1487 2201 m 1275 2193 l 1324 2074 l 1324 2074 l 1487 2201 l P NP 1699 2209 m 1487 2201 l 1561 2071 l 1561 2071 l 1699 2209 l P NP 1699 2209 m 1561 2071 l 1799 2068 l 1799 2068 l 1699 2209 l P 0.87149 G NP 1912 2216 m 1799 2068 l 2036 2065 l 2036 2065 l 1912 2216 l P 0.869176 G NP 1912 2216 m 1699 2209 l 1799 2068 l 1799 2068 l 1912 2216 l P 0.87149 G NP 2124 2224 m 2036 2065 l 2273 2062 l 2273 2062 l 2124 2224 l P NP 2124 2224 m 1912 2216 l 2036 2065 l 2124 2224 l P NP 2336 2232 m 2273 2062 l 2510 2059 l 2510 2059 l 2336 2232 l P NP 2336 2232 m 2124 2224 l 2273 2062 l 2273 2062 l 2336 2232 l P NP 2548 2240 m 2510 2059 l 2747 2056 l 2747 2056 l 2548 2240 l P NP 2548 2240 m 2336 2232 l 2510 2059 l 2510 2059 l 2548 2240 l P NP 2761 2247 m 2548 2240 l 2747 2056 l 2747 2056 l 2761 2247 l P NP 2761 2247 m 2747 2056 l 2984 2053 l 2984 2053 l 2761 2247 l P 0.867647 G NP 996 2255 m 909 2247 l 913 2240 l 913 2240 l 996 2255 l P NP 996 2255 m 905 2255 l 909 2247 l 909 2247 l 996 2255 l P 0.87149 G NP 2973 2255 m 2984 2053 l 3222 2050 l 2973 2255 l P NP 2973 2255 m 2761 2247 l 2984 2053 l 2984 2053 l 2973 2255 l P NP 3185 2263 m 3222 2050 l 3459 2047 l 3185 2263 l P NP 3185 2263 m 2973 2255 l 3222 2050 l 3222 2050 l 3185 2263 l P NP 3397 2270 m 3185 2263 l 3459 2047 l 3397 2270 l P NP 3397 2270 m 3459 2047 l 3634 2045 l 3656 2074 l 3397 2270 l P 0.887686 G NP 1070 2272 m 996 2255 l 925 2242 l 943 2218 l 943 2218 l 1070 2272 l P NP 1073 2273 m 1070 2272 l 943 2218 l 969 2182 l 969 2182 l 1062 2186 l 1073 2273 l P 0.87149 G NP 3610 2278 m 3397 2270 l 3656 2074 l 3656 2074 l 3676 2099 l 3610 2278 l P NP 3610 2278 m 3676 2099 l 3739 2183 l 3610 2278 l P NP 3817 2285 m 3610 2278 l 3739 2183 l 3739 2183 l 3817 2285 l P 0.887686 G NP 1262 2290 m 1077 2274 l 1073 2273 l 1062 2186 l 1062 2186 l 1262 2290 l P NP 1262 2290 m 1062 2186 l 1275 2193 l 1262 2290 l P 0.867647 G NP 1085 2299 m 892 2280 l 901 2263 l 901 2263 l 1085 2299 l P NP 1085 2299 m 901 2263 l 905 2255 l 905 2255 l 996 2255 l 1085 2299 l P NP 1085 2299 m 968 2299 l 890 2282 l 892 2280 l 892 2280 l 1085 2299 l P 0.887686 G NP 1450 2308 m 1275 2193 l 1487 2201 l 1450 2308 l P NP 1450 2308 m 1262 2290 l 1275 2193 l 1275 2193 l 1450 2308 l P 0.886471 G NP 971 2319 m 876 2307 l 880 2301 l 880 2301 l 971 2319 l P 0.887647 G NP 971 2319 m 880 2301 l 889 2286 l 889 2286 l 971 2319 l P NP 971 2319 m 886 2319 l 872 2315 l 876 2307 l 876 2307 l 971 2319 l P 0.89 G NP 1269 2320 m 1077 2274 l 1262 2290 l 1269 2320 l P 0.887686 G NP 1639 2325 m 1487 2201 l 1699 2209 l 1639 2325 l P NP 1639 2325 m 1450 2308 l 1487 2201 l 1487 2201 l 1639 2325 l P 0.844118 G NP 1312 2330 m 1085 2299 l 996 2255 l 996 2255 l 1312 2330 l P NP 1312 2330 m 1203 2330 l 1085 2299 l 1085 2299 l 1312 2330 l P 0.89 G NP 906 2333 m 866 2325 l 868 2322 l 868 2322 l 906 2333 l P NP 906 2333 m 866 2326 l 866 2325 l 866 2325 l 906 2333 l P NP 906 2333 m 874 2333 l 863 2330 l 866 2326 l 866 2326 l 906 2333 l P 0.887686 G NP 1828 2342 m 1699 2209 l 1912 2216 l 1828 2342 l P NP 1828 2342 m 1639 2325 l 1699 2209 l 1699 2209 l 1828 2342 l P 0.867647 G NP 1173 2344 m 968 2299 l 1085 2299 l 1173 2344 l P NP 1173 2344 m 971 2319 l 889 2286 l 890 2282 l 890 2282 l 1173 2344 l P NP 1173 2344 m 1077 2344 l 971 2319 l 971 2319 l 1173 2344 l P 0.89 G NP 1358 2344 m 1312 2330 l 1269 2320 l 1262 2290 l 1262 2290 l 1358 2344 l P 0.887686 G NP 2016 2359 m 1912 2216 l 2124 2224 l 2016 2359 l P NP 2016 2359 m 1828 2342 l 1912 2216 l 1912 2216 l 2016 2359 l P 0.887647 G NP 1077 2360 m 886 2319 l 971 2319 l 1077 2360 l P 0.886471 G NP 1077 2360 m 906 2333 l 868 2322 l 872 2315 l 872 2315 l 1077 2360 l P NP 1077 2360 m 1015 2360 l 906 2333 l 906 2333 l 1077 2360 l P 0.89 G NP 1031 2371 m 874 2333 l 906 2333 l 1031 2371 l P NP 1031 2371 m 1021 2371 l 966 2355 l 966 2355 l 1031 2371 l P NP 1448 2372 m 1358 2344 l 1262 2290 l 1262 2290 l 1450 2308 l 1448 2372 l P 0.887686 G NP 2205 2376 m 2016 2359 l 2124 2224 l 2124 2224 l 2205 2376 l P NP 2205 2376 m 2124 2224 l 2336 2232 l 2205 2376 l P 0.844118 G NP 1384 2376 m 1203 2330 l 1312 2330 l 1384 2376 l P NP 1384 2376 m 1173 2344 l 1085 2299 l 1384 2376 l P NP 1384 2376 m 1289 2376 l 1173 2344 l 1173 2344 l 1384 2376 l P 0.867647 G NP 1262 2389 m 1077 2344 l 1173 2344 l 1262 2389 l P NP 1262 2389 m 1077 2360 l 971 2319 l 1262 2389 l P NP 1262 2389 m 1184 2389 l 1077 2360 l 1077 2360 l 1262 2389 l P 0.887686 G NP 2393 2393 m 2336 2232 l 2548 2240 l 2393 2393 l P NP 2393 2393 m 2205 2376 l 2336 2232 l 2336 2232 l 2393 2393 l P 0.886471 G NP 1183 2402 m 1015 2360 l 1077 2360 l 1183 2402 l P 0.884118 G NP 1183 2402 m 1031 2371 l 906 2333 l 906 2333 l 1183 2402 l P 0.886471 G NP 1183 2402 m 1142 2402 l 1031 2371 l 1031 2371 l 1183 2402 l P 0.89 G NP 1155 2408 m 1021 2371 l 1031 2371 l 1155 2408 l P 0.887686 G NP 2582 2410 m 2393 2393 l 2548 2240 l 2548 2240 l 2582 2410 l P NP 2582 2410 m 2548 2240 l 2761 2247 l 2582 2410 l P 0.818235 G NP 1600 2419 m 1384 2376 l 1312 2330 l 1312 2330 l 1600 2419 l P 0.89 G NP 1614 2419 m 1450 2308 l 1639 2325 l 1614 2419 l P NP 1614 2419 m 1586 2415 l 1448 2372 l 1450 2308 l 1450 2308 l 1614 2419 l P NP 1614 2419 m 1600 2419 l 1600 2419 l 1600 2419 l 1586 2415 l 1614 2419 l P 0.844118 G NP 1456 2423 m 1262 2389 l 1173 2344 l 1173 2344 l 1456 2423 l P NP 1456 2423 m 1289 2376 l 1384 2376 l 1456 2423 l P NP 1456 2423 m 1374 2423 l 1262 2389 l 1262 2389 l 1456 2423 l P 0.89 G NP 1614 2425 m 1600 2419 l 1614 2419 l 1614 2425 l P 0.887686 G NP 2770 2427 m 2582 2410 l 2761 2247 l 2761 2247 l 2770 2427 l P NP 2770 2427 m 2761 2247 l 2973 2255 l 2770 2427 l P 0.89 G NP 1631 2431 m 1614 2425 l 1614 2419 l 1614 2419 l 1631 2431 l P 0.867647 G NP 1351 2434 m 1184 2389 l 1262 2389 l 1351 2434 l P 0.866471 G NP 1351 2434 m 1183 2402 l 1077 2360 l 1351 2434 l P 0.867647 G NP 1351 2434 m 1289 2434 l 1183 2402 l 1183 2402 l 1351 2434 l P 0.887686 G NP 2959 2444 m 2973 2255 l 3185 2263 l 2959 2444 l P NP 2959 2444 m 2770 2427 l 2973 2255 l 2973 2255 l 2959 2444 l P 0.886471 G NP 1289 2444 m 1142 2402 l 1183 2402 l 1289 2444 l P 0.877059 G NP 1289 2444 m 1155 2408 l 1031 2371 l 1031 2371 l 1289 2444 l P 0.885294 G NP 1289 2444 m 1266 2444 l 1155 2408 l 1155 2408 l 1289 2444 l P 0.89 G NP 1780 2444 m 1639 2325 l 1828 2342 l 1780 2444 l P NP 1780 2444 m 1614 2419 l 1639 2325 l 1639 2325 l 1780 2444 l P 0.887686 G NP 3148 2461 m 2959 2444 l 3185 2263 l 3185 2263 l 3148 2461 l P NP 3148 2461 m 3185 2263 l 3397 2270 l 3148 2461 l P 0.818235 G NP 1657 2466 m 1384 2376 l 1600 2419 l 1657 2466 l P NP 1657 2466 m 1456 2423 l 1384 2376 l 1657 2466 l P NP 1657 2466 m 1579 2466 l 1456 2423 l 1456 2423 l 1657 2466 l P 0.89 G NP 1946 2469 m 1828 2342 l 2016 2359 l 1946 2469 l P NP 1946 2469 m 1780 2444 l 1828 2342 l 1828 2342 l 1946 2469 l P 0.844118 G NP 1529 2469 m 1374 2423 l 1456 2423 l 1529 2469 l P NP 1529 2469 m 1351 2434 l 1262 2389 l 1262 2389 l 1529 2469 l P NP 1529 2469 m 1459 2469 l 1351 2434 l 1351 2434 l 1529 2469 l P 0.887686 G NP 3336 2478 m 3397 2270 l 3610 2278 l 3336 2478 l P NP 3336 2478 m 3148 2461 l 3397 2270 l 3397 2270 l 3336 2478 l P 0.867647 G NP 1439 2479 m 1289 2434 l 1351 2434 l 1439 2479 l P 0.866471 G NP 1439 2479 m 1289 2444 l 1183 2402 l 1183 2402 l 1439 2479 l P NP 1439 2479 m 1393 2479 l 1289 2444 l 1289 2444 l 1439 2479 l P 0.885294 G NP 1396 2486 m 1266 2444 l 1289 2444 l 1396 2486 l P 0.879412 G NP 1396 2486 m 1389 2486 l 1340 2468 l 1340 2468 l 1396 2486 l P 0.89 G NP 1772 2487 m 1631 2431 l 1614 2419 l 1614 2419 l 1780 2444 l 1772 2487 l P NP 2112 2494 m 1946 2469 l 2016 2359 l 2016 2359 l 2112 2494 l P NP 2112 2494 m 2016 2359 l 2205 2376 l 2112 2494 l P 0.887686 G NP 3525 2495 m 3610 2278 l 3817 2285 l 3819 2288 l 3525 2495 l P NP 3525 2495 m 3336 2478 l 3610 2278 l 3610 2278 l 3525 2495 l P NP 3713 2512 m 3820 2290 l 3893 2389 l 3713 2512 l P NP 3713 2512 m 3525 2495 l 3819 2288 l 3819 2288 l 3820 2290 l 3713 2512 l P 0.817059 G NP 1714 2513 m 1529 2469 l 1456 2423 l 1456 2423 l 1714 2513 l P 0.818235 G NP 1714 2513 m 1579 2466 l 1657 2466 l 1714 2513 l P NP 1714 2513 m 1645 2513 l 1529 2469 l 1529 2469 l 1714 2513 l P 0.844118 G NP 1601 2516 m 1459 2469 l 1529 2469 l 1601 2516 l P NP 1601 2516 m 1439 2479 l 1351 2434 l 1601 2516 l P NP 1601 2516 m 1543 2516 l 1439 2479 l 1439 2479 l 1601 2516 l P 0.89 G NP 2278 2519 m 2112 2494 l 2205 2376 l 2205 2376 l 2278 2519 l P NP 2278 2519 m 2205 2376 l 2393 2393 l 2278 2519 l P 0.788824 G NP 1859 2522 m 1657 2466 l 1600 2419 l 1859 2522 l P NP 1859 2522 m 1791 2522 l 1657 2466 l 1657 2466 l 1859 2522 l P 0.864118 G NP 1528 2523 m 1396 2486 l 1289 2444 l 1528 2523 l P 0.866471 G NP 1528 2523 m 1393 2479 l 1439 2479 l 1528 2523 l P NP 1528 2523 m 1496 2523 l 1396 2486 l 1396 2486 l 1528 2523 l P 0.879412 G NP 1502 2528 m 1389 2486 l 1396 2486 l 1502 2528 l P 0.887686 G NP 3902 2529 m 3713 2512 l 3893 2389 l 3893 2389 l 3943 2456 l 3902 2529 l P NP 3902 2529 m 3943 2456 l 3965 2487 l 3902 2529 l P 0.89 G NP 1879 2531 m 1859 2522 l 1772 2487 l 1780 2444 l 1780 2444 l 1879 2531 l P 0.887686 G NP 4003 2538 m 3902 2529 l 3965 2487 l 3965 2487 l 4003 2538 l P 0.89 G NP 2445 2545 m 2393 2393 l 2582 2410 l 2445 2545 l P NP 2445 2545 m 2278 2519 l 2393 2393 l 2393 2393 l 2445 2545 l P NP 1914 2549 m 1879 2531 l 1780 2444 l 1780 2444 l 1946 2469 l 1914 2549 l P 0.817059 G NP 1771 2560 m 1601 2516 l 1529 2469 l 1529 2469 l 1771 2560 l P 0.818235 G NP 1771 2560 m 1645 2513 l 1714 2513 l 1771 2560 l P NP 1771 2560 m 1711 2560 l 1601 2516 l 1601 2516 l 1771 2560 l P 0.842941 G NP 1673 2562 m 1528 2523 l 1439 2479 l 1673 2562 l P 0.844118 G NP 1673 2562 m 1543 2516 l 1601 2516 l 1673 2562 l P NP 1673 2562 m 1627 2562 l 1528 2523 l 1528 2523 l 1673 2562 l P 0.788824 G NP 1903 2568 m 1714 2513 l 1657 2466 l 1903 2568 l P NP 1903 2568 m 1791 2522 l 1859 2522 l 1903 2568 l P NP 1903 2568 m 1840 2568 l 1714 2513 l 1714 2513 l 1903 2568 l P 0.866471 G NP 1616 2568 m 1496 2523 l 1528 2523 l 1616 2568 l P 0.855882 G NP 1616 2568 m 1502 2528 l 1396 2486 l 1396 2486 l 1616 2568 l P 0.865294 G NP 1616 2568 m 1598 2568 l 1502 2528 l 1502 2528 l 1616 2568 l P 0.89 G NP 2611 2570 m 2445 2545 l 2582 2410 l 2582 2410 l 2611 2570 l P NP 2611 2570 m 2582 2410 l 2770 2427 l 2611 2570 l P NP 2055 2591 m 1946 2469 l 2112 2494 l 2055 2591 l P NP 2055 2591 m 1953 2569 l 1914 2549 l 1946 2469 l 1946 2469 l 2055 2591 l P NP 2777 2595 m 2611 2570 l 2770 2427 l 2770 2427 l 2777 2595 l P NP 2777 2595 m 2770 2427 l 2959 2444 l 2777 2595 l P 0.818235 G NP 1828 2607 m 1711 2560 l 1771 2560 l 1828 2607 l P 0.817059 G NP 1828 2607 m 1673 2562 l 1601 2516 l 1828 2607 l P 0.818235 G NP 1828 2607 m 1776 2607 l 1673 2562 l 1673 2562 l 1828 2607 l P 0.844118 G NP 1745 2609 m 1627 2562 l 1673 2562 l 1745 2609 l P 0.842941 G NP 1745 2609 m 1616 2568 l 1528 2523 l 1528 2523 l 1745 2609 l P 0.844118 G NP 1745 2609 m 1710 2609 l 1616 2568 l 1616 2568 l 1745 2609 l P 0.865294 G NP 1705 2613 m 1598 2568 l 1616 2568 l 1705 2613 l P 0.858235 G NP 1705 2613 m 1699 2613 l 1658 2593 l 1658 2593 l 1705 2613 l P 0.788824 G NP 1946 2614 m 1771 2560 l 1714 2513 l 1946 2614 l P NP 1946 2614 m 1840 2568 l 1903 2568 l 1946 2614 l P NP 1946 2614 m 1889 2614 l 1771 2560 l 1771 2560 l 1946 2614 l P 0.89 G NP 2047 2616 m 1953 2569 l 2055 2591 l 2047 2616 l P NP 2943 2620 m 2959 2444 l 3148 2461 l 2943 2620 l P NP 2943 2620 m 2777 2595 l 2959 2444 l 2959 2444 l 2943 2620 l P NP 2200 2623 m 2055 2591 l 2112 2494 l 2112 2494 l 2200 2623 l P NP 2200 2623 m 2112 2494 l 2278 2519 l 2200 2623 l P 0.760588 G NP 2091 2638 m 1903 2568 l 1859 2522 l 1859 2522 l 2091 2638 l P 0.89 G NP 2100 2644 m 2091 2638 l 2047 2616 l 2055 2591 l 2055 2591 l 2100 2644 l P NP 3109 2645 m 2943 2620 l 3148 2461 l 3148 2461 l 3109 2645 l P NP 3109 2645 m 3148 2461 l 3336 2478 l 3109 2645 l P 0.817059 G NP 1885 2654 m 1745 2609 l 1673 2562 l 1673 2562 l 1885 2654 l P 0.818235 G NP 1885 2654 m 1776 2607 l 1828 2607 l 1885 2654 l P 0.817059 G NP 1885 2654 m 1842 2654 l 1745 2609 l 1745 2609 l 1885 2654 l P 0.89 G NP 2345 2655 m 2200 2623 l 2278 2519 l 2278 2519 l 2345 2655 l P NP 2345 2655 m 2278 2519 l 2445 2545 l 2345 2655 l P 0.844118 G NP 1818 2655 m 1710 2609 l 1745 2609 l 1818 2655 l P 0.840588 G NP 1818 2655 m 1705 2613 l 1616 2568 l 1818 2655 l P 0.842941 G NP 1818 2655 m 1793 2655 l 1705 2613 l 1705 2613 l 1818 2655 l P 0.858235 G NP 1793 2658 m 1699 2613 l 1705 2613 l 1793 2658 l P 0.788824 G NP 1989 2660 m 1889 2614 l 1946 2614 l 1989 2660 l P NP 1989 2660 m 1828 2607 l 1771 2560 l 1989 2660 l P NP 1989 2660 m 1938 2660 l 1828 2607 l 1828 2607 l 1989 2660 l P 0.89 G NP 3275 2670 m 3336 2478 l 3525 2495 l 3275 2670 l P NP 3275 2670 m 3109 2645 l 3336 2478 l 3336 2478 l 3275 2670 l P 0.760588 G NP 2122 2682 m 1946 2614 l 1903 2568 l 2122 2682 l P NP 2122 2682 m 1903 2568 l 2091 2638 l 2122 2682 l P NP 2122 2682 m 2071 2682 l 1946 2614 l 1946 2614 l 2122 2682 l P 0.89 G NP 2166 2687 m 2100 2644 l 2055 2591 l 2055 2591 l 2200 2623 l 2166 2687 l P NP 2490 2687 m 2445 2545 l 2611 2570 l 2490 2687 l P NP 2490 2687 m 2345 2655 l 2445 2545 l 2445 2545 l 2490 2687 l P NP 3441 2695 m 3525 2495 l 3713 2512 l 3441 2695 l P NP 3441 2695 m 3275 2670 l 3525 2495 l 3525 2495 l 3441 2695 l P 0.817059 G NP 1942 2701 m 1842 2654 l 1885 2654 l 1942 2701 l P 0.815882 G NP 1942 2701 m 1818 2655 l 1745 2609 l 1942 2701 l P 0.817059 G NP 1942 2701 m 1907 2701 l 1818 2655 l 1818 2655 l 1942 2701 l P 0.83 G NP 1890 2702 m 1793 2658 l 1705 2613 l 1705 2613 l 1890 2702 l P 0.842941 G NP 1890 2702 m 1793 2655 l 1818 2655 l 1890 2702 l P 0.841765 G NP 1890 2702 m 1876 2702 l 1793 2658 l 1793 2658 l 1890 2702 l P 0.788824 G NP 2032 2706 m 1938 2660 l 1989 2660 l 2032 2706 l P NP 2032 2706 m 1885 2654 l 1828 2607 l 1828 2607 l 2032 2706 l P NP 2032 2706 m 1987 2706 l 1885 2654 l 1885 2654 l 2032 2706 l P 0.89 G NP 2635 2719 m 2611 2570 l 2777 2595 l 2635 2719 l P NP 2635 2719 m 2490 2687 l 2611 2570 l 2611 2570 l 2635 2719 l P NP 3608 2721 m 3713 2512 l 3902 2529 l 3608 2721 l P NP 3608 2721 m 3441 2695 l 3713 2512 l 3713 2512 l 3608 2721 l P 0.760588 G NP 2152 2726 m 1989 2660 l 1946 2614 l 1946 2614 l 2152 2726 l P NP 2152 2726 m 2071 2682 l 2122 2682 l 2152 2726 l P NP 2152 2726 m 2106 2726 l 1989 2660 l 1989 2660 l 2152 2726 l P 0.89 G NP 3774 2746 m 3902 2529 l 4003 2538 l 4034 2582 l 3774 2746 l P NP 3774 2746 m 3608 2721 l 3902 2529 l 3902 2529 l 3774 2746 l P NP 2280 2746 m 2200 2623 l 2345 2655 l 2280 2746 l P NP 2280 2746 m 2241 2734 l 2166 2687 l 2200 2623 l 2200 2623 l 2280 2746 l P 0.815882 G NP 2000 2748 m 1890 2702 l 1818 2655 l 1818 2655 l 2000 2748 l P 0.817059 G NP 2000 2748 m 1907 2701 l 1942 2701 l 2000 2748 l P NP 2000 2748 m 1972 2748 l 1890 2702 l 1890 2702 l 2000 2748 l P 0.841765 G NP 1962 2749 m 1876 2702 l 1890 2702 l 1962 2749 l P 0.833529 G NP 1962 2749 m 1958 2749 l 1924 2728 l 1924 2728 l 1962 2749 l P 0.89 G NP 2780 2751 m 2777 2595 l 2943 2620 l 2780 2751 l P NP 2780 2751 m 2635 2719 l 2777 2595 l 2777 2595 l 2780 2751 l P 0.788824 G NP 2076 2752 m 1987 2706 l 2032 2706 l 2076 2752 l P NP 2076 2752 m 1942 2701 l 1885 2654 l 1885 2654 l 2076 2752 l P NP 2076 2752 m 2036 2752 l 1942 2701 l 1942 2701 l 2076 2752 l P 0.888824 G NP 2275 2756 m 2241 2734 l 2280 2746 l 2275 2756 l P NP 2293 2768 m 2275 2756 l 2280 2746 l 2280 2746 l 2293 2768 l P 0.740588 G NP 2295 2769 m 2122 2682 l 2091 2638 l 2295 2769 l P NP 2295 2769 m 2252 2769 l 2122 2682 l 2122 2682 l 2295 2769 l P 0.760588 G NP 2183 2771 m 2032 2706 l 1989 2660 l 1989 2660 l 2183 2771 l P NP 2183 2771 m 2106 2726 l 2152 2726 l 2183 2771 l P NP 2183 2771 m 2140 2771 l 2032 2706 l 2032 2706 l 2183 2771 l P 0.89 G NP 3940 2771 m 4051 2606 l 4099 2674 l 3940 2771 l P NP 3940 2771 m 3774 2746 l 4034 2582 l 4034 2582 l 4051 2606 l 3940 2771 l P NP 2925 2783 m 2780 2751 l 2943 2620 l 2943 2620 l 2925 2783 l P NP 2925 2783 m 2943 2620 l 3109 2645 l 2925 2783 l P NP 2405 2783 m 2280 2746 l 2345 2655 l 2345 2655 l 2405 2783 l P NP 2405 2783 m 2345 2655 l 2490 2687 l 2405 2783 l P 0.813529 G NP 2057 2795 m 1962 2749 l 1890 2702 l 2057 2795 l P 0.817059 G NP 2057 2795 m 1972 2748 l 2000 2748 l 2057 2795 l P 0.815882 G NP 2057 2795 m 2037 2795 l 1962 2749 l 1962 2749 l 2057 2795 l P 0.833529 G NP 2034 2795 m 1958 2749 l 1962 2749 l 2034 2795 l P 0.89 G NP 4106 2796 m 4147 2741 l 4162 2763 l 4106 2796 l P NP 4106 2796 m 3940 2771 l 4099 2674 l 4099 2674 l 4147 2741 l 4106 2796 l P 0.787647 G NP 2119 2799 m 2000 2748 l 1942 2701 l 1942 2701 l 2119 2799 l P 0.788824 G NP 2119 2799 m 2036 2752 l 2076 2752 l 2119 2799 l P NP 2119 2799 m 2085 2799 l 2000 2748 l 2000 2748 l 2119 2799 l P 0.89 G NP 4195 2810 m 4106 2796 l 4162 2763 l 4162 2763 l 4195 2810 l P 0.740588 G NP 2314 2810 m 2252 2769 l 2295 2769 l 2314 2810 l P NP 2314 2810 m 2152 2726 l 2122 2682 l 2314 2810 l P NP 2314 2810 m 2274 2810 l 2152 2726 l 2152 2726 l 2314 2810 l P 0.760588 G NP 2214 2815 m 2140 2771 l 2183 2771 l 2214 2815 l P NP 2214 2815 m 2076 2752 l 2032 2706 l 2032 2706 l 2214 2815 l P NP 2214 2815 m 2174 2815 l 2076 2752 l 2076 2752 l 2214 2815 l P 0.89 G NP 3070 2815 m 3109 2645 l 3275 2670 l 3070 2815 l P NP 3070 2815 m 2925 2783 l 3109 2645 l 3109 2645 l 3070 2815 l P NP 2529 2821 m 2405 2783 l 2490 2687 l 2490 2687 l 2529 2821 l P NP 2529 2821 m 2490 2687 l 2635 2719 l 2529 2821 l P 0.888824 G NP 2371 2832 m 2295 2769 l 2293 2768 l 2280 2746 l 2280 2746 l 2405 2783 l 2371 2832 l P 0.801765 G NP 2114 2842 m 2034 2795 l 1962 2749 l 1962 2749 l 2114 2842 l P 0.815882 G NP 2114 2842 m 2037 2795 l 2057 2795 l 2114 2842 l P 0.814706 G NP 2114 2842 m 2102 2842 l 2034 2795 l 2034 2795 l 2114 2842 l P 0.788824 G NP 2162 2845 m 2085 2799 l 2119 2799 l 2162 2845 l P 0.787647 G NP 2162 2845 m 2057 2795 l 2000 2748 l 2162 2845 l P 0.788824 G NP 2162 2845 m 2134 2845 l 2057 2795 l 2057 2795 l 2162 2845 l P 0.89 G NP 3215 2847 m 3275 2670 l 3441 2695 l 3215 2847 l P NP 3215 2847 m 3070 2815 l 3275 2670 l 3275 2670 l 3215 2847 l P 0.740588 G NP 2333 2850 m 2274 2810 l 2314 2810 l 2333 2850 l P NP 2333 2850 m 2183 2771 l 2152 2726 l 2152 2726 l 2333 2850 l P NP 2333 2850 m 2295 2850 l 2183 2771 l 2183 2771 l 2333 2850 l P 0.89 G NP 2654 2858 m 2635 2719 l 2780 2751 l 2654 2858 l P NP 2654 2858 m 2529 2821 l 2635 2719 l 2635 2719 l 2654 2858 l P 0.760588 G NP 2244 2859 m 2174 2815 l 2214 2815 l 2244 2859 l P 0.759412 G NP 2244 2859 m 2119 2799 l 2076 2752 l 2244 2859 l P 0.760588 G NP 2244 2859 m 2209 2859 l 2119 2799 l 2119 2799 l 2244 2859 l P 0.89 G NP 3359 2879 m 3441 2695 l 3608 2721 l 3359 2879 l P NP 3359 2879 m 3215 2847 l 3441 2695 l 3441 2695 l 3359 2879 l P 0.814706 G NP 2171 2889 m 2102 2842 l 2114 2842 l 2171 2889 l P 0.805294 G NP 2171 2889 m 2167 2889 l 2140 2867 l 2140 2867 l 2171 2889 l P 0.740588 G NP 2352 2891 m 2295 2850 l 2333 2850 l 2352 2891 l P NP 2352 2891 m 2214 2815 l 2183 2771 l 2352 2891 l P NP 2352 2891 m 2317 2891 l 2214 2815 l 2214 2815 l 2352 2891 l P 0.788824 G NP 2205 2891 m 2134 2845 l 2162 2845 l 2205 2891 l P 0.786471 G NP 2205 2891 m 2114 2842 l 2057 2795 l 2205 2891 l P 0.787647 G NP 2205 2891 m 2184 2891 l 2114 2842 l 2114 2842 l 2205 2891 l P 0.89 G NP 2779 2896 m 2780 2751 l 2925 2783 l 2779 2896 l P NP 2779 2896 m 2654 2858 l 2780 2751 l 2780 2751 l 2779 2896 l P 0.888824 G NP 2456 2901 m 2371 2832 l 2405 2783 l 2405 2783 l 2456 2901 l P NP 2458 2903 m 2456 2901 l 2405 2783 l 2405 2783 l 2529 2821 l 2458 2903 l P 0.759412 G NP 2275 2903 m 2162 2845 l 2119 2799 l 2275 2903 l P 0.760588 G NP 2275 2903 m 2209 2859 l 2244 2859 l 2275 2903 l P NP 2275 2903 m 2244 2903 l 2162 2845 l 2162 2845 l 2275 2903 l P 0.89 G NP 3504 2911 m 3359 2879 l 3608 2721 l 3608 2721 l 3504 2911 l P NP 3504 2911 m 3608 2721 l 3774 2746 l 3504 2911 l P 0.733647 G NP 2471 2913 m 2314 2810 l 2295 2769 l 2295 2769 l 2471 2913 l P NP 2471 2913 m 2437 2913 l 2314 2810 l 2314 2810 l 2471 2913 l P 0.740588 G NP 2372 2932 m 2244 2859 l 2214 2815 l 2214 2815 l 2372 2932 l P NP 2372 2932 m 2317 2891 l 2352 2891 l 2372 2932 l P NP 2372 2932 m 2338 2932 l 2244 2859 l 2244 2859 l 2372 2932 l P 0.89 G NP 2904 2933 m 2925 2783 l 3070 2815 l 2904 2933 l P NP 2904 2933 m 2779 2896 l 2925 2783 l 2925 2783 l 2904 2933 l P 0.805294 G NP 2228 2936 m 2167 2889 l 2171 2889 l 2228 2936 l P 0.784118 G NP 2249 2937 m 2171 2889 l 2114 2842 l 2114 2842 l 2249 2937 l P 0.787647 G NP 2249 2937 m 2184 2891 l 2205 2891 l 2249 2937 l P NP 2249 2937 m 2233 2937 l 2171 2889 l 2171 2889 l 2249 2937 l P 0.596157 G NP 1998 2940 m 1993 2918 l 1993 2918 l 1993 2918 l 1998 2940 l P NP 1998 2940 m 1991 2922 l 1993 2918 l 1993 2918 l 1998 2940 l P NP 1998 2940 m 1990 2940 l 1986 2928 l 1991 2922 l 1991 2922 l 1998 2940 l P 0.89 G NP 3649 2943 m 3504 2911 l 3774 2746 l 3774 2746 l 3649 2943 l P NP 3649 2943 m 3774 2746 l 3940 2771 l 3649 2943 l P 0.888824 G NP 2563 2945 m 2461 2905 l 2458 2903 l 2529 2821 l 2529 2821 l 2563 2945 l P NP 2563 2945 m 2529 2821 l 2654 2858 l 2563 2945 l P 0.759412 G NP 2305 2947 m 2205 2891 l 2162 2845 l 2162 2845 l 2305 2947 l P 0.760588 G NP 2305 2947 m 2244 2903 l 2275 2903 l 2305 2947 l P NP 2305 2947 m 2278 2947 l 2205 2891 l 2205 2891 l 2305 2947 l P 0.733647 G NP 2480 2949 m 2333 2850 l 2314 2810 l 2480 2949 l P NP 2480 2949 m 2437 2913 l 2471 2913 l 2480 2949 l P NP 2480 2949 m 2447 2949 l 2333 2850 l 2333 2850 l 2480 2949 l P 0.89 G NP 3029 2971 m 3070 2815 l 3215 2847 l 3029 2971 l P NP 3029 2971 m 2904 2933 l 3070 2815 l 3070 2815 l 3029 2971 l P 0.740588 G NP 2391 2972 m 2338 2932 l 2372 2932 l 2391 2972 l P NP 2391 2972 m 2275 2903 l 2244 2859 l 2391 2972 l P NP 2391 2972 m 2360 2972 l 2275 2903 l 2275 2903 l 2391 2972 l P 0.89 G NP 3794 2975 m 3940 2771 l 4106 2796 l 3794 2975 l P NP 3794 2975 m 3649 2943 l 3940 2771 l 3940 2771 l 3794 2975 l P 0.868824 G NP 2533 2980 m 2471 2913 l 2461 2905 l 2563 2945 l 2533 2980 l P 0.772353 G NP 2292 2983 m 2228 2936 l 2171 2889 l 2292 2983 l P 0.787647 G NP 2292 2983 m 2233 2937 l 2249 2937 l 2292 2983 l P 0.785294 G NP 2292 2983 m 2283 2983 l 2228 2936 l 2228 2936 l 2292 2983 l P 0.733647 G NP 2489 2985 m 2447 2949 l 2480 2949 l 2489 2985 l P NP 2489 2985 m 2352 2891 l 2333 2850 l 2489 2985 l P NP 2489 2985 m 2457 2985 l 2352 2891 l 2352 2891 l 2489 2985 l P 0.596157 G NP 2012 2986 m 1998 2940 l 1993 2918 l 1995 2916 l 1995 2916 l 2012 2986 l P 0.888824 G NP 2669 2987 m 2654 2858 l 2779 2896 l 2669 2987 l P NP 2669 2987 m 2563 2945 l 2654 2858 l 2654 2858 l 2669 2987 l P 0.596157 G NP 1987 2990 m 1972 2948 l 1979 2939 l 1979 2939 l 1987 2990 l P NP 1987 2990 m 1955 2973 l 1972 2948 l 1972 2948 l 1987 2990 l P NP 1987 2990 m 1953 2990 l 1949 2981 l 1955 2973 l 1955 2973 l 1987 2990 l P 0.758235 G NP 2336 2991 m 2249 2937 l 2205 2891 l 2205 2891 l 2336 2991 l P 0.760588 G NP 2336 2991 m 2278 2947 l 2305 2947 l 2336 2991 l P 0.759412 G NP 2336 2991 m 2313 2991 l 2249 2937 l 2249 2937 l 2336 2991 l P 0.89 G NP 3939 3007 m 4106 2796 l 4195 2810 l 4222 2849 l 3939 3007 l P NP 3939 3007 m 3794 2975 l 4106 2796 l 4106 2796 l 3939 3007 l P NP 3154 3009 m 3215 2847 l 3359 2879 l 3154 3009 l P NP 3154 3009 m 3029 2971 l 3215 2847 l 3215 2847 l 3154 3009 l P 0.740588 G NP 2410 3013 m 2305 2947 l 2275 2903 l 2410 3013 l P NP 2410 3013 m 2360 2972 l 2391 2972 l 2410 3013 l P NP 2410 3013 m 2382 3013 l 2305 2947 l 2305 2947 l 2410 3013 l P 0.733647 G NP 2498 3021 m 2372 2932 l 2352 2891 l 2498 3021 l P NP 2498 3021 m 2457 2985 l 2489 2985 l 2498 3021 l P NP 2498 3021 m 2468 3021 l 2372 2932 l 2372 2932 l 2498 3021 l P 0.888824 G NP 2775 3029 m 2779 2896 l 2904 2933 l 2775 3029 l P NP 2775 3029 m 2669 2987 l 2779 2896 l 2779 2896 l 2775 3029 l P 0.785294 G NP 2335 3030 m 2283 2983 l 2292 2983 l 2335 3030 l P 0.775882 G NP 2335 3030 m 2332 3030 l 2311 3008 l 2311 3008 l 2335 3030 l P 0.868824 G NP 2584 3035 m 2533 2980 l 2563 2945 l 2563 2945 l 2584 3035 l P 0.758235 G NP 2367 3035 m 2292 2983 l 2249 2937 l 2367 3035 l P 0.759412 G NP 2367 3035 m 2313 2991 l 2336 2991 l 2367 3035 l P NP 2367 3035 m 2349 3035 l 2292 2983 l 2292 2983 l 2367 3035 l P 0.596157 G NP 2021 3039 m 1990 2940 l 1998 2940 l 2021 3039 l P NP 2021 3039 m 1987 2990 l 1979 2939 l 1986 2928 l 1986 2928 l 2021 3039 l P NP 2021 3039 m 2005 3039 l 1987 2990 l 1987 2990 l 2021 3039 l P 0.89 G NP 4084 3039 m 4234 2866 l 4279 2933 l 4084 3039 l P NP 4084 3039 m 3939 3007 l 4222 2849 l 4222 2849 l 4234 2866 l 4084 3039 l P NP 3279 3046 m 3359 2879 l 3504 2911 l 3279 3046 l P NP 3279 3046 m 3154 3009 l 3359 2879 l 3359 2879 l 3279 3046 l P 0.868824 G NP 2600 3052 m 2584 3035 l 2563 2945 l 2563 2945 l 2669 2987 l 2600 3052 l P 0.740588 G NP 2429 3054 m 2382 3013 l 2410 3013 l 2429 3054 l P NP 2429 3054 m 2336 2991 l 2305 2947 l 2305 2947 l 2429 3054 l P NP 2429 3054 m 2404 3054 l 2336 2991 l 2336 2991 l 2429 3054 l P 0.733647 G NP 2507 3058 m 2468 3021 l 2498 3021 l 2507 3058 l P NP 2507 3058 m 2391 2972 l 2372 2932 l 2507 3058 l P NP 2507 3058 m 2478 3058 l 2391 2972 l 2391 2972 l 2507 3058 l P 0.888824 G NP 2881 3071 m 2775 3029 l 2904 2933 l 2904 2933 l 2881 3071 l P NP 2881 3071 m 2904 2933 l 3029 2971 l 2881 3071 l P 0.89 G NP 4229 3071 m 4084 3039 l 4279 2933 l 4279 2933 l 4312 2981 l 4229 3071 l P NP 4229 3071 m 4312 2981 l 4335 3015 l 4229 3071 l P 0.659608 G NP 2618 3071 m 2480 2949 l 2471 2913 l 2471 2913 l 2618 3071 l P 0.775882 G NP 2378 3076 m 2332 3030 l 2335 3030 l 2378 3076 l P 0.759412 G NP 2397 3079 m 2349 3035 l 2367 3035 l 2397 3079 l P 0.755882 G NP 2397 3079 m 2335 3030 l 2292 2983 l 2397 3079 l P 0.758235 G NP 2397 3079 m 2384 3079 l 2335 3030 l 2335 3030 l 2397 3079 l P 0.89 G NP 3404 3084 m 3279 3046 l 3504 2911 l 3504 2911 l 3404 3084 l P NP 3404 3084 m 3504 2911 l 3649 2943 l 3404 3084 l P 0.733647 G NP 2516 3094 m 2478 3058 l 2507 3058 l 2516 3094 l P NP 2516 3094 m 2410 3013 l 2391 2972 l 2391 2972 l 2516 3094 l P NP 2516 3094 m 2489 3094 l 2410 3013 l 2410 3013 l 2516 3094 l P 0.740588 G NP 2448 3095 m 2404 3054 l 2429 3054 l 2448 3095 l P NP 2448 3095 m 2367 3035 l 2336 2991 l 2336 2991 l 2448 3095 l P NP 2448 3095 m 2426 3095 l 2367 3035 l 2367 3035 l 2448 3095 l P 0.659608 G NP 2618 3102 m 2480 2949 l 2618 3071 l 2618 3102 l P NP 2618 3102 m 2489 2985 l 2480 2949 l 2618 3102 l P 0.89 G NP 4374 3103 m 4229 3071 l 4335 3015 l 4335 3015 l 4386 3090 l 4374 3103 l P NP 4374 3103 m 4386 3090 l 4389 3095 l 4374 3103 l P 0.868824 G NP 2679 3106 m 2622 3077 l 2618 3071 l 2618 3071 l 2600 3052 l 2669 2987 l 2669 2987 l 2679 3106 l P NP 2679 3106 m 2669 2987 l 2775 3029 l 2679 3106 l P 0.596157 G NP 1955 3107 m 1914 3028 l 1945 2987 l 1945 2987 l 1955 3107 l P 0.89 G NP 4398 3108 m 4374 3103 l 4389 3095 l 4389 3095 l 4398 3108 l P 0.596157 G NP 2005 3109 m 1953 2990 l 1987 2990 l 2005 3109 l P NP 2005 3109 m 1955 3107 l 1945 2987 l 1949 2981 l 1949 2981 l 2005 3109 l P NP 2005 3109 m 1956 3109 l 1955 3107 l 1955 3107 l 2005 3109 l P NP 2048 3109 m 2021 3039 l 1998 2940 l 1998 2940 l 2048 3109 l P 0.888824 G NP 2988 3113 m 2881 3071 l 3029 2971 l 3029 2971 l 2988 3113 l P NP 2988 3113 m 3029 2971 l 3154 3009 l 2988 3113 l P 0.89 G NP 3529 3121 m 3404 3084 l 3649 2943 l 3649 2943 l 3529 3121 l P NP 3529 3121 m 3649 2943 l 3794 2975 l 3529 3121 l P 0.744118 G NP 2428 3123 m 2378 3076 l 2335 3030 l 2335 3030 l 2428 3123 l P 0.758235 G NP 2428 3123 m 2384 3079 l 2397 3079 l 2428 3123 l P 0.757059 G NP 2428 3123 m 2420 3123 l 2378 3076 l 2378 3076 l 2428 3123 l P 0.846471 G NP 2657 3126 m 2622 3077 l 2679 3106 l 2657 3126 l P 0.733647 G NP 2525 3130 m 2489 3094 l 2516 3094 l 2525 3130 l P NP 2525 3130 m 2429 3054 l 2410 3013 l 2525 3130 l P NP 2525 3130 m 2500 3130 l 2429 3054 l 2429 3054 l 2525 3130 l P 0.659608 G NP 2618 3132 m 2489 2985 l 2618 3102 l 2618 3132 l P NP 2618 3132 m 2498 3021 l 2489 2985 l 2489 2985 l 2618 3132 l P 0.740588 G NP 2467 3135 m 2426 3095 l 2448 3095 l 2467 3135 l P NP 2467 3135 m 2397 3079 l 2367 3035 l 2467 3135 l P NP 2467 3135 m 2449 3135 l 2397 3079 l 2397 3079 l 2467 3135 l P 0.596157 G NP 1890 3135 m 1864 3095 l 1888 3064 l 1888 3064 l 1890 3135 l P NP 1890 3135 m 1888 3064 l 1914 3028 l 1914 3028 l 1955 3107 l 1890 3135 l P NP 2043 3138 m 2005 3109 l 1987 2990 l 2043 3138 l P NP 2043 3138 m 2005 3039 l 2021 3039 l 2043 3138 l P NP 2043 3138 m 2019 3138 l 2005 3109 l 2005 3109 l 2043 3138 l P 0.868824 G NP 2768 3151 m 2679 3106 l 2775 3029 l 2775 3029 l 2768 3151 l P NP 2768 3151 m 2775 3029 l 2881 3071 l 2768 3151 l P 0.888824 G NP 3094 3154 m 2988 3113 l 3154 3009 l 3154 3009 l 3094 3154 l P NP 3094 3154 m 3154 3009 l 3279 3046 l 3094 3154 l P 0.89 G NP 3654 3159 m 3529 3121 l 3794 2975 l 3794 2975 l 3654 3159 l P NP 3654 3159 m 3794 2975 l 3939 3007 l 3654 3159 l P 0.659608 G NP 2618 3162 m 2498 3021 l 2618 3132 l 2618 3162 l P NP 2618 3162 m 2507 3058 l 2498 3021 l 2618 3162 l P 0.846471 G NP 2682 3163 m 2657 3126 l 2679 3106 l 2679 3106 l 2682 3163 l P 0.733647 G NP 2533 3166 m 2500 3130 l 2525 3130 l 2533 3166 l P NP 2533 3166 m 2448 3095 l 2429 3054 l 2533 3166 l P NP 2533 3166 m 2511 3166 l 2448 3095 l 2448 3095 l 2533 3166 l P 0.757059 G NP 2458 3167 m 2420 3123 l 2428 3123 l 2458 3167 l P 0.748824 G NP 2458 3167 m 2456 3167 l 2440 3147 l 2440 3147 l 2458 3167 l P 0.740588 G NP 2487 3176 m 2428 3123 l 2397 3079 l 2487 3176 l P NP 2487 3176 m 2449 3135 l 2467 3135 l 2487 3176 l P NP 2487 3176 m 2472 3176 l 2428 3123 l 2428 3123 l 2487 3176 l P 0.596157 G NP 2074 3190 m 2021 3039 l 2048 3109 l 2074 3190 l P NP 2074 3190 m 2043 3138 l 2021 3039 l 2074 3190 l P NP 2074 3190 m 2065 3190 l 2043 3138 l 2043 3138 l 2074 3190 l P 0.657294 G NP 2618 3192 m 2516 3094 l 2507 3058 l 2507 3058 l 2618 3192 l P 0.659608 G NP 2618 3192 m 2507 3058 l 2618 3162 l 2618 3192 l P 0.868824 G NP 2856 3195 m 2768 3151 l 2881 3071 l 2881 3071 l 2856 3195 l P NP 2856 3195 m 2881 3071 l 2988 3113 l 2856 3195 l P 0.89 G NP 3779 3196 m 3654 3159 l 3939 3007 l 3939 3007 l 3779 3196 l P NP 3779 3196 m 3939 3007 l 4084 3039 l 3779 3196 l P 0.888824 G NP 3200 3196 m 3279 3046 l 3404 3084 l 3200 3196 l P NP 3200 3196 m 3094 3154 l 3279 3046 l 3279 3046 l 3200 3196 l P 0.846471 G NP 2706 3197 m 2682 3163 l 2679 3106 l 2679 3106 l 2768 3151 l 2706 3197 l P 0.596157 G NP 1807 3199 m 1808 3169 l 1864 3095 l 1890 3135 l 1807 3199 l P NP 1807 3199 m 1795 3185 l 1808 3169 l 1808 3169 l 1807 3199 l P 0.733647 G NP 2542 3202 m 2467 3135 l 2448 3095 l 2448 3095 l 2542 3202 l P NP 2542 3202 m 2511 3166 l 2533 3166 l 2542 3202 l P NP 2542 3202 m 2522 3202 l 2467 3135 l 2467 3135 l 2542 3202 l P 0.748824 G NP 2489 3211 m 2456 3167 l 2458 3167 l 2489 3211 l P 0.740588 G NP 2506 3217 m 2472 3176 l 2487 3176 l 2506 3217 l P NP 2506 3217 m 2458 3167 l 2428 3123 l 2506 3217 l P NP 2506 3217 m 2495 3217 l 2458 3167 l 2458 3167 l 2506 3217 l P 0.657294 G NP 2618 3222 m 2525 3130 l 2516 3094 l 2618 3222 l P 0.659608 G NP 2618 3222 m 2516 3094 l 2618 3192 l 2618 3222 l P 0.596157 G NP 2089 3227 m 2074 3190 l 2048 3109 l 2089 3227 l P NP 2023 3228 m 1956 3109 l 2005 3109 l 2023 3228 l P 0.89 G NP 3903 3234 m 4084 3039 l 4229 3071 l 3903 3234 l P NP 3903 3234 m 3779 3196 l 4084 3039 l 4084 3039 l 3903 3234 l P 0.596157 G NP 2066 3237 m 2019 3138 l 2043 3138 l 2066 3237 l P NP 2066 3237 m 2023 3228 l 2005 3109 l 2005 3109 l 2066 3237 l P NP 2066 3237 m 2028 3237 l 2023 3228 l 2023 3228 l 2066 3237 l P 0.733647 G NP 2551 3238 m 2522 3202 l 2542 3202 l 2551 3238 l P 0.731333 G NP 2551 3238 m 2487 3176 l 2467 3135 l 2467 3135 l 2551 3238 l P 0.733647 G NP 2551 3238 m 2533 3238 l 2487 3176 l 2487 3176 l 2551 3238 l P 0.888824 G NP 3306 3238 m 3404 3084 l 3529 3121 l 3306 3238 l P NP 3306 3238 m 3200 3196 l 3404 3084 l 3404 3084 l 3306 3238 l P 0.868824 G NP 2945 3240 m 2988 3113 l 3094 3154 l 2945 3240 l P NP 2945 3240 m 2856 3195 l 2988 3113 l 2988 3113 l 2945 3240 l P 0.574 G NP 2738 3243 m 2618 3102 l 2618 3071 l 2618 3071 l 2738 3243 l P 0.596157 G NP 1967 3247 m 1890 3135 l 1955 3107 l 1955 3107 l 1967 3247 l P NP 1967 3247 m 1955 3107 l 2023 3228 l 1967 3247 l P 0.657294 G NP 2618 3252 m 2533 3166 l 2525 3130 l 2525 3130 l 2618 3252 l P NP 2618 3252 m 2525 3130 l 2618 3222 l 2618 3252 l P 0.740588 G NP 2525 3257 m 2489 3211 l 2458 3167 l 2458 3167 l 2525 3257 l P NP 2525 3257 m 2495 3217 l 2506 3217 l 2525 3257 l P NP 2525 3257 m 2518 3257 l 2489 3211 l 2489 3211 l 2525 3257 l P 0.845294 G NP 2757 3260 m 2768 3151 l 2856 3195 l 2757 3260 l P NP 2757 3260 m 2742 3251 l 2738 3243 l 2738 3243 l 2706 3197 l 2768 3151 l 2768 3151 l 2757 3260 l P 0.819412 G NP 2749 3266 m 2742 3251 l 2757 3260 l 2749 3266 l P 0.574 G NP 2731 3266 m 2618 3102 l 2738 3243 l 2731 3266 l P NP 2731 3266 m 2618 3132 l 2618 3102 l 2618 3102 l 2731 3266 l P 0.596157 G NP 2100 3270 m 2066 3237 l 2043 3138 l 2043 3138 l 2100 3270 l P NP 2100 3270 m 2065 3190 l 2074 3190 l 2100 3270 l P NP 2100 3270 m 2083 3270 l 2066 3237 l 2066 3237 l 2100 3270 l P 0.89 G NP 4028 3271 m 3903 3234 l 4229 3071 l 4229 3071 l 4028 3271 l P NP 4028 3271 m 4229 3071 l 4374 3103 l 4028 3271 l P 0.733647 G NP 2560 3274 m 2533 3238 l 2551 3238 l 2560 3274 l P 0.731333 G NP 2560 3274 m 2506 3217 l 2487 3176 l 2487 3176 l 2560 3274 l P NP 2560 3274 m 2545 3274 l 2506 3217 l 2506 3217 l 2560 3274 l P 0.819412 G NP 2755 3276 m 2749 3266 l 2757 3260 l 2757 3260 l 2755 3276 l P 0.888824 G NP 3412 3280 m 3529 3121 l 3654 3159 l 3412 3280 l P NP 3412 3280 m 3306 3238 l 3529 3121 l 3529 3121 l 3412 3280 l P 0.657294 G NP 2618 3283 m 2533 3166 l 2618 3252 l 2618 3283 l P NP 2618 3283 m 2542 3202 l 2533 3166 l 2533 3166 l 2618 3283 l P 0.868824 G NP 3034 3285 m 3094 3154 l 3200 3196 l 3034 3285 l P NP 3034 3285 m 2945 3240 l 3094 3154 l 3094 3154 l 3034 3285 l P 0.596157 G NP 1714 3289 m 1795 3185 l 1807 3199 l 1714 3289 l P 0.574 G NP 2723 3289 m 2618 3132 l 2731 3266 l 2723 3289 l P NP 2723 3289 m 2618 3162 l 2618 3132 l 2618 3132 l 2723 3289 l P 0.596157 G NP 1896 3298 m 1807 3199 l 1890 3135 l 1890 3135 l 1896 3298 l P NP 1896 3298 m 1890 3135 l 1967 3247 l 1896 3298 l P 0.740588 G NP 2544 3298 m 2518 3257 l 2525 3257 l 2544 3298 l P NP 2544 3298 m 2542 3298 l 2532 3279 l 2532 3279 l 2544 3298 l P 0.845294 G NP 2829 3307 m 2856 3195 l 2945 3240 l 2829 3307 l P NP 2829 3307 m 2757 3260 l 2856 3195 l 2856 3195 l 2829 3307 l P 0.89 G NP 4153 3309 m 4028 3271 l 4374 3103 l 4374 3103 l 4153 3309 l P NP 4153 3309 m 4374 3103 l 4398 3108 l 4440 3172 l 4153 3309 l P 0.72902 G NP 2569 3310 m 2525 3257 l 2506 3217 l 2506 3217 l 2569 3310 l P 0.731333 G NP 2569 3310 m 2545 3274 l 2560 3274 l 2569 3310 l P NP 2569 3310 m 2557 3310 l 2525 3257 l 2525 3257 l 2569 3310 l P 0.574 G NP 2715 3312 m 2618 3162 l 2723 3289 l 2715 3312 l P NP 2715 3312 m 2618 3192 l 2618 3162 l 2618 3162 l 2715 3312 l P 0.657294 G NP 2618 3313 m 2542 3202 l 2618 3283 l 2618 3313 l P NP 2618 3313 m 2551 3238 l 2542 3202 l 2618 3313 l P 0.888824 G NP 3518 3322 m 3412 3280 l 3654 3159 l 3654 3159 l 3518 3322 l P NP 3518 3322 m 3654 3159 l 3779 3196 l 3518 3322 l P 0.596157 G NP 2129 3324 m 2100 3270 l 2074 3190 l 2074 3190 l 2129 3324 l P 0.868824 G NP 3122 3330 m 3034 3285 l 3200 3196 l 3200 3196 l 3122 3330 l P NP 3122 3330 m 3200 3196 l 3306 3238 l 3122 3330 l P 0.819412 G NP 2783 3334 m 2755 3276 l 2757 3260 l 2757 3260 l 2829 3307 l 2783 3334 l P 0.574 G NP 2707 3335 m 2618 3222 l 2618 3192 l 2618 3192 l 2707 3335 l P NP 2707 3335 m 2618 3192 l 2715 3312 l 2707 3335 l P 0.596157 G NP 2088 3335 m 2028 3237 l 2066 3237 l 2088 3335 l P 0.740588 G NP 2563 3339 m 2542 3298 l 2544 3298 l 2563 3339 l P 0.657294 G NP 2618 3343 m 2560 3274 l 2551 3238 l 2551 3238 l 2618 3343 l P NP 2618 3343 m 2551 3238 l 2618 3313 l 2618 3343 l P 0.731333 G NP 2578 3346 m 2557 3310 l 2569 3310 l 2578 3346 l P 0.724392 G NP 2578 3346 m 2544 3298 l 2525 3257 l 2525 3257 l 2578 3346 l P 0.72902 G NP 2578 3346 m 2569 3346 l 2544 3298 l 2544 3298 l 2578 3346 l P 0.89 G NP 4278 3346 m 4153 3309 l 4440 3172 l 4440 3172 l 4453 3193 l 4278 3346 l P 0.596157 G NP 2041 3346 m 2023 3228 l 2088 3335 l 2041 3346 l P NP 2041 3346 m 1967 3247 l 2023 3228 l 2023 3228 l 2041 3346 l P NP 2125 3350 m 2088 3335 l 2066 3237 l 2066 3237 l 2125 3350 l P NP 2125 3350 m 2083 3270 l 2100 3270 l 2125 3350 l P NP 2125 3350 m 2098 3350 l 2088 3335 l 2088 3335 l 2125 3350 l P 0.845294 G NP 2901 3353 m 2945 3240 l 3034 3285 l 2901 3353 l P NP 2901 3353 m 2829 3307 l 2945 3240 l 2945 3240 l 2901 3353 l P 0.574 G NP 2700 3358 m 2618 3252 l 2618 3222 l 2618 3222 l 2700 3358 l P NP 2700 3358 m 2618 3222 l 2707 3335 l 2700 3358 l P 0.888824 G NP 3624 3363 m 3518 3322 l 3779 3196 l 3779 3196 l 3624 3363 l P NP 3624 3363 m 3779 3196 l 3903 3234 l 3624 3363 l P 0.657294 G NP 2618 3373 m 2560 3274 l 2618 3343 l 2618 3373 l P 0.65498 G NP 2618 3373 m 2569 3310 l 2560 3274 l 2560 3274 l 2618 3373 l P 0.868824 G NP 3211 3374 m 3306 3238 l 3412 3280 l 3211 3374 l P NP 3211 3374 m 3122 3330 l 3306 3238 l 3306 3238 l 3211 3374 l P 0.574 G NP 2692 3381 m 2618 3283 l 2618 3252 l 2618 3252 l 2692 3381 l P NP 2692 3381 m 2618 3252 l 2700 3358 l 2692 3381 l P 0.819412 G NP 2807 3382 m 2783 3334 l 2829 3307 l 2829 3307 l 2807 3382 l P 0.698941 G NP 2587 3382 m 2563 3339 l 2544 3298 l 2544 3298 l 2587 3382 l P 0.72902 G NP 2587 3382 m 2569 3346 l 2578 3346 l 2587 3382 l P 0.724392 G NP 2587 3382 m 2581 3382 l 2563 3339 l 2563 3339 l 2587 3382 l P 0.596157 G NP 1804 3385 m 1807 3199 l 1896 3298 l 1804 3385 l P NP 1804 3385 m 1704 3302 l 1714 3289 l 1807 3199 l 1807 3199 l 1804 3385 l P NP 1979 3387 m 1967 3247 l 2041 3346 l 1979 3387 l P NP 1979 3387 m 1896 3298 l 1967 3247 l 1967 3247 l 1979 3387 l P NP 2157 3387 m 2100 3270 l 2129 3324 l 2157 3387 l P NP 2157 3387 m 2125 3350 l 2100 3270 l 2100 3270 l 2157 3387 l P NP 2157 3387 m 2147 3387 l 2125 3350 l 2125 3350 l 2157 3387 l P 0.819412 G NP 2815 3398 m 2807 3382 l 2829 3307 l 2829 3307 l 2901 3353 l 2815 3398 l P 0.845294 G NP 2974 3400 m 2901 3353 l 3034 3285 l 3034 3285 l 2974 3400 l P NP 2974 3400 m 3034 3285 l 3122 3330 l 2974 3400 l P 0.657294 G NP 2618 3403 m 2569 3310 l 2618 3373 l 2618 3403 l P 0.65498 G NP 2618 3403 m 2578 3346 l 2569 3310 l 2618 3403 l P 0.574 G NP 2684 3404 m 2618 3313 l 2618 3283 l 2618 3283 l 2684 3404 l P NP 2684 3404 m 2618 3283 l 2692 3381 l 2684 3404 l P 0.888824 G NP 3730 3405 m 3903 3234 l 4028 3271 l 3730 3405 l P NP 3730 3405 m 3624 3363 l 3903 3234 l 3903 3234 l 3730 3405 l P 0.596157 G NP 2172 3415 m 2157 3387 l 2129 3324 l 2172 3415 l P 0.724392 G NP 2596 3418 m 2581 3382 l 2587 3382 l 2596 3418 l P 0.698941 G NP 2596 3418 m 2594 3418 l 2589 3400 l 2589 3400 l 2596 3418 l P 0.868824 G NP 3299 3419 m 3412 3280 l 3518 3322 l 3299 3419 l P NP 3299 3419 m 3211 3374 l 3412 3280 l 3412 3280 l 3299 3419 l P 0.574 G NP 2676 3427 m 2618 3343 l 2618 3313 l 2618 3313 l 2676 3427 l P NP 2676 3427 m 2618 3313 l 2684 3404 l 2676 3427 l P 0.488392 G NP 2830 3429 m 2731 3266 l 2738 3243 l 2738 3243 l 2830 3429 l P 0.596157 G NP 2151 3431 m 2098 3350 l 2125 3350 l 2151 3431 l P 0.65498 G NP 2618 3433 m 2578 3346 l 2618 3403 l 2618 3433 l P 0.652667 G NP 2618 3433 m 2587 3382 l 2578 3346 l 2618 3433 l P 0.596157 G NP 2110 3434 m 2041 3346 l 2088 3335 l 2088 3335 l 2110 3434 l P NP 2110 3434 m 2088 3335 l 2151 3431 l 2110 3434 l P 0.488392 G NP 2816 3444 m 2723 3289 l 2731 3266 l 2731 3266 l 2816 3444 l P NP 2816 3444 m 2731 3266 l 2830 3429 l 2816 3444 l P 0.845294 G NP 3046 3446 m 3122 3330 l 3211 3374 l 3046 3446 l P NP 3046 3446 m 2974 3400 l 3122 3330 l 3122 3330 l 3046 3446 l P 0.887647 G NP 3837 3447 m 3730 3405 l 4028 3271 l 4028 3271 l 3837 3447 l P 0.888824 G NP 3837 3447 m 4028 3271 l 4153 3309 l 3837 3447 l P 0.596157 G NP 2184 3450 m 2151 3431 l 2125 3350 l 2125 3350 l 2184 3450 l P NP 2184 3450 m 2147 3387 l 2157 3387 l 2184 3450 l P NP 2184 3450 m 2165 3450 l 2151 3431 l 2151 3431 l 2184 3450 l P 0.574 G NP 2669 3450 m 2618 3343 l 2676 3427 l 2669 3450 l P 0.571686 G NP 2669 3450 m 2618 3373 l 2618 3343 l 2618 3343 l 2669 3450 l P 0.819412 G NP 2857 3452 m 2831 3431 l 2830 3429 l 2830 3429 l 2815 3398 l 2901 3353 l 2901 3353 l 2857 3452 l P NP 2857 3452 m 2901 3353 l 2974 3400 l 2857 3452 l P 0.698941 G NP 2605 3454 m 2594 3418 l 2596 3418 l 2605 3454 l P 0.488392 G NP 2802 3458 m 2723 3289 l 2816 3444 l 2802 3458 l P NP 2802 3458 m 2715 3312 l 2723 3289 l 2723 3289 l 2802 3458 l P 0.791176 G NP 2840 3460 m 2831 3431 l 2857 3452 l 2840 3460 l P 0.596157 G NP 1901 3460 m 1896 3298 l 1979 3387 l 1901 3460 l P NP 1901 3460 m 1804 3385 l 1896 3298 l 1896 3298 l 1901 3460 l P 0.645725 G NP 2618 3464 m 2596 3418 l 2587 3382 l 2618 3464 l P 0.65498 G NP 2618 3464 m 2587 3382 l 2618 3433 l 2618 3464 l P 0.868824 G NP 3388 3464 m 3518 3322 l 3624 3363 l 3388 3464 l P NP 3388 3464 m 3299 3419 l 3518 3322 l 3518 3322 l 3388 3464 l P 0.596157 G NP 2059 3465 m 1979 3387 l 2041 3346 l 2041 3346 l 2059 3465 l P NP 2059 3465 m 2041 3346 l 2110 3434 l 2059 3465 l P 0.488392 G NP 2787 3473 m 2715 3312 l 2802 3458 l 2787 3473 l P NP 2787 3473 m 2707 3335 l 2715 3312 l 2715 3312 l 2787 3473 l P 0.571686 G NP 2661 3473 m 2618 3403 l 2618 3373 l 2618 3373 l 2661 3473 l P 0.574 G NP 2661 3473 m 2618 3373 l 2669 3450 l 2661 3473 l P 0.79 G NP 2845 3475 m 2840 3460 l 2857 3452 l 2857 3452 l 2845 3475 l P 0.488392 G NP 2773 3487 m 2707 3335 l 2787 3473 l 2773 3487 l P NP 2773 3487 m 2700 3358 l 2707 3335 l 2707 3335 l 2773 3487 l P 0.596157 G NP 2214 3488 m 2184 3450 l 2157 3387 l 2157 3387 l 2214 3488 l P 0.888824 G NP 3943 3489 m 4153 3309 l 4278 3346 l 3943 3489 l P 0.887647 G NP 3943 3489 m 3837 3447 l 4153 3309 l 4153 3309 l 3943 3489 l P 0.845294 G NP 3118 3493 m 3211 3374 l 3299 3419 l 3118 3493 l P NP 3118 3493 m 3046 3446 l 3211 3374 l 3211 3374 l 3118 3493 l P 0.652667 G NP 2618 3494 m 2596 3418 l 2618 3464 l 2618 3494 l P 0.617961 G NP 2618 3494 m 2605 3454 l 2596 3418 l 2596 3418 l 2618 3494 l P 0.571686 G NP 2653 3496 m 2618 3403 l 2661 3473 l 2653 3496 l P NP 2653 3496 m 2618 3433 l 2618 3403 l 2618 3403 l 2653 3496 l P 0.819412 G NP 2914 3499 m 2974 3400 l 3046 3446 l 2914 3499 l P NP 2914 3499 m 2857 3452 l 2974 3400 l 2974 3400 l 2914 3499 l P 0.488392 G NP 2759 3502 m 2692 3381 l 2700 3358 l 2700 3358 l 2759 3502 l P NP 2759 3502 m 2700 3358 l 2773 3487 l 2759 3502 l P 0.868824 G NP 3476 3509 m 3388 3464 l 3624 3363 l 3624 3363 l 3476 3509 l P NP 3476 3509 m 3624 3363 l 3730 3405 l 3476 3509 l P 0.596157 G NP 2177 3511 m 2110 3434 l 2151 3431 l 2151 3431 l 2177 3511 l P NP 1690 3512 m 1630 3478 l 1699 3308 l 1702 3304 l 1702 3304 l 1690 3512 l P NP 1690 3512 m 1702 3304 l 1704 3302 l 1704 3302 l 1804 3385 l 1690 3512 l P NP 2212 3513 m 2165 3450 l 2184 3450 l 2212 3513 l P NP 2212 3513 m 2177 3511 l 2151 3431 l 2151 3431 l 2212 3513 l P NP 2212 3513 m 2179 3513 l 2177 3511 l 2177 3511 l 2212 3513 l P 0.488392 G NP 2745 3516 m 2692 3381 l 2759 3502 l 2745 3516 l P NP 2745 3516 m 2684 3404 l 2692 3381 l 2692 3381 l 2745 3516 l P 0.571686 G NP 2645 3519 m 2618 3433 l 2653 3496 l 2645 3519 l P 0.569373 G NP 2645 3519 m 2618 3464 l 2618 3433 l 2618 3433 l 2645 3519 l P 0.791176 G NP 2859 3520 m 2845 3475 l 2857 3452 l 2857 3452 l 2914 3499 l 2859 3520 l P 0.645725 G NP 2618 3524 m 2605 3454 l 2618 3494 l 2618 3524 l P 0.596157 G NP 1992 3527 m 1901 3460 l 1979 3387 l 1979 3387 l 1992 3527 l P NP 1992 3527 m 1979 3387 l 2059 3465 l 1992 3527 l P 0.887647 G NP 4049 3531 m 3943 3489 l 4278 3346 l 4278 3346 l 4049 3531 l P 0.488392 G NP 2730 3531 m 2684 3404 l 2745 3516 l 2730 3531 l P NP 2730 3531 m 2676 3427 l 2684 3404 l 2684 3404 l 2730 3531 l P 0.596157 G NP 2133 3533 m 2059 3465 l 2110 3434 l 2110 3434 l 2133 3533 l P NP 2133 3533 m 2110 3434 l 2177 3511 l 2133 3533 l P NP 2242 3535 m 2212 3513 l 2184 3450 l 2242 3535 l P NP 2242 3535 m 2184 3450 l 2214 3488 l 2242 3535 l P 0.845294 G NP 3190 3539 m 3118 3493 l 3299 3419 l 3299 3419 l 3190 3539 l P NP 3190 3539 m 3299 3419 l 3388 3464 l 3190 3539 l P 0.567059 G NP 2638 3542 m 2618 3494 l 2618 3464 l 2618 3464 l 2638 3542 l P 0.571686 G NP 2638 3542 m 2618 3464 l 2645 3519 l 2638 3542 l P 0.488392 G NP 2716 3545 m 2676 3427 l 2730 3531 l 2716 3545 l P NP 2716 3545 m 2669 3450 l 2676 3427 l 2676 3427 l 2716 3545 l P 0.819412 G NP 2971 3546 m 3046 3446 l 3118 3493 l 2971 3546 l P NP 2971 3546 m 2914 3499 l 3046 3446 l 3046 3446 l 2971 3546 l P 0.868824 G NP 3565 3554 m 3476 3509 l 3730 3405 l 3730 3405 l 3565 3554 l P NP 3565 3554 m 3730 3405 l 3837 3447 l 3565 3554 l P 0.617961 G NP 2618 3554 m 2615 3508 l 2618 3524 l 2618 3554 l P 0.59498 G NP 2258 3555 m 2242 3535 l 2214 3488 l 2258 3555 l P 0.79 G NP 2872 3559 m 2859 3520 l 2914 3499 l 2914 3499 l 2872 3559 l P 0.486078 G NP 2702 3560 m 2661 3473 l 2669 3450 l 2669 3450 l 2702 3560 l P 0.488392 G NP 2702 3560 m 2669 3450 l 2716 3545 l 2702 3560 l P 0.569373 G NP 2630 3565 m 2618 3494 l 2638 3542 l 2630 3565 l P 0.562431 G NP 2630 3565 m 2618 3524 l 2618 3494 l 2618 3494 l 2630 3565 l P 0.596157 G NP 1801 3571 m 1690 3512 l 1804 3385 l 1804 3385 l 1801 3571 l P NP 1801 3571 m 1804 3385 l 1901 3460 l 1801 3571 l P 0.488392 G NP 2687 3574 m 2661 3473 l 2702 3560 l 2687 3574 l P 0.486078 G NP 2687 3574 m 2653 3496 l 2661 3473 l 2661 3473 l 2687 3574 l P 0.596157 G NP 2240 3576 m 2179 3513 l 2212 3513 l 2240 3576 l P 0.79 G NP 2877 3576 m 2872 3559 l 2914 3499 l 2914 3499 l 2971 3546 l 2877 3576 l P 0.596157 G NP 2271 3582 m 2240 3576 l 2212 3513 l 2212 3513 l 2271 3582 l P NP 2271 3582 m 2212 3513 l 2242 3535 l 2271 3582 l P NP 2271 3582 m 2247 3582 l 2240 3576 l 2240 3576 l 2271 3582 l P NP 2076 3584 m 1992 3527 l 2059 3465 l 2059 3465 l 2076 3584 l P NP 2076 3584 m 2059 3465 l 2133 3533 l 2076 3584 l P 0.845294 G NP 3263 3586 m 3190 3539 l 3388 3464 l 3388 3464 l 3263 3586 l P NP 3263 3586 m 3388 3464 l 3476 3509 l 3263 3586 l P 0.532353 G NP 2622 3588 m 2618 3554 l 2618 3524 l 2618 3524 l 2622 3588 l P 0.567059 G NP 2622 3588 m 2618 3524 l 2630 3565 l 2622 3588 l P 0.486078 G NP 2673 3589 m 2653 3496 l 2687 3574 l 2673 3589 l P NP 2673 3589 m 2645 3519 l 2653 3496 l 2653 3496 l 2673 3589 l P 0.596157 G NP 2203 3591 m 2133 3533 l 2177 3511 l 2177 3511 l 2203 3591 l P NP 2203 3591 m 2177 3511 l 2240 3576 l 2203 3591 l P 0.819412 G NP 3028 3593 m 3118 3493 l 3190 3539 l 3028 3593 l P NP 3028 3593 m 2971 3546 l 3118 3493 l 3118 3493 l 3028 3593 l P 0.868824 G NP 3654 3598 m 3837 3447 l 3943 3489 l 3654 3598 l P NP 3654 3598 m 3565 3554 l 3837 3447 l 3837 3447 l 3654 3598 l P 0.486078 G NP 2659 3604 m 2638 3542 l 2645 3519 l 2645 3519 l 2659 3604 l P NP 2659 3604 m 2645 3519 l 2673 3589 l 2659 3604 l P 0.586745 G NP 2299 3606 m 2271 3582 l 2242 3535 l 2299 3606 l P 0.562431 G NP 2614 3611 m 2618 3554 l 2622 3588 l 2614 3611 l P 0.486078 G NP 2645 3618 m 2638 3542 l 2659 3604 l 2645 3618 l P 0.483765 G NP 2645 3618 m 2630 3565 l 2638 3542 l 2638 3542 l 2645 3618 l P 0.596157 G NP 1906 3623 m 1801 3571 l 1901 3460 l 1901 3460 l 1906 3623 l P NP 1906 3623 m 1901 3460 l 1992 3527 l 1906 3623 l P NP 2299 3629 m 2247 3582 l 2271 3582 l 2299 3629 l P 0.460863 G NP 2894 3629 m 2816 3444 l 2830 3429 l 2830 3429 l 2894 3629 l P 0.79 G NP 2898 3630 m 2971 3546 l 3028 3593 l 2898 3630 l P NP 2898 3630 m 2893 3625 l 2877 3576 l 2971 3546 l 2971 3546 l 2898 3630 l P 0.761765 G NP 2895 3630 m 2894 3629 l 2893 3625 l 2898 3630 l 2895 3630 l P NP 2895 3632 m 2895 3630 l 2898 3630 l 2898 3630 l 2895 3632 l P 0.596157 G NP 2155 3632 m 2133 3533 l 2203 3591 l 2155 3632 l P NP 2155 3632 m 2076 3584 l 2133 3533 l 2133 3533 l 2155 3632 l P 0.845294 G NP 3335 3633 m 3263 3586 l 3476 3509 l 3476 3509 l 3335 3633 l P NP 3335 3633 m 3476 3509 l 3565 3554 l 3335 3633 l P 0.486078 G NP 2630 3633 m 2630 3565 l 2645 3618 l 2630 3633 l P 0.481451 G NP 2630 3633 m 2622 3588 l 2630 3565 l 2630 3565 l 2630 3633 l P 0.460863 G NP 2875 3634 m 2816 3444 l 2894 3629 l 2875 3634 l P NP 2875 3634 m 2802 3458 l 2816 3444 l 2816 3444 l 2875 3634 l P 0.532353 G NP 2607 3634 m 2615 3598 l 2614 3611 l 2607 3634 l P 0.592627 G NP 2327 3638 m 2271 3582 l 2299 3606 l 2327 3638 l P 0.585569 G NP 2327 3638 m 2299 3629 l 2271 3582 l 2271 3582 l 2327 3638 l P NP 2327 3638 m 2312 3638 l 2299 3629 l 2299 3629 l 2327 3638 l P 0.460863 G NP 2855 3638 m 2802 3458 l 2875 3634 l 2855 3638 l P NP 2855 3638 m 2787 3473 l 2802 3458 l 2802 3458 l 2855 3638 l P 0.596157 G NP 2268 3639 m 2203 3591 l 2240 3576 l 2240 3576 l 2268 3639 l P NP 2268 3639 m 2240 3576 l 2299 3629 l 2268 3639 l P 0.819412 G NP 3085 3640 m 3190 3539 l 3263 3586 l 3085 3640 l P NP 3085 3640 m 3028 3593 l 3190 3539 l 3190 3539 l 3085 3640 l P 0.460863 G NP 2835 3643 m 2787 3473 l 2855 3638 l 2835 3643 l P NP 2835 3643 m 2773 3487 l 2787 3473 l 2787 3473 l 2835 3643 l P 0.868824 G NP 3742 3643 m 3654 3598 l 3943 3489 l 3943 3489 l 3742 3643 l P NP 3742 3643 m 3943 3489 l 4049 3531 l 3742 3643 l P 0.483765 G NP 2616 3647 m 2622 3588 l 2630 3633 l 2616 3647 l P 0.476824 G NP 2616 3647 m 2614 3611 l 2622 3588 l 2622 3588 l 2616 3647 l P 0.460863 G NP 2816 3648 m 2759 3502 l 2773 3487 l 2773 3487 l 2816 3648 l P NP 2816 3648 m 2773 3487 l 2835 3643 l 2816 3648 l P 0.570275 G NP 2342 3650 m 2327 3638 l 2299 3606 l 2342 3650 l P 0.460863 G NP 2796 3653 m 2745 3516 l 2759 3502 l 2759 3502 l 2796 3653 l P NP 2796 3653 m 2759 3502 l 2816 3648 l 2796 3653 l P NP 2777 3658 m 2745 3516 l 2796 3653 l 2777 3658 l P NP 2777 3658 m 2730 3531 l 2745 3516 l 2745 3516 l 2777 3658 l P 0.449059 G NP 2602 3662 m 2607 3634 l 2614 3611 l 2614 3611 l 2602 3662 l P 0.481451 G NP 2602 3662 m 2614 3611 l 2616 3647 l 2602 3662 l P 0.460863 G NP 2757 3662 m 2730 3531 l 2777 3658 l 2757 3662 l P NP 2757 3662 m 2716 3545 l 2730 3531 l 2730 3531 l 2757 3662 l P 0.596157 G NP 2004 3667 m 1906 3623 l 1992 3527 l 1992 3527 l 2004 3667 l P NP 2004 3667 m 1992 3527 l 2076 3584 l 2004 3667 l P 0.460863 G NP 2738 3667 m 2702 3560 l 2716 3545 l 2716 3545 l 2738 3667 l P 0.462039 G NP 2738 3667 m 2716 3545 l 2757 3662 l 2738 3667 l P 0.585569 G NP 2355 3670 m 2312 3638 l 2327 3638 l 2355 3670 l P 0.596157 G NP 2228 3671 m 2155 3632 l 2203 3591 l 2203 3591 l 2228 3671 l P NP 2228 3671 m 2203 3591 l 2268 3639 l 2228 3671 l P 0.462039 G NP 2718 3672 m 2687 3574 l 2702 3560 l 2702 3560 l 2718 3672 l P NP 2718 3672 m 2702 3560 l 2738 3667 l 2718 3672 l P 0.596157 G NP 2328 3676 m 2268 3639 l 2299 3629 l 2299 3629 l 2328 3676 l P 0.584392 G NP 2328 3676 m 2299 3629 l 2355 3670 l 2328 3676 l P 0.79 G NP 2941 3676 m 2898 3630 l 3028 3593 l 3028 3593 l 2941 3676 l P NP 2941 3676 m 3028 3593 l 3085 3640 l 2941 3676 l P 0.47451 G NP 2587 3676 m 2607 3634 l 2602 3662 l 2587 3676 l P 0.451412 G NP 2587 3676 m 2584 3676 l 2592 3666 l 2592 3666 l 2587 3676 l P 0.462039 G NP 2698 3677 m 2673 3589 l 2687 3574 l 2687 3574 l 2698 3677 l P NP 2698 3677 m 2687 3574 l 2718 3672 l 2698 3677 l P 0.845294 G NP 3407 3679 m 3565 3554 l 3654 3598 l 3407 3679 l P NP 3407 3679 m 3335 3633 l 3565 3554 l 3565 3554 l 3407 3679 l P 0.560863 G NP 2381 3682 m 2355 3670 l 2327 3638 l 2327 3638 l 2381 3682 l P 0.566745 G NP 2381 3682 m 2375 3682 l 2355 3670 l 2355 3670 l 2381 3682 l P 0.462039 G NP 2679 3682 m 2659 3604 l 2673 3589 l 2673 3589 l 2679 3682 l P NP 2679 3682 m 2673 3589 l 2698 3677 l 2679 3682 l P 0.596157 G NP 1548 3682 m 1630 3478 l 1690 3512 l 1548 3682 l P 0.761765 G NP 2903 3683 m 2895 3632 l 2898 3630 l 2898 3630 l 2941 3676 l 2903 3683 l P 0.462039 G NP 2659 3686 m 2659 3604 l 2679 3682 l 2659 3686 l P NP 2659 3686 m 2645 3618 l 2659 3604 l 2659 3604 l 2659 3686 l P 0.819412 G NP 3142 3687 m 3085 3640 l 3263 3586 l 3263 3586 l 3142 3687 l P NP 3142 3687 m 3263 3586 l 3335 3633 l 3142 3687 l P 0.868824 G NP 3831 3688 m 3742 3643 l 4049 3531 l 4049 3531 l 3831 3688 l P 0.451412 G NP 2573 3691 m 2584 3676 l 2587 3676 l 2587 3676 l 2573 3691 l P 0.462039 G NP 2640 3691 m 2645 3618 l 2659 3686 l 2640 3691 l P NP 2640 3691 m 2630 3633 l 2645 3618 l 2645 3618 l 2640 3691 l P NP 2620 3696 m 2616 3647 l 2630 3633 l 2630 3633 l 2620 3696 l P NP 2620 3696 m 2630 3633 l 2640 3691 l 2620 3696 l P 0.566745 G NP 2408 3700 m 2375 3682 l 2381 3682 l 2408 3700 l P 0.463216 G NP 2600 3701 m 2602 3662 l 2616 3647 l 2600 3701 l P NP 2600 3701 m 2616 3647 l 2620 3696 l 2600 3701 l P 0.596157 G NP 2295 3702 m 2228 3671 l 2268 3639 l 2268 3639 l 2295 3702 l P NP 2295 3702 m 2268 3639 l 2328 3676 l 2295 3702 l P 0.761765 G NP 2907 3702 m 2903 3683 l 2941 3676 l 2941 3676 l 2907 3702 l P 0.55851 G NP 2383 3702 m 2355 3670 l 2408 3700 l 2383 3702 l P 0.584392 G NP 2383 3702 m 2328 3676 l 2355 3670 l 2355 3670 l 2383 3702 l P 0.596157 G NP 2094 3703 m 2076 3584 l 2155 3632 l 2094 3703 l P NP 2094 3703 m 2004 3667 l 2076 3584 l 2076 3584 l 2094 3703 l P 0.540863 G NP 2421 3705 m 2408 3700 l 2381 3682 l 2421 3705 l P 0.463216 G NP 2581 3706 m 2602 3662 l 2600 3701 l 2581 3706 l P 0.464392 G NP 2581 3706 m 2587 3676 l 2602 3662 l 2581 3706 l P 0.471451 G NP 2561 3710 m 2573 3691 l 2587 3676 l 2561 3710 l P 0.464392 G NP 2561 3710 m 2587 3676 l 2581 3706 l 2561 3710 l P 0.466745 G NP 2542 3715 m 2573 3691 l 2561 3710 l 2542 3715 l P 0.477333 G NP 2542 3715 m 2534 3715 l 2550 3709 l 2550 3709 l 2542 3715 l P 0.559686 G NP 2434 3719 m 2383 3702 l 2408 3700 l 2408 3700 l 2434 3719 l P 0.530275 G NP 2434 3719 m 2408 3700 l 2457 3718 l 2434 3719 l P 0.477333 G NP 2522 3720 m 2534 3715 l 2542 3715 l 2542 3715 l 2522 3720 l P 0.79 G NP 2984 3722 m 3085 3640 l 3142 3687 l 2984 3722 l P NP 2984 3722 m 2941 3676 l 3085 3640 l 3085 3640 l 2984 3722 l P 0.596157 G NP 2356 3722 m 2295 3702 l 2328 3676 l 2328 3676 l 2356 3722 l P 0.584392 G NP 2356 3722 m 2328 3676 l 2383 3702 l 2356 3722 l P 0.596157 G NP 1677 3723 m 1690 3512 l 1801 3571 l 1677 3723 l P NP 1677 3723 m 1548 3682 l 1690 3512 l 1690 3512 l 1677 3723 l P 0.537333 G NP 2481 3725 m 2434 3719 l 2457 3718 l 2457 3718 l 2481 3725 l P 0.509098 G NP 2481 3725 m 2457 3718 l 2492 3723 l 2481 3725 l P 0.845294 G NP 3479 3726 m 3407 3679 l 3654 3598 l 3654 3598 l 3479 3726 l P NP 3479 3726 m 3654 3598 l 3742 3643 l 3479 3726 l P 0.497333 G NP 2504 3731 m 2481 3725 l 2522 3720 l 2522 3720 l 2504 3731 l P 0.504392 G NP 2504 3731 m 2522 3720 l 2542 3715 l 2542 3715 l 2504 3731 l P 0.596157 G NP 2178 3731 m 2094 3703 l 2155 3632 l 2155 3632 l 2178 3731 l P NP 2178 3731 m 2155 3632 l 2228 3671 l 2178 3731 l P 0.761765 G NP 2912 3732 m 2907 3702 l 2941 3676 l 2941 3676 l 2984 3722 l 2912 3732 l P 0.557333 G NP 2412 3734 m 2383 3702 l 2434 3719 l 2412 3734 l P 0.584392 G NP 2412 3734 m 2356 3722 l 2383 3702 l 2383 3702 l 2412 3734 l P 0.819412 G NP 3200 3734 m 3335 3633 l 3407 3679 l 3200 3734 l P NP 3200 3734 m 3142 3687 l 3335 3633 l 3335 3633 l 3200 3734 l P 0.496157 G NP 2528 3737 m 2542 3715 l 2561 3710 l 2528 3737 l P 0.49498 G NP 2528 3737 m 2504 3731 l 2542 3715 l 2542 3715 l 2528 3737 l P 0.529098 G NP 2461 3737 m 2481 3725 l 2504 3731 l 2461 3737 l P 0.557333 G NP 2461 3737 m 2412 3734 l 2434 3719 l 2434 3719 l 2461 3737 l P 0.527922 G NP 2461 3737 m 2434 3719 l 2481 3725 l 2461 3737 l P 0.493804 G NP 2552 3743 m 2561 3710 l 2581 3706 l 2552 3743 l P NP 2552 3743 m 2528 3737 l 2561 3710 l 2561 3710 l 2552 3743 l P NP 2575 3749 m 2552 3743 l 2581 3706 l 2581 3706 l 2575 3749 l P NP 2575 3749 m 2581 3706 l 2600 3701 l 2575 3749 l P 0.596157 G NP 2254 3752 m 2228 3671 l 2295 3702 l 2254 3752 l P NP 2254 3752 m 2178 3731 l 2228 3671 l 2228 3671 l 2254 3752 l P 0.492627 G NP 2599 3756 m 2575 3749 l 2600 3701 l 2600 3701 l 2599 3756 l P NP 2599 3756 m 2600 3701 l 2620 3696 l 2599 3756 l P 0.526745 G NP 2487 3756 m 2504 3731 l 2528 3737 l 2487 3756 l P NP 2487 3756 m 2461 3737 l 2504 3731 l 2504 3731 l 2487 3756 l P 0.596157 G NP 1799 3757 m 1801 3571 l 1906 3623 l 1799 3757 l P NP 1799 3757 m 1677 3723 l 1801 3571 l 1801 3571 l 1799 3757 l P 0.492627 G NP 2623 3762 m 2620 3696 l 2640 3691 l 2640 3691 l 2623 3762 l P NP 2623 3762 m 2599 3756 l 2620 3696 l 2620 3696 l 2623 3762 l P 0.596157 G NP 2323 3764 m 2295 3702 l 2356 3722 l 2323 3764 l P NP 2323 3764 m 2254 3752 l 2295 3702 l 2295 3702 l 2323 3764 l P 0.761765 G NP 2917 3765 m 2912 3732 l 2984 3722 l 2984 3722 l 2917 3765 l P 0.557333 G NP 2440 3766 m 2461 3737 l 2487 3756 l 2440 3766 l P NP 2440 3766 m 2412 3734 l 2461 3737 l 2440 3766 l P 0.492627 G NP 2646 3768 m 2623 3762 l 2640 3691 l 2646 3768 l P NP 2646 3768 m 2640 3691 l 2659 3686 l 2659 3686 l 2646 3768 l P 0.79 G NP 3027 3768 m 2984 3722 l 3142 3687 l 3142 3687 l 3027 3768 l P NP 3027 3768 m 3142 3687 l 3200 3734 l 3027 3768 l P 0.583216 G NP 2385 3769 m 2356 3722 l 2412 3734 l 2385 3769 l P 0.596157 G NP 2385 3769 m 2323 3764 l 2356 3722 l 2356 3722 l 2385 3769 l P 0.583216 G NP 2385 3769 m 2412 3734 l 2440 3766 l 2385 3769 l P 0.845294 G NP 3552 3772 m 3742 3643 l 3831 3688 l 3552 3772 l P NP 3552 3772 m 3479 3726 l 3742 3643 l 3742 3643 l 3552 3772 l P 0.525569 G NP 2514 3774 m 2528 3737 l 2552 3743 l 2514 3774 l P NP 2514 3774 m 2487 3756 l 2528 3737 l 2528 3737 l 2514 3774 l P 0.492627 G NP 2670 3774 m 2646 3768 l 2659 3686 l 2670 3774 l P NP 2670 3774 m 2659 3686 l 2679 3682 l 2679 3682 l 2670 3774 l P NP 2694 3780 m 2679 3682 l 2698 3677 l 2698 3677 l 2694 3780 l P NP 2694 3780 m 2670 3774 l 2679 3682 l 2694 3780 l P 0.761765 G NP 2920 3781 m 2917 3765 l 2984 3722 l 2984 3722 l 3027 3768 l 2920 3781 l P 0.819412 G NP 3257 3781 m 3200 3734 l 3407 3679 l 3407 3679 l 3257 3781 l P NP 3257 3781 m 3407 3679 l 3479 3726 l 3257 3781 l P 0.596157 G NP 1912 3785 m 1906 3623 l 2004 3667 l 1912 3785 l P NP 1912 3785 m 1799 3757 l 1906 3623 l 1906 3623 l 1912 3785 l P 0.492627 G NP 2717 3786 m 2694 3780 l 2698 3677 l 2698 3677 l 2717 3786 l P NP 2717 3786 m 2698 3677 l 2718 3672 l 2718 3672 l 2717 3786 l P 0.525569 G NP 2540 3792 m 2514 3774 l 2552 3743 l 2552 3743 l 2540 3792 l P NP 2540 3792 m 2552 3743 l 2575 3749 l 2540 3792 l P 0.492627 G NP 2741 3793 m 2718 3672 l 2738 3667 l 2738 3667 l 2741 3793 l P NP 2741 3793 m 2717 3786 l 2718 3672 l 2718 3672 l 2741 3793 l P 0.556157 G NP 2468 3798 m 2440 3766 l 2487 3756 l 2487 3756 l 2468 3798 l P NP 2468 3798 m 2487 3756 l 2514 3774 l 2468 3798 l P 0.492627 G NP 2765 3799 m 2738 3667 l 2757 3662 l 2757 3662 l 2765 3799 l P NP 2765 3799 m 2741 3793 l 2738 3667 l 2738 3667 l 2765 3799 l P NP 2788 3805 m 2757 3662 l 2777 3658 l 2777 3658 l 2788 3805 l P NP 2788 3805 m 2765 3799 l 2757 3662 l 2788 3805 l P 0.596157 G NP 2016 3807 m 1912 3785 l 2004 3667 l 2004 3667 l 2016 3807 l P NP 2016 3807 m 2004 3667 l 2094 3703 l 2016 3807 l P 0.525569 G NP 2567 3811 m 2575 3749 l 2599 3756 l 2567 3811 l P NP 2567 3811 m 2540 3792 l 2575 3749 l 2575 3749 l 2567 3811 l P 0.492627 G NP 2812 3811 m 2777 3658 l 2796 3653 l 2796 3653 l 2812 3811 l P NP 2812 3811 m 2788 3805 l 2777 3658 l 2777 3658 l 2812 3811 l P 0.79 G NP 3071 3814 m 3027 3768 l 3200 3734 l 3200 3734 l 3071 3814 l P NP 3071 3814 m 3200 3734 l 3257 3781 l 3071 3814 l P 0.583216 G NP 2414 3816 m 2440 3766 l 2468 3798 l 2414 3816 l P NP 2414 3816 m 2385 3769 l 2440 3766 l 2440 3766 l 2414 3816 l P 0.492627 G NP 2836 3817 m 2812 3811 l 2796 3653 l 2796 3653 l 2836 3817 l P 0.491451 G NP 2836 3817 m 2796 3653 l 2816 3648 l 2816 3648 l 2836 3817 l P 0.845294 G NP 3624 3819 m 3552 3772 l 3831 3688 l 3831 3688 l 3624 3819 l P 0.596157 G NP 2112 3822 m 2094 3703 l 2178 3731 l 2112 3822 l P NP 2112 3822 m 2016 3807 l 2094 3703 l 2094 3703 l 2112 3822 l P 0.761765 G NP 2927 3823 m 2920 3781 l 3027 3768 l 3027 3768 l 2927 3823 l P 0.491451 G NP 2859 3824 m 2816 3648 l 2835 3643 l 2835 3643 l 2859 3824 l P 0.492627 G NP 2859 3824 m 2836 3817 l 2816 3648 l 2816 3648 l 2859 3824 l P 0.596157 G NP 2351 3827 m 2385 3769 l 2414 3816 l 2351 3827 l P NP 2351 3827 m 2323 3764 l 2385 3769 l 2351 3827 l P 0.761765 G NP 2928 3828 m 2927 3823 l 3027 3768 l 3027 3768 l 3071 3814 l 2928 3828 l P 0.819412 G NP 3314 3828 m 3257 3781 l 3479 3726 l 3479 3726 l 3314 3828 l P NP 3314 3828 m 3479 3726 l 3552 3772 l 3314 3828 l P 0.525569 G NP 2593 3829 m 2599 3756 l 2623 3762 l 2593 3829 l P NP 2593 3829 m 2567 3811 l 2599 3756 l 2599 3756 l 2593 3829 l P 0.491451 G NP 2883 3830 m 2835 3643 l 2855 3638 l 2855 3638 l 2883 3830 l P 0.492627 G NP 2883 3830 m 2859 3824 l 2835 3643 l 2835 3643 l 2883 3830 l P 0.596157 G NP 2200 3830 m 2112 3822 l 2178 3731 l 2178 3731 l 2200 3830 l P NP 2200 3830 m 2178 3731 l 2254 3752 l 2200 3830 l P 0.556157 G NP 2496 3830 m 2468 3798 l 2514 3774 l 2514 3774 l 2496 3830 l P NP 2496 3830 m 2514 3774 l 2540 3792 l 2496 3830 l P 0.596157 G NP 2280 3832 m 2200 3830 l 2254 3752 l 2254 3752 l 2280 3832 l P NP 2280 3832 m 2254 3752 l 2323 3764 l 2280 3832 l P NP 2280 3832 m 2323 3764 l 2351 3827 l 2280 3832 l P 0.491451 G NP 2907 3836 m 2855 3638 l 2875 3634 l 2875 3634 l 2907 3836 l P NP 2907 3836 m 2883 3830 l 2855 3638 l 2855 3638 l 2907 3836 l P NP 2930 3842 m 2907 3836 l 2875 3634 l 2875 3634 l 2930 3842 l P NP 2930 3842 m 2875 3634 l 2894 3629 l 2894 3629 l 2930 3842 l P 0.525569 G NP 2620 3848 m 2623 3762 l 2646 3768 l 2620 3848 l P NP 2620 3848 m 2593 3829 l 2623 3762 l 2623 3762 l 2620 3848 l P 0.79 G NP 3114 3861 m 3257 3781 l 3314 3828 l 3114 3861 l P NP 3114 3861 m 3071 3814 l 3257 3781 l 3257 3781 l 3114 3861 l P 0.556157 G NP 2524 3862 m 2496 3830 l 2540 3792 l 2540 3792 l 2524 3862 l P NP 2524 3862 m 2540 3792 l 2567 3811 l 2524 3862 l P 0.583216 G NP 2442 3863 m 2414 3816 l 2468 3798 l 2468 3798 l 2442 3863 l P NP 2442 3863 m 2468 3798 l 2496 3830 l 2442 3863 l P 0.525569 G NP 2647 3866 m 2620 3848 l 2646 3768 l 2646 3768 l 2647 3866 l P 0.524392 G NP 2647 3866 m 2646 3768 l 2670 3774 l 2647 3866 l P 0.740588 G NP 2948 3873 m 2931 3872 l 2930 3847 l 2930 3847 l 2948 3873 l P 0.761765 G NP 2948 3873 m 3071 3814 l 3114 3861 l 2948 3873 l P NP 2948 3873 m 2930 3847 l 2930 3842 l 2930 3842 l 2928 3828 l 3071 3814 l 3071 3814 l 2948 3873 l P 0.819412 G NP 3371 3875 m 3552 3772 l 3624 3819 l 3371 3875 l P NP 3371 3875 m 3314 3828 l 3552 3772 l 3552 3772 l 3371 3875 l P 0.740588 G NP 2931 3878 m 2931 3872 l 2948 3873 l 2931 3878 l P 0.524392 G NP 2673 3885 m 2647 3866 l 2670 3774 l 2670 3774 l 2673 3885 l P NP 2673 3885 m 2670 3774 l 2694 3780 l 2673 3885 l P 0.596157 G NP 2378 3890 m 2414 3816 l 2442 3863 l 2378 3890 l P NP 2378 3890 m 2351 3827 l 2414 3816 l 2414 3816 l 2378 3890 l P 0.556157 G NP 2552 3894 m 2524 3862 l 2567 3811 l 2567 3811 l 2552 3894 l P NP 2552 3894 m 2567 3811 l 2593 3829 l 2552 3894 l P 0.524392 G NP 2700 3903 m 2673 3885 l 2694 3780 l 2694 3780 l 2700 3903 l P NP 2700 3903 m 2694 3780 l 2717 3786 l 2700 3903 l P 0.79 G NP 3157 3907 m 3314 3828 l 3371 3875 l 3157 3907 l P NP 3157 3907 m 3114 3861 l 3314 3828 l 3314 3828 l 3157 3907 l P 0.583216 G NP 2471 3910 m 2442 3863 l 2496 3830 l 2496 3830 l 2471 3910 l P NP 2471 3910 m 2496 3830 l 2524 3862 l 2471 3910 l P 0.596157 G NP 2305 3912 m 2280 3832 l 2351 3827 l 2351 3827 l 2305 3912 l P NP 2305 3912 m 2351 3827 l 2378 3890 l 2305 3912 l P 0.761765 G NP 2979 3917 m 2948 3873 l 3114 3861 l 3114 3861 l 2979 3917 l P 0.740588 G NP 2979 3917 m 2933 3914 l 2931 3878 l 2948 3873 l 2948 3873 l 2979 3917 l P 0.761765 G NP 2979 3917 m 3114 3861 l 3157 3907 l 2979 3917 l P 0.596157 G NP 1525 3919 m 1548 3682 l 1677 3723 l 1525 3919 l P 0.524392 G NP 2726 3922 m 2717 3786 l 2741 3793 l 2726 3922 l P NP 2726 3922 m 2700 3903 l 2717 3786 l 2717 3786 l 2726 3922 l P 0.819412 G NP 3428 3922 m 3371 3875 l 3624 3819 l 3624 3819 l 3428 3922 l P 0.556157 G NP 2581 3926 m 2552 3894 l 2593 3829 l 2593 3829 l 2581 3926 l P NP 2581 3926 m 2593 3829 l 2620 3848 l 2581 3926 l P 0.740588 G NP 2933 3928 m 2933 3914 l 2979 3917 l 2933 3928 l P 0.596157 G NP 2223 3929 m 2280 3832 l 2305 3912 l 2223 3929 l P NP 2223 3929 m 2200 3830 l 2280 3832 l 2223 3929 l P NP 1665 3934 m 1677 3723 l 1799 3757 l 1665 3934 l P NP 1665 3934 m 1525 3919 l 1677 3723 l 1677 3723 l 1665 3934 l P 0.524392 G NP 2753 3940 m 2726 3922 l 2741 3793 l 2741 3793 l 2753 3940 l P NP 2753 3940 m 2741 3793 l 2765 3799 l 2753 3940 l P 0.596157 G NP 2130 3941 m 2200 3830 l 2223 3929 l 2130 3941 l P NP 2130 3941 m 2112 3822 l 2200 3830 l 2130 3941 l P NP 1796 3944 m 1665 3934 l 1799 3757 l 1799 3757 l 1796 3944 l P NP 1796 3944 m 1799 3757 l 1912 3785 l 1796 3944 l P NP 2028 3947 m 2112 3822 l 2130 3941 l 2028 3947 l P NP 2028 3947 m 2016 3807 l 2112 3822 l 2028 3947 l P NP 1917 3948 m 1796 3944 l 1912 3785 l 1912 3785 l 1917 3948 l P NP 1917 3948 m 2016 3807 l 2028 3947 l 1917 3948 l P NP 1917 3948 m 1912 3785 l 2016 3807 l 1917 3948 l P 0.79 G NP 3201 3953 m 3157 3907 l 3371 3875 l 3371 3875 l 3201 3953 l P NP 3201 3953 m 3371 3875 l 3428 3922 l 3201 3953 l P 0.596157 G NP 2406 3953 m 2442 3863 l 2471 3910 l 2406 3953 l P NP 2406 3953 m 2378 3890 l 2442 3863 l 2442 3863 l 2406 3953 l P 0.583216 G NP 2499 3957 m 2524 3862 l 2552 3894 l 2499 3957 l P NP 2499 3957 m 2471 3910 l 2524 3862 l 2524 3862 l 2499 3957 l P 0.556157 G NP 2609 3958 m 2581 3926 l 2620 3848 l 2620 3848 l 2609 3958 l P NP 2609 3958 m 2620 3848 l 2647 3866 l 2609 3958 l P 0.524392 G NP 2779 3959 m 2753 3940 l 2765 3799 l 2765 3799 l 2779 3959 l P NP 2779 3959 m 2765 3799 l 2788 3805 l 2779 3959 l P 0.740588 G NP 3009 3961 m 2934 3955 l 2933 3928 l 2979 3917 l 2979 3917 l 3009 3961 l P 0.761765 G NP 3009 3961 m 3157 3907 l 3201 3953 l 3009 3961 l P NP 3009 3961 m 2979 3917 l 3157 3907 l 3157 3907 l 3009 3961 l P 0.740588 G NP 2935 3977 m 2934 3955 l 3009 3961 l 2935 3977 l P 0.524392 G NP 2806 3977 m 2779 3959 l 2788 3805 l 2788 3805 l 2806 3977 l P NP 2806 3977 m 2788 3805 l 2812 3811 l 2806 3977 l P 0.556157 G NP 2637 3990 m 2647 3866 l 2673 3885 l 2637 3990 l P NP 2637 3990 m 2609 3958 l 2647 3866 l 2647 3866 l 2637 3990 l P 0.596157 G NP 2331 3993 m 2378 3890 l 2406 3953 l 2331 3993 l P NP 2331 3993 m 2305 3912 l 2378 3890 l 2378 3890 l 2331 3993 l P 0.524392 G NP 2832 3996 m 2806 3977 l 2812 3811 l 2812 3811 l 2832 3996 l P NP 2832 3996 m 2812 3811 l 2836 3817 l 2832 3996 l P 0.79 G NP 3244 3999 m 3201 3953 l 3428 3922 l 3428 3922 l 3244 3999 l P 0.583216 G NP 2528 4004 m 2552 3894 l 2581 3926 l 2528 4004 l P NP 2528 4004 m 2499 3957 l 2552 3894 l 2552 3894 l 2528 4004 l P 0.740588 G NP 3040 4005 m 2936 3996 l 2935 3977 l 3009 3961 l 3009 3961 l 3040 4005 l P 0.761765 G NP 3040 4005 m 3201 3953 l 3244 3999 l 3040 4005 l P NP 3040 4005 m 3009 3961 l 3201 3953 l 3201 3953 l 3040 4005 l P 0.524392 G NP 2859 4014 m 2832 3996 l 2836 3817 l 2836 3817 l 2859 4014 l P NP 2859 4014 m 2836 3817 l 2859 3824 l 2859 4014 l P 0.596157 G NP 2434 4016 m 2406 3953 l 2471 3910 l 2471 3910 l 2434 4016 l P NP 2434 4016 m 2471 3910 l 2499 3957 l 2434 4016 l P 0.556157 G NP 2665 4022 m 2673 3885 l 2700 3903 l 2665 4022 l P NP 2665 4022 m 2637 3990 l 2673 3885 l 2673 3885 l 2665 4022 l P 0.740588 G NP 2937 4023 m 2936 3996 l 3040 4005 l 2937 4023 l P 0.596157 G NP 2245 4028 m 2223 3929 l 2305 3912 l 2305 3912 l 2245 4028 l P NP 2245 4028 m 2305 3912 l 2331 3993 l 2245 4028 l P 0.524392 G NP 2885 4033 m 2859 4014 l 2859 3824 l 2859 3824 l 2885 4033 l P NP 2885 4033 m 2859 3824 l 2883 3830 l 2885 4033 l P 0.761765 G NP 3071 4049 m 3040 4005 l 3244 3999 l 3244 3999 l 3071 4049 l P 0.740588 G NP 3071 4049 m 2937 4036 l 2937 4023 l 3040 4005 l 3040 4005 l 3071 4049 l P 0.583216 G NP 2556 4050 m 2528 4004 l 2581 3926 l 2581 3926 l 2556 4050 l P NP 2556 4050 m 2581 3926 l 2609 3958 l 2556 4050 l P 0.524392 G NP 2912 4051 m 2885 4033 l 2883 3830 l 2912 4051 l P NP 2912 4051 m 2883 3830 l 2907 3836 l 2912 4051 l P 0.556157 G NP 2693 4055 m 2665 4022 l 2700 3903 l 2700 3903 l 2693 4055 l P 0.55498 G NP 2693 4055 m 2700 3903 l 2726 3922 l 2693 4055 l P 0.596157 G NP 2148 4060 m 2130 3941 l 2223 3929 l 2223 3929 l 2148 4060 l P NP 2148 4060 m 2223 3929 l 2245 4028 l 2148 4060 l P 0.740588 G NP 2938 4068 m 2937 4036 l 3071 4049 l 2938 4068 l P 0.524392 G NP 2938 4070 m 2907 3836 l 2930 3842 l 2938 4070 l P NP 2938 4070 m 2912 4051 l 2907 3836 l 2938 4070 l P 0.596157 G NP 2357 4073 m 2331 3993 l 2406 3953 l 2406 3953 l 2357 4073 l P NP 2357 4073 m 2406 3953 l 2434 4016 l 2357 4073 l P NP 2462 4079 m 2499 3957 l 2528 4004 l 2462 4079 l P NP 2462 4079 m 2434 4016 l 2499 3957 l 2499 3957 l 2462 4079 l P 0.556157 G NP 2721 4087 m 2693 4055 l 2726 3922 l 2726 3922 l 2721 4087 l P 0.55498 G NP 2721 4087 m 2726 3922 l 2753 3940 l 2721 4087 l P 0.596157 G NP 2041 4087 m 2130 3941 l 2148 4060 l 2041 4087 l P NP 2041 4087 m 2028 3947 l 2130 3941 l 2130 3941 l 2041 4087 l P 0.583216 G NP 2585 4097 m 2609 3958 l 2637 3990 l 2585 4097 l P NP 2585 4097 m 2556 4050 l 2609 3958 l 2609 3958 l 2585 4097 l P 0.596157 G NP 1922 4110 m 2028 3947 l 2041 4087 l 1922 4110 l P NP 1922 4110 m 1917 3948 l 2028 3947 l 2028 3947 l 1922 4110 l P 0.55498 G NP 2749 4119 m 2721 4087 l 2753 3940 l 2753 3940 l 2749 4119 l P NP 2749 4119 m 2753 3940 l 2779 3959 l 2749 4119 l P 0.596157 G NP 2267 4127 m 2245 4028 l 2331 3993 l 2331 3993 l 2267 4127 l P NP 2267 4127 m 2331 3993 l 2357 4073 l 2267 4127 l P NP 1793 4130 m 1796 3944 l 1917 3948 l 1793 4130 l P NP 1793 4130 m 1917 3948 l 1922 4110 l 1793 4130 l P NP 2489 4142 m 2462 4079 l 2528 4004 l 2528 4004 l 2489 4142 l P NP 2489 4142 m 2528 4004 l 2556 4050 l 2489 4142 l P 0.583216 G NP 2613 4144 m 2637 3990 l 2665 4022 l 2613 4144 l P NP 2613 4144 m 2585 4097 l 2637 3990 l 2637 3990 l 2613 4144 l P 0.596157 G NP 1653 4145 m 1665 3934 l 1796 3944 l 1653 4145 l P NP 1653 4145 m 1796 3944 l 1793 4130 l 1653 4145 l P 0.55498 G NP 2778 4151 m 2749 4119 l 2779 3959 l 2779 3959 l 2778 4151 l P NP 2778 4151 m 2779 3959 l 2806 3977 l 2778 4151 l P 0.596157 G NP 2382 4153 m 2357 4073 l 2434 4016 l 2434 4016 l 2382 4153 l P NP 2382 4153 m 2434 4016 l 2462 4079 l 2382 4153 l P NP 1502 4157 m 1525 3919 l 1665 3934 l 1502 4157 l P NP 1502 4157 m 1665 3934 l 1653 4145 l 1502 4157 l P NP 2166 4178 m 2148 4060 l 2245 4028 l 2245 4028 l 2166 4178 l P NP 2166 4178 m 2245 4028 l 2267 4127 l 2166 4178 l P 0.55498 G NP 2806 4183 m 2806 3977 l 2832 3996 l 2806 4183 l P NP 2806 4183 m 2778 4151 l 2806 3977 l 2806 3977 l 2806 4183 l P 0.583216 G NP 2642 4191 m 2665 4022 l 2693 4055 l 2642 4191 l P NP 2642 4191 m 2613 4144 l 2665 4022 l 2665 4022 l 2642 4191 l P 0.596157 G NP 2517 4205 m 2556 4050 l 2585 4097 l 2517 4205 l P NP 2517 4205 m 2489 4142 l 2556 4050 l 2556 4050 l 2517 4205 l P 0.55498 G NP 2834 4215 m 2806 4183 l 2832 3996 l 2832 3996 l 2834 4215 l P NP 2834 4215 m 2832 3996 l 2859 4014 l 2834 4215 l P 0.596157 G NP 2290 4226 m 2267 4127 l 2357 4073 l 2357 4073 l 2290 4226 l P NP 2290 4226 m 2357 4073 l 2382 4153 l 2290 4226 l P NP 2053 4227 m 2041 4087 l 2148 4060 l 2148 4060 l 2053 4227 l P NP 2053 4227 m 2148 4060 l 2166 4178 l 2053 4227 l P NP 2408 4234 m 2382 4153 l 2462 4079 l 2462 4079 l 2408 4234 l P NP 2408 4234 m 2462 4079 l 2489 4142 l 2408 4234 l P 0.583216 G NP 2671 4238 m 2693 4055 l 2721 4087 l 2671 4238 l P NP 2671 4238 m 2642 4191 l 2693 4055 l 2693 4055 l 2671 4238 l P 0.55498 G NP 2862 4247 m 2859 4014 l 2885 4033 l 2862 4247 l P NP 2862 4247 m 2834 4215 l 2859 4014 l 2859 4014 l 2862 4247 l P 0.596157 G NP 2545 4268 m 2517 4205 l 2585 4097 l 2585 4097 l 2545 4268 l P NP 2545 4268 m 2585 4097 l 2613 4144 l 2545 4268 l P NP 1927 4273 m 1922 4110 l 2041 4087 l 2041 4087 l 1927 4273 l P NP 1927 4273 m 2041 4087 l 2053 4227 l 1927 4273 l P 0.55498 G NP 2890 4279 m 2862 4247 l 2885 4033 l 2885 4033 l 2890 4279 l P NP 2890 4279 m 2885 4033 l 2912 4051 l 2890 4279 l P 0.583216 G NP 2699 4285 m 2671 4238 l 2721 4087 l 2721 4087 l 2699 4285 l P NP 2699 4285 m 2721 4087 l 2749 4119 l 2699 4285 l P 0.596157 G NP 2184 4297 m 2166 4178 l 2267 4127 l 2267 4127 l 2184 4297 l P NP 2184 4297 m 2267 4127 l 2290 4226 l 2184 4297 l P 0.55498 G NP 2918 4311 m 2890 4279 l 2912 4051 l 2912 4051 l 2918 4311 l P NP 2918 4311 m 2912 4051 l 2938 4070 l 2918 4311 l P 0.596157 G NP 2434 4314 m 2489 4142 l 2517 4205 l 2434 4314 l P NP 2434 4314 m 2408 4234 l 2489 4142 l 2489 4142 l 2434 4314 l P NP 1790 4316 m 1922 4110 l 1927 4273 l 1790 4316 l P NP 1790 4316 m 1793 4130 l 1922 4110 l 1790 4316 l P NP 2312 4325 m 2382 4153 l 2408 4234 l 2312 4325 l P NP 2312 4325 m 2290 4226 l 2382 4153 l 2382 4153 l 2312 4325 l P NP 2573 4331 m 2545 4268 l 2613 4144 l 2613 4144 l 2573 4331 l P NP 2573 4331 m 2613 4144 l 2642 4191 l 2573 4331 l P 0.583216 G NP 2728 4331 m 2749 4119 l 2778 4151 l 2728 4331 l P NP 2728 4331 m 2699 4285 l 2749 4119 l 2749 4119 l 2728 4331 l P 0.596157 G NP 1640 4357 m 1793 4130 l 1790 4316 l 1640 4357 l P NP 1640 4357 m 1653 4145 l 1793 4130 l 1640 4357 l P NP 2065 4367 m 2053 4227 l 2166 4178 l 2166 4178 l 2065 4367 l P NP 2065 4367 m 2166 4178 l 2184 4297 l 2065 4367 l P 0.583216 G NP 2756 4378 m 2778 4151 l 2806 4183 l 2756 4378 l P NP 2756 4378 m 2728 4331 l 2778 4151 l 2778 4151 l 2756 4378 l P 0.596157 G NP 2600 4394 m 2573 4331 l 2642 4191 l 2642 4191 l 2600 4394 l P NP 2600 4394 m 2642 4191 l 2671 4238 l 2600 4394 l P NP 2459 4394 m 2434 4314 l 2517 4205 l 2517 4205 l 2459 4394 l P NP 2459 4394 m 2517 4205 l 2545 4268 l 2459 4394 l P NP 1479 4394 m 1502 4157 l 1653 4145 l 1653 4145 l 1479 4394 l P NP 1479 4394 m 1653 4145 l 1640 4357 l 1479 4394 l P NP 2202 4416 m 2290 4226 l 2312 4325 l 2202 4416 l P NP 2202 4416 m 2184 4297 l 2290 4226 l 2290 4226 l 2202 4416 l P NP 2335 4424 m 2312 4325 l 2408 4234 l 2408 4234 l 2335 4424 l P NP 2335 4424 m 2408 4234 l 2434 4314 l 2335 4424 l P 0.583216 G NP 2785 4425 m 2756 4378 l 2806 4183 l 2806 4183 l 2785 4425 l P NP 2785 4425 m 2806 4183 l 2834 4215 l 2785 4425 l P 0.596157 G NP 1933 4436 m 2053 4227 l 2065 4367 l 1933 4436 l P NP 1933 4436 m 1927 4273 l 2053 4227 l 2053 4227 l 1933 4436 l P NP 2628 4457 m 2671 4238 l 2699 4285 l 2628 4457 l P NP 2628 4457 m 2600 4394 l 2671 4238 l 2671 4238 l 2628 4457 l P 0.583216 G NP 2813 4472 m 2834 4215 l 2862 4247 l 2813 4472 l P NP 2813 4472 m 2785 4425 l 2834 4215 l 2834 4215 l 2813 4472 l P 0.596157 G NP 2485 4475 m 2459 4394 l 2545 4268 l 2545 4268 l 2485 4475 l P NP 2485 4475 m 2545 4268 l 2573 4331 l 2485 4475 l P NP 1787 4502 m 1927 4273 l 1933 4436 l 1787 4502 l P NP 1787 4502 m 1790 4316 l 1927 4273 l 1787 4502 l P NP 2077 4507 m 2184 4297 l 2202 4416 l 2077 4507 l P NP 2077 4507 m 2065 4367 l 2184 4297 l 2184 4297 l 2077 4507 l P 0.583216 G NP 2842 4519 m 2813 4472 l 2862 4247 l 2862 4247 l 2842 4519 l P NP 2842 4519 m 2862 4247 l 2890 4279 l 2842 4519 l P 0.596157 G NP 2656 4520 m 2628 4457 l 2699 4285 l 2699 4285 l 2656 4520 l P NP 2656 4520 m 2699 4285 l 2728 4331 l 2656 4520 l P NP 2357 4523 m 2434 4314 l 2459 4394 l 2357 4523 l P NP 2357 4523 m 2335 4424 l 2434 4314 l 2434 4314 l 2357 4523 l P NP 2220 4535 m 2312 4325 l 2335 4424 l 2220 4535 l P NP 2220 4535 m 2202 4416 l 2312 4325 l 2312 4325 l 2220 4535 l P NP 2511 4555 m 2573 4331 l 2600 4394 l 2511 4555 l P NP 2511 4555 m 2485 4475 l 2573 4331 l 2573 4331 l 2511 4555 l P 0.583216 G NP 2870 4566 m 2890 4279 l 2918 4311 l 2870 4566 l P NP 2870 4566 m 2842 4519 l 2890 4279 l 2890 4279 l 2870 4566 l P 0.596157 G NP 1628 4568 m 1790 4316 l 1787 4502 l 1628 4568 l P NP 1628 4568 m 1640 4357 l 1790 4316 l 1628 4568 l P NP 2684 4583 m 2728 4331 l 2756 4378 l 2684 4583 l P NP 2684 4583 m 2656 4520 l 2728 4331 l 2728 4331 l 2684 4583 l P NP 1938 4598 m 2065 4367 l 2077 4507 l 1938 4598 l P NP 1938 4598 m 1933 4436 l 2065 4367 l 2065 4367 l 1938 4598 l P NP 2380 4622 m 2357 4523 l 2459 4394 l 2459 4394 l 2380 4622 l P NP 2380 4622 m 2459 4394 l 2485 4475 l 2380 4622 l P NP 1456 4632 m 1479 4394 l 1640 4357 l 1640 4357 l 1456 4632 l P NP 1456 4632 m 1640 4357 l 1628 4568 l 1456 4632 l P NP 2537 4635 m 2511 4555 l 2600 4394 l 2600 4394 l 2537 4635 l P NP 2537 4635 m 2600 4394 l 2628 4457 l 2537 4635 l P NP 2711 4646 m 2756 4378 l 2785 4425 l 2711 4646 l P NP 2711 4646 m 2684 4583 l 2756 4378 l 2756 4378 l 2711 4646 l P NP 2089 4647 m 2202 4416 l 2220 4535 l 2089 4647 l P NP 2089 4647 m 2077 4507 l 2202 4416 l 2202 4416 l 2089 4647 l P NP 2238 4654 m 2220 4535 l 2335 4424 l 2335 4424 l 2238 4654 l P NP 2238 4654 m 2335 4424 l 2357 4523 l 2238 4654 l P NP 1784 4689 m 1933 4436 l 1938 4598 l 1784 4689 l P NP 1784 4689 m 1787 4502 l 1933 4436 l 1784 4689 l P NP 2739 4709 m 2785 4425 l 2813 4472 l 2739 4709 l P NP 2739 4709 m 2711 4646 l 2785 4425 l 2785 4425 l 2739 4709 l P NP 2562 4715 m 2537 4635 l 2628 4457 l 2628 4457 l 2562 4715 l P NP 2562 4715 m 2628 4457 l 2656 4520 l 2562 4715 l P NP 2402 4721 m 2485 4475 l 2511 4555 l 2402 4721 l P NP 2402 4721 m 2380 4622 l 2485 4475 l 2485 4475 l 2402 4721 l P NP 1943 4761 m 1938 4598 l 2077 4507 l 2077 4507 l 1943 4761 l P NP 1943 4761 m 2077 4507 l 2089 4647 l 1943 4761 l P NP 2767 4771 m 2739 4709 l 2813 4472 l 2813 4472 l 2767 4771 l P NP 2767 4771 m 2813 4472 l 2842 4519 l 2767 4771 l P NP 2256 4773 m 2238 4654 l 2357 4523 l 2357 4523 l 2256 4773 l P NP 2256 4773 m 2357 4523 l 2380 4622 l 2256 4773 l P NP 1616 4779 m 1787 4502 l 1784 4689 l 1616 4779 l P NP 1616 4779 m 1628 4568 l 1787 4502 l 1616 4779 l P NP 2102 4787 m 2089 4647 l 2220 4535 l 2220 4535 l 2102 4787 l P NP 2102 4787 m 2220 4535 l 2238 4654 l 2102 4787 l P NP 2588 4796 m 2562 4715 l 2656 4520 l 2656 4520 l 2588 4796 l P NP 2588 4796 m 2656 4520 l 2684 4583 l 2588 4796 l P NP 2424 4820 m 2511 4555 l 2537 4635 l 2424 4820 l P NP 2424 4820 m 2402 4721 l 2511 4555 l 2511 4555 l 2424 4820 l P NP 2795 4834 m 2842 4519 l 2870 4566 l 2795 4834 l P NP 2795 4834 m 2767 4771 l 2842 4519 l 2842 4519 l 2795 4834 l P NP 1433 4869 m 1456 4632 l 1628 4568 l 1628 4568 l 1433 4869 l P NP 1433 4869 m 1628 4568 l 1616 4779 l 1433 4869 l P NP 1781 4875 m 1938 4598 l 1943 4761 l 1781 4875 l P NP 1781 4875 m 1784 4689 l 1938 4598 l 1781 4875 l P NP 2614 4876 m 2684 4583 l 2711 4646 l 2614 4876 l P NP 2614 4876 m 2588 4796 l 2684 4583 l 2684 4583 l 2614 4876 l P NP 2274 4892 m 2256 4773 l 2380 4622 l 2380 4622 l 2274 4892 l P NP 2274 4892 m 2380 4622 l 2402 4721 l 2274 4892 l P NP 2447 4919 m 2537 4635 l 2562 4715 l 2447 4919 l P NP 2447 4919 m 2424 4820 l 2537 4635 l 2537 4635 l 2447 4919 l P NP 1948 4923 m 2089 4647 l 2102 4787 l 1948 4923 l P NP 1948 4923 m 1943 4761 l 2089 4647 l 2089 4647 l 1948 4923 l P NP 2114 4927 m 2238 4654 l 2256 4773 l 2114 4927 l P NP 2114 4927 m 2102 4787 l 2238 4654 l 2238 4654 l 2114 4927 l P NP 2639 4956 m 2711 4646 l 2739 4709 l 2639 4956 l P NP 2639 4956 m 2614 4876 l 2711 4646 l 2711 4646 l 2639 4956 l P NP 1603 4990 m 1784 4689 l 1781 4875 l 1603 4990 l P NP 1603 4990 m 1616 4779 l 1784 4689 l 1603 4990 l P NP 2292 5011 m 2274 4892 l 2402 4721 l 2402 4721 l 2292 5011 l P NP 2292 5011 m 2402 4721 l 2424 4820 l 2292 5011 l P NP 2469 5018 m 2562 4715 l 2588 4796 l 2469 5018 l P NP 2469 5018 m 2447 4919 l 2562 4715 l 2562 4715 l 2469 5018 l P NP 2665 5037 m 2739 4709 l 2767 4771 l 2665 5037 l P NP 2665 5037 m 2639 4956 l 2739 4709 l 2739 4709 l 2665 5037 l P NP 1778 5061 m 1943 4761 l 1948 4923 l 1778 5061 l P NP 1778 5061 m 1781 4875 l 1943 4761 l 1778 5061 l P NP 2126 5067 m 2256 4773 l 2274 4892 l 2126 5067 l P NP 2126 5067 m 2114 4927 l 2256 4773 l 2256 4773 l 2126 5067 l P NP 1954 5086 m 1948 4923 l 2102 4787 l 2102 4787 l 1954 5086 l P NP 1954 5086 m 2102 4787 l 2114 4927 l 1954 5086 l P NP 1420 5092 m 1412 5089 l 1433 4869 l 1616 4779 l 1616 4779 l 1420 5092 l P NP 1430 5095 m 1420 5092 l 1616 4779 l 1603 4990 l 1430 5095 l P NP 2492 5117 m 2469 5018 l 2588 4796 l 2588 4796 l 2492 5117 l P NP 2492 5117 m 2588 4796 l 2614 4876 l 2492 5117 l P NP 2691 5117 m 2665 5037 l 2767 4771 l 2767 4771 l 2691 5117 l P NP 2691 5117 m 2767 4771 l 2795 4834 l 2691 5117 l P NP 1522 5124 m 1430 5095 l 1603 4990 l 1603 4990 l 1522 5124 l P NP 2310 5129 m 2424 4820 l 2447 4919 l 2310 5129 l P NP 2310 5129 m 2292 5011 l 2424 4820 l 2424 4820 l 2310 5129 l P NP 1594 5147 m 1522 5124 l 1603 4990 l 1603 4990 l 1594 5147 l P NP 1619 5154 m 1594 5147 l 1603 4990 l 1781 4875 l 1619 5154 l P NP 1645 5162 m 1619 5154 l 1781 4875 l 1781 4875 l 1778 5061 l 1645 5162 l P NP 1711 5180 m 1645 5162 l 1778 5061 l 1778 5061 l 1711 5180 l P NP 1776 5199 m 1711 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5275 l P NP 2145 5282 m 2107 5275 l 2138 5207 l 2145 5282 l P NP 2176 5287 m 2145 5282 l 2138 5207 l 2292 5011 l 2292 5011 l 2176 5287 l P NP 2193 5290 m 2176 5287 l 2292 5011 l 2310 5129 l 2193 5290 l P NP 2241 5297 m 2193 5290 l 2310 5129 l 2310 5129 l 2241 5297 l P NP 2289 5305 m 2241 5297 l 2310 5129 l 2310 5129 l 2327 5248 l 2289 5305 l P NP 2304 5307 m 2289 5305 l 2327 5248 l 2327 5248 l 2304 5307 l P NP 2337 5312 m 2304 5307 l 2327 5248 l 2327 5248 l 2337 5312 l P NP 2537 5315 m 2514 5216 l 2639 4956 l 2639 4956 l 2537 5315 l P NP 2537 5315 m 2639 4956 l 2665 5037 l 2537 5315 l P NP 2364 5315 m 2337 5312 l 2327 5248 l 2469 5018 l 2469 5018 l 2364 5315 l P NP 2375 5316 m 2364 5315 l 2469 5018 l 2469 5018 l 2492 5117 l 2375 5316 l P NP 2421 5320 m 2375 5316 l 2492 5117 l 2492 5117 l 2421 5320 l P NP 2454 5323 m 2421 5320 l 2492 5117 l 2492 5117 l 2514 5216 l 2454 5323 l P NP 2477 5325 m 2454 5323 l 2514 5216 l 2514 5216 l 2477 5325 l P NP 2528 5330 m 2477 5325 l 2514 5216 l 2514 5216 l 2537 5315 l 2528 5330 l P NP 2531 5331 m 2528 5330 l 2537 5315 l 2537 5315 l 2531 5331 l P NP 2540 5331 m 2531 5331 l 2537 5315 l 2537 5315 l 2540 5331 l P NP 2582 5332 m 2540 5331 l 2537 5315 l 2665 5037 l 2665 5037 l 2582 5332 l P NP 2595 5333 m 2582 5332 l 2665 5037 l 2691 5117 l 2595 5333 l P defaultFont 0.3 G 30 setlinewidth [] 0 setdash NP 862 2331 m 973 2357 l 1195 2421 l S NP 1266 2444 m 1449 2507 l S NP 1470 2516 m 1632 2582 l S NP 1654 2592 m 1815 2669 l S NP 1862 2695 m 1989 2767 l S NP 2006 2778 m 2104 2843 l S NP 2132 2862 m 2238 2944 l S NP 1995 2915 m 2013 2988 l S NP 2265 2968 m 2346 3042 l S NP 2357 3055 m 2413 3115 l S NP 2016 2999 m 2063 3153 l S NP 2433 3138 m 2492 3216 l S NP 2063 3153 m 2107 3271 l S NP 2503 3234 m 2546 3304 l 2576 3368 l S NP 2115 3292 m 2170 3410 l S NP 2585 3390 m 2606 3456 l S NP 2170 3410 m 2221 3499 l S NP 2607 3463 m 2617 3524 l 2617 3572 l S NP 2226 3508 m 2274 3574 l S NP 2616 3589 m 2607 3633 l S NP 2282 3586 m 2337 3645 l S NP 2607 3633 m 2588 3673 l S NP 2337 3645 m 2386 3684 l S NP 2588 3673 m 2565 3697 l S NP 2389 3687 m 2433 3709 l S NP 2556 3704 m 2526 3719 l S NP 2439 3713 m 2485 3723 l S NP 2524 3719 m 2485 3723 l S 0 G boundarythick setlinewidth % The following draws a box around the plot, % if the variable drawborder is true drawborder { /bd boundarythick 2 idiv def [] 0 setdash NP bd bd m bd 6923 bd sub l 5000 bd sub 6923 bd sub l 5000 bd sub bd l bd bd l S } if % end of if to draw the border showpage grestore end %%Trailer %%BoundingBox: 152 220 488 571 %%EOF %%EndDocument @endspecial 292 4061 a(Figure)g(2:)42 b(P)m(ejorativ)m(e)32 b(manifolds)f(of)f(p)s(olynomials)h(with)f(degree)h(3)g(\(view)f(from)g (t)m(w)m(o)i(angles\))94 4252 y(As)f(a)g(sp)s(ecial)g(case,)g(\005)886 4271 y Fx([1)p FL(;)p Fx(1)p FL(;)p FE(\001\001\001)o FL(;)p Fx(1])1178 4252 y FN(=)25 b FO(C)1350 4219 y FL(n)1427 4252 y FN(is)31 b(the)f(v)m(ector)i(space)f(of)g(all)g(monic)g(p)s (olynomials)f(with)g(degree)h FM(n)p FN(.)94 4491 y FP(3.3)113 b(Solving)38 b(the)g(nonsingular)h(least)f(squares)g(problem)94 4663 y FN(Let)46 b FM(`)k FN(=)g([)p FM(`)544 4677 y Fx(1)583 4663 y FM(;)15 b Fy(\001)g(\001)g(\001)i FM(;)e(`)823 4677 y FL(m)890 4663 y FN(])45 b(b)s(e)g(a)g(m)m(ultiplicit)m(y)j (structure)c(of)i(degree)g FM(n)e FN(and)h(\005)2872 4678 y FL(`)2950 4663 y FN(b)s(e)g(the)g(corresp)s(onding)94 4776 y(p)s(ejorativ)m(e)36 b(manifold.)52 b(If)34 b(the)h(p)s (olynomial)f FM(p)e Fy(\030)g FO(a)f Fy(2)h FN(\005)2099 4791 y FL(`)2132 4776 y FN(,)j(then)f(there)h(is)f(a)h(v)m(ector)h FO(z)c Fy(2)f FO(C)3338 4743 y FL(m)3439 4776 y FN(suc)m(h)j(that)94 4889 y FM(G)165 4904 y FL(`)199 4889 y FN(\()p FO(z)p FN(\))26 b(=)f FO(a)p FN(.)41 b(In)29 b(general,)j(the)f(p)s(olynomial) f(system)1032 4985 y Fu(8)1032 5059 y(>)1032 5084 y(>)1032 5109 y(>)1032 5134 y(>)1032 5159 y(<)1032 5308 y(>)1032 5333 y(>)1032 5358 y(>)1032 5383 y(>)1032 5408 y(:)1151 5063 y FM(g)1194 5077 y Fx(1)1234 5063 y FN(\()p FM(z)1311 5077 y Fx(1)1351 5063 y FM(;)15 b Fy(\001)g(\001)g(\001)i FM(;)e(z)1595 5077 y FL(m)1662 5063 y FN(\))87 b(=)f FM(a)1989 5077 y Fx(1)1151 5176 y FM(g)1194 5190 y Fx(2)1234 5176 y FN(\()p FM(z)1311 5190 y Fx(1)1351 5176 y FM(;)15 b Fy(\001)g(\001)g(\001)i FM(;)e(z)1595 5190 y FL(m)1662 5176 y FN(\))87 b(=)f FM(a)1989 5190 y Fx(2)1411 5270 y FN(.)1411 5303 y(.)1411 5336 y(.)1807 5270 y(.)1807 5303 y(.)1807 5336 y(.)1973 5270 y(.)1973 5303 y(.)1973 5336 y(.)1147 5449 y FM(g)1190 5463 y FL(n)1238 5449 y FN(\()p FM(z)1315 5463 y Fx(1)1355 5449 y FM(;)15 b Fy(\001)g(\001)g(\001)h FM(;)f(z)1598 5463 y FL(m)1666 5449 y FN(\))83 b(=)g FM(a)1986 5463 y FL(n)2251 5256 y FN(or)151 b FM(G)2554 5271 y FL(`)2588 5256 y FN(\()p FO(z)p FN(\))26 b(=)f FO(a)776 b FN(\(10\))1932 5686 y(9)p eop %%Page: 10 12 10 11 bop 94 99 a FN(is)29 b(o)m(v)m(erdetermined)h(except)g(for)e(the) h(plain)g(structure)f FM(`)d FN(=)g([1)p FM(;)15 b FN(1)p FM(;)g Fy(\001)g(\001)g(\001)k FM(;)c FN(1].)41 b(Let)29 b FM(W)38 b FN(=)25 b 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1336 y FM(!)1289 1350 y FL(j)1351 1336 y FN(=)25 b(min)1613 1241 y Fu(n)1684 1336 y FN(1)p FM(;)41 b Fy(j)p FM(a)1868 1350 y FL(j)1905 1336 y Fy(j)1930 1298 y FE(\000)p Fx(1)2024 1241 y Fu(o)2095 1336 y FM(;)91 b(j)31 b FN(=)25 b(1)p FM(;)15 b Fy(\001)g(\001)g(\001)i FM(;)e(n)976 b FN(\(12\))94 1550 y(lead)37 b(to)f(minimization)h(of)f(the)g(relativ) m(e)i(bac)m(kw)m(ard)e(error)g(at)g(ev)m(ery)h(co)s(e\016cien)m(t)g (larger)g(than)f(one.)57 b(All)94 1663 y(our)30 b(n)m(umerical)h(exp)s (erimen)m(ts)g(for)f(Algorithm)h(I)f(are)h(conducted)f(using)g(the)h(w) m(eigh)m(ts)g(\(12\).)94 1871 y(F)-8 b(rom)34 b(Lemma)f(2.5,)i(let)e FM(J)9 b FN(\()p FO(z)p FN(\))34 b(b)s(e)f(the)g(Jacobian)g(of)g FM(G)2025 1886 y FL(`)2059 1871 y FN(\()p FO(z)p FN(\).)49 b(In)32 b(order)g(to)i(\014nd)d(a)i(lo)s(cal)h(minim)m(um)e(p)s(oin)m (t)94 1999 y(of)f FM(F)13 b FN(\()p FO(z)p FN(\))27 b Fy(\021)d FM(W)606 1905 y Fu(h)645 1999 y FM(G)716 2014 y FL(`)750 1999 y FN(\()p FO(z)p FN(\))d Fy(\000)f FO(a)1029 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FO(De\014nition)36 b(3.2)46 b FC(L)-5 b(et)33 b FM(p)27 b Fy(\030)f FO(a)33 b FC(b)-5 b(e)34 b(a)g(p)-5 b(olynomial)36 b(of)e(de)-5 b(gr)g(e)g(e)34 b FM(n)p FC(.)44 b(F)-7 b(or)34 b(any)g(given)f (multiplicity)h(structur)-5 b(e)35 b FM(`)94 2698 y FC(of)g(the)g(same) g(de)-5 b(gr)g(e)g(e,)36 b(the)f(ve)-5 b(ctor)1315 2697 y FN(~)1314 2698 y FO(z)35 b FC(satisfying)g(\(13\))h(is)e(c)-5 b(al)5 b(le)-5 b(d)36 b(a)f Fr(p)-5 b(ejor)g(ative)39 b(r)-5 b(o)g(ot)39 b(ve)-5 b(ctor)34 b FC(or)h(simply)94 2811 y Fr(p)-5 b(ejor)g(ative)38 b(r)-5 b(o)g(ot)31 b FC(of)i FM(p)f FC(c)-5 b(orr)g(esp)g(onding)36 b(to)d(the)g (multiplicity)h(structur)-5 b(e)33 b FM(`)f FC(and)i(weight)f FM(W)13 b FC(.)94 3047 y FN(Our)33 b(algorithms)i(emanate)g(from)f(the) g(follo)m(wing)i(fundamen)m(tal)e(theorem)g(b)m(y)g(whic)m(h)g(one)g (ma)m(y)h(con)m(v)m(ert)94 3160 y(the)27 b(singular)g(problem)f(of)h (computing)g(m)m(ultiple)h(ro)s(ots)f(with)f(standard)g(metho)s(ds)g (to)i(a)f(regular)g(problem)94 3273 y(b)m(y)k(seeking)g(the)f(least)i (squares)e(solution)h(of)g(\(10\).)94 3509 y FO(Theorem)k(3.1)124 b FC(L)-5 b(et)51 b FM(G)1041 3524 y FL(`)1115 3509 y FN(:)41 b FO(C)1257 3476 y FL(m)1348 3509 y Fy(\000)-15 b(!)25 b FO(C)1596 3476 y FL(n)1668 3509 y FC(b)-5 b(e)26 b(the)g(c)-5 b(o)g(e\016cient)26 b(op)-5 b(er)g(ator)29 b(asso)-5 b(ciate)g(d)29 b(with)d(a)g(multiplicity)94 3622 y(structur)-5 b(e)30 b FM(`)25 b FN(=)g([)p FM(`)696 3636 y Fx(1)736 3622 y FM(;)15 b Fy(\001)g(\001)g(\001)i FM(;)e(`)976 3636 y FL(m)1042 3622 y FN(])p FC(.)41 b(Then)30 b(the)f(Jac)-5 b(obian)30 b FM(J)9 b FN(\()p FO(z)p FN(\))31 b FC(of)e FM(G)2271 3637 y FL(`)2305 3622 y FN(\()p FO(z)p FN(\))h FC(is)f(of)g(ful)5 b(l)29 b(\(c)-5 b(olumn\))31 b(r)-5 b(ank)30 b(if)e(and)j(only)94 3735 y(if)i(the)g(entries)g(of)65 b FO(z)26 b FN(=)f(\()p FM(z)1019 3749 y Fx(1)1059 3735 y FM(;)15 b Fy(\001)g(\001)g(\001)h FM(;)f(z)1302 3749 y FL(m)1370 3735 y FN(\))1405 3702 y FE(>)1497 3735 y FC(ar)-5 b(e)33 b(distinct.)94 3971 y FO(Pro)s(of.)88 b FN(Let)32 b FM(z)670 3985 y Fx(1)709 3971 y FM(;)15 b Fy(\001)g(\001)g(\001)i FM(;)e(z)953 3985 y FL(m)1052 3971 y FN(b)s(e)30 b(distinct.)45 b(T)-8 b(o)32 b(pro)m(v)m(e)g FM(J)9 b FN(\()p FO(z)p FN(\))32 b(is)g(of)g(full)f(\(column\))h(rank,) f(or)h(the)f(columns)g(of)94 4169 y FM(J)9 b FN(\()p FO(z)p FN(\))30 b(are)g(linearly)f(indep)s(enden)m(t,)f(write)h(the)g FM(j)5 b FN(-th)29 b(column)g(of)g FM(J)9 b FN(\()p FO(z)p FN(\))30 b(as)e FM(J)2669 4183 y FL(j)2732 4169 y FN(=)2828 4025 y Fu( )2903 4107 y FM(@)5 b(g)2999 4121 y Fx(1)3039 4107 y FN(\()p FO(z)p FN(\))p 2903 4148 254 4 v 2964 4231 a FM(@)g(z)3059 4245 y FL(j)3167 4169 y FM(;)15 b Fy(\001)g(\001)g(\001)h FM(;)3379 4107 y(@)5 b(g)3475 4121 y FL(n)3522 4107 y FN(\()p FO(z)p FN(\))p 3379 4148 261 4 v 3443 4231 a FM(@)g(z)3538 4245 y FL(j)3649 4025 y Fu(!)3715 4048 y FE(>)3789 4169 y FM(:)94 4347 y FN(F)-8 b(or)32 b FM(j)f FN(=)24 b(1)p FM(;)15 b Fy(\001)g(\001)g(\001)j FM(;)d(m)p FN(,)30 b(let)i FM(q)977 4361 y FL(j)1013 4347 y FN(\()p FM(x)p FN(\),)f(a)g(p)s(olynomial)f(in)g FM(x)p FN(,)h(b)s(e)f(de\014ned)f(as)h(follo)m(ws,)376 4621 y FM(q)417 4635 y FL(j)453 4621 y FN(\()p FM(x)p FN(\))83 b(=)812 4477 y Fu( )888 4559 y FM(@)5 b(g)984 4573 y Fx(1)1024 4559 y FN(\()p FO(z)p FN(\))p 888 4599 254 4 v 948 4683 a FM(@)g(z)1043 4697 y FL(j)1151 4477 y Fu(!)1232 4621 y FM(x)1284 4583 y FL(n)p FE(\000)p Fx(1)1441 4621 y FN(+)20 b Fy(\001)15 b(\001)g(\001)21 b FN(+)1749 4477 y Fu( )1825 4559 y FM(@)5 b(g)1921 4573 y FL(n)p FE(\000)p Fx(1)2059 4559 y FN(\()p FO(z)p FN(\))p 1825 4599 351 4 v 1934 4683 a FM(@)g(z)2029 4697 y FL(j)2186 4477 y Fu(!)2267 4621 y FM(x)20 b FN(+)2430 4477 y Fu( )2506 4559 y FM(@)5 b(g)2602 4573 y FL(n)2649 4559 y FN(\()p FO(z)p FN(\))p 2506 4599 261 4 v 2570 4683 a FM(@)g(z)2665 4697 y FL(j)2776 4477 y Fu(!)658 4880 y FN(=)862 4819 y FM(@)p 822 4859 133 4 v 822 4943 a(@)g(z)917 4957 y FL(j)964 4786 y Fu(h)1003 4880 y FM(x)1055 4843 y FL(n)1123 4880 y FN(+)19 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Fu(3)2027 5155 y(5)2097 5174 y FM(:)1531 b FN(\(14\))1909 5686 y(10)p eop %%Page: 11 13 11 12 bop 94 99 a FN(If)30 b FM(c)224 113 y Fx(1)264 99 y FM(J)314 113 y Fx(1)374 99 y FN(+)20 b Fy(\001)15 b(\001)g(\001)21 b FN(+)f FM(c)721 113 y FL(m)788 99 y FM(J)838 113 y FL(m)930 99 y FN(=)25 b(0)61 b(for)31 b(constan)m(ts)g FM(c)1713 113 y Fx(1)1753 99 y FM(;)15 b Fy(\001)g(\001)g(\001)i FM(;)e(c)1994 113 y FL(m)2061 99 y FN(,)30 b(then)94 280 y FM(q)s FN(\()p FM(x)p FN(\))c Fy(\021)f FM(c)421 294 y Fx(1)476 280 y FM(q)517 294 y Fx(1)556 280 y FN(\()p FM(x)p FN(\))c(+)f Fy(\001)15 b(\001)g(\001)21 b FN(+)f FM(c)1046 294 y FL(m)1128 280 y FM(q)1169 294 y FL(m)1235 280 y FN(\()p FM(x)p FN(\))261 508 y(=)k Fy(\000)472 402 y FL(m)444 427 y Fu(X)442 609 y FL(j)t Fx(=1)580 336 y Fu(8)580 410 y(<)580 560 y(:)654 508 y FM(c)693 522 y FL(j)730 508 y FM(`)768 522 y FL(j)819 508 y FN(\()p FM(x)d Fy(\000)f FM(z)1060 522 y FL(j)1096 508 y FN(\))1131 470 y FL(`)1160 480 y Fs(j)1193 470 y FE(\000)p Fx(1)1303 339 y Fu(2)1303 488 y(4)1368 427 y(Y)1358 611 y FL(k)r FE(6)p Fx(=)p FL(j)1484 508 y FN(\()p FM(x)g Fy(\000)g FM(z)1724 523 y FL(k)1767 508 y FN(\))1802 470 y FL(`)1831 482 y Fs(k)1874 339 y Fu(3)1874 488 y(5)1929 336 y(9)1929 410 y(=)1929 560 y(;)2028 508 y FN(=)25 b Fy(\000)2210 364 y Fu(")2309 402 y FL(m)2287 427 y Fu(Y)2273 608 y FL(\033)r Fx(=1)2406 508 y FN(\()p FM(x)c Fy(\000)f FM(z)2647 522 y FL(\033)2694 508 y FN(\))2729 470 y FL(`)2758 478 y Fs(\033)2800 470 y FE(\000)p Fx(1)2909 364 y Fu(#)3003 402 y FL(m)2974 427 y Fu(X)2973 609 y FL(j)t Fx(=1)3110 339 y Fu(2)3110 488 y(4)3181 508 y FM(c)3220 522 y FL(j)3257 508 y FM(`)3295 522 y FL(j)3356 427 y Fu(Y)3346 611 y FL(k)r FE(6)p Fx(=)p FL(j)3472 508 y FN(\()p FM(x)h Fy(\000)e FM(z)3712 523 y FL(k)3755 508 y FN(\))3790 339 y Fu(3)3790 488 y(5)94 808 y FN(is)47 b(a)f(zero)h(p)s(olynomial,)k(yielding)93 b FM(r)s FN(\()p FM(x)p FN(\))52 b(=)1760 744 y Fu(P)1847 770 y FL(m)1847 831 y(j)t Fx(=1)1989 808 y FM(c)2028 822 y FL(j)2065 808 y FM(`)2103 822 y FL(j)2154 714 y Fu(h)2194 744 y(Q)2272 831 y FL(k)r FE(6)p Fx(=)p FL(j)2402 808 y FN(\()p FM(x)20 b Fy(\000)g FM(z)2642 823 y FL(k)2685 808 y FN(\))2720 714 y Fu(i)2812 808 y Fy(\021)51 b FN(0)p FM(:)c FN(Therefore,)j(for)c FM(l)54 b FN(=)94 966 y(1)p FM(;)15 b Fy(\001)g(\001)g(\001)j FM(;)d(m)p FN(,)42 b FM(r)s FN(\()p FM(z)610 981 y FL(l)637 966 y FN(\))f(=)g FM(c)864 981 y FL(l)906 872 y Fu(h)945 966 y FM(`)983 981 y FL(l)1024 902 y Fu(Q)1102 989 y FL(k)r FE(6)p Fx(=)p FL(l)1222 966 y FN(\()p FM(z)1299 981 y FL(l)1346 966 y Fy(\000)20 b FM(z)1479 981 y FL(k)1522 966 y FN(\))1557 872 y Fu(i)1637 966 y FN(=)42 b(0)80 b(implies)g FM(c)2272 981 y FL(l)2340 966 y FN(=)41 b(0)80 b(since)41 b FM(`)2848 981 y FL(l)2873 966 y FN('s)g(are)f(p)s(ositiv)m (e)h(and)e FM(z)3710 981 y FL(k)3753 966 y FN('s)94 1090 y(are)31 b(distinct.)41 b(Therefore,)31 b FM(J)1098 1104 y FL(j)1135 1090 y FN('s)f(are)h(linearly)g(indep)s(enden)m(t.)94 1297 y(On)j(the)h(other)f(hand,)h(supp)s(ose)e FM(z)1298 1311 y Fx(1)1338 1297 y FM(;)15 b Fy(\001)g(\001)g(\001)h FM(;)f(z)1581 1311 y FL(m)1683 1297 y FN(are)35 b(not)f(distinct,)i(sa) m(y)-8 b(,)37 b(for)d(instance,)j FM(z)3113 1311 y Fx(1)3185 1297 y FN(=)31 b FM(z)3329 1311 y Fx(2)3369 1297 y FN(.)53 b(Then)33 b(the)94 1410 y(\014rst)d(t)m(w)m(o)i(columns)e(of)g FM(J)9 b FN(\()p FO(z)p FN(\))32 b(are)f(co)s(e\016cien)m(ts)h(of)e(p)s (olynomials)h FM(h)2391 1424 y Fx(1)2431 1410 y FN(\()p FM(x)p FN(\))g(and)f FM(h)2813 1424 y Fx(2)2852 1410 y FN(\()p FM(x)p FN(\))h(de\014ned)e(as)232 1652 y Fy(\000)p FM(`)341 1666 y Fx(1)395 1652 y FN(\()p FM(x)21 b Fy(\000)f FM(z)636 1666 y Fx(1)675 1652 y FN(\))710 1614 y FL(`)739 1623 y Fw(1)774 1614 y FE(\000)p Fx(1)869 1652 y FN(\()p FM(x)g Fy(\000)g FM(z)1109 1666 y Fx(2)1149 1652 y FN(\))1184 1614 y FL(`)1213 1623 y Fw(2)1300 1546 y FL(m)1278 1571 y Fu(Y)1267 1755 y FL(k)r Fx(=3)1395 1652 y FN(\()p FM(x)h Fy(\000)f FM(z)1636 1667 y FL(k)1679 1652 y FN(\))1714 1614 y FL(`)1743 1626 y Fs(k)1846 1652 y FN(and)110 b Fy(\000)20 b FM(`)2232 1666 y Fx(2)2287 1652 y FN(\()p FM(x)g Fy(\000)g FM(z)2527 1666 y Fx(1)2567 1652 y FN(\))2602 1614 y FL(`)2631 1623 y Fw(1)2670 1652 y FN(\()p FM(x)g Fy(\000)g FM(z)2910 1666 y Fx(2)2950 1652 y FN(\))2985 1614 y FL(`)3014 1623 y Fw(2)3049 1614 y FE(\000)p Fx(1)3191 1546 y FL(m)3170 1571 y Fu(Y)3158 1755 y FL(k)r Fx(=3)3287 1652 y FN(\()p FM(x)g Fy(\000)g FM(z)3527 1667 y FL(k)3570 1652 y FN(\))3605 1614 y FL(`)3634 1626 y Fs(k)94 1903 y FN(resp)s(ectiv)m(ely)-8 b(.)77 b(Since)42 b FM(z)943 1917 y Fx(1)1027 1903 y FN(=)i FM(z)1184 1917 y Fx(2)1224 1903 y FN(,)h(these)d(t)m(w)m(o)i(p)s(olynomials)e(di\013er)f(b)m(y)h (constan)m(t)h(m)m(ultiples)g FM(`)3443 1917 y Fx(1)3524 1903 y FN(and)e FM(`)3750 1917 y Fx(2)3789 1903 y FN(.)94 2016 y(Therefore)31 b FM(J)9 b FN(\()p FO(z)p FN(\))31 b(is)g(\(column\))g(rank)e(de\014cien)m(t.)1779 b(Q.E.D.)94 2224 y(With)31 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3061 y Fy(\000)2190 3060 y FN(~)2189 3061 y FO(z)2250 2964 y Fu(\015)2250 3014 y(\015)2250 3064 y(\015)2296 3118 y Fx(2)2361 3061 y FM(<)25 b(\017)p FC(,)30 b(the)g(iter)-5 b(ation)32 b(\(15\))f(is)e(wel)5 b(l)30 b(de\014ne)-5 b(d)94 3194 y(and)39 b(c)-5 b(onver)g(ges)38 b(to)g(the)g(p)-5 b(ejor)g(ative)40 b(r)-5 b(o)g(ot)1564 3193 y FN(~)1563 3194 y FO(z)38 b FC(with)h(at)f(le)-5 b(ast)38 b(a)g(line)-5 b(ar)39 b(r)-5 b(ate.)58 b(If)37 b(we)h(have)75 b FO(a)34 b FN(=)g FM(G)3510 3209 y FL(`)3543 3194 y FN(\()3579 3193 y(~)3578 3194 y FO(z)q FN(\))75 b FC(in)94 3307 y(addition,)35 b(then)e(the)g(c)-5 b(onver)g(genc)g(e)33 b(is)g(quadr)-5 b(atic.)94 3531 y FO(Pro)s(of.)75 b FN(Let)26 b FM(F)13 b FN(\()p FO(z)p FN(\))26 b(=)f FM(W)1017 3436 y Fu(h)1056 3531 y FM(G)1127 3546 y FL(`)1160 3531 y FN(\()p FO(z)p FN(\))10 b Fy(\000)g FO(a)1418 3436 y Fu(i)1483 3531 y FN(and)25 b Fy(J)16 b FN(\()p FO(z)p FN(\))27 b(b)s(e)d(its)i (Jacobian.)40 b FM(F)13 b FN(\()p FO(z)p FN(\))26 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m(b)s(er)d(can)h(b)s(e)g(c)m(haracterized)1909 5686 y(11)p eop %%Page: 12 14 12 13 bop 94 99 a FN(as)31 b(the)g(smallest)g(n)m(um)m(b)s(er)e (satisfying)573 202 y Fu(h)612 296 y FM(f)10 b(or)s(w)r(ar)s(d)p 966 296 28 4 v 32 w(er)s(r)s(or)1211 202 y Fu(i)1275 296 y Fy(\024)1371 202 y Fu(h)1410 296 y FM(condition)p 1794 296 V 33 w(number)2134 202 y Fu(i)2193 296 y Fy(\002)2284 202 y Fu(h)2323 296 y FM(back)s(w)r(ar)s(d)p 2710 296 V 33 w(er)s(r)s(or)2956 202 y Fu(i)3015 296 y FN(+)20 b FM(h:o:t:;)318 b FN(\(16\))94 494 y(where)22 b FM(h:o:t)g FN(represen)m(ts)f(higher)h(order)f(terms)g(of)h(the)g(bac)m(kw)m(ard)g (error.)37 b(F)-8 b(or)23 b(a)f(p)s(olynomial)f(with)h(m)m(ultiple)94 607 y(ro)s(ots,)40 b(under)35 b FC(unr)-5 b(estricte)g(d)39 b FN(p)s(erturbation,)f(the)f(only)h(condition)f(n)m(um)m(b)s(er)f (satisfying)i(\(16\))h(is)e(in\014nit)m(y)-8 b(.)94 719 y(F)g(or)33 b(a)e(simple)h(example,)g(let)h(p)s(olynomial)63 b FM(p)p FN(\()p FM(x)p FN(\))27 b(=)f FM(x)1981 686 y Fx(2)2021 719 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b(and)d(the)g(m)m(ultiplicit)m(y)j(structure)c(is)i(preserv)m (ed.)64 b(With)38 b(this)h(shift)f(in)g(computing)g(strategy)-8 b(,)42 b(the)94 1491 y(sensitivit)m(y)32 b(of)f(the)f(ro)s(ots)h(can)g (b)s(e)e(analyzed)j(di\013eren)m(tly)-8 b(.)94 1699 y(Let's)42 b(consider)e(the)g(ro)s(ot)h(v)m(ector)h FO(z)f FN(of)81 b FM(p)42 b Fy(\030)f FO(a)h FN(=)g FM(G)2065 1714 y FL(`)2098 1699 y FN(\()p FO(z)p FN(\).)72 b(The)40 b(p)s(olynomial)h FM(p)f FN(is)g(p)s(erturb)s(ed,)h(with)94 1812 y(m)m(ultiplicit)m(y)35 b(structure)e FM(`)f FN(b)s(eing)g(preserv)m(ed,)i(to)f(b)s(e)73 b(^)-53 b FM(p)29 b Fy(\030)2177 1811 y FN(^)2175 1812 y FO(a)g FN(=)g FM(G)2426 1827 y FL(`)2459 1812 y FN(\()2495 1811 y(^)2494 1812 y FO(z)q FN(\).)48 b(In)32 b(other)h(w)m(ords,)g(b)s (oth)f FM(p)h FN(and)39 b(^)-52 b FM(p)94 1925 y FN(are)31 b(on)g(the)f(same)h(p)s(ejorativ)m(e)g(manifold)g(\005)1626 1940 y FL(`)1659 1925 y FN(.)40 b(Then)881 2121 y(^)878 2122 y FO(a)20 b Fy(\000)g FO(a)55 b FN(=)25 b FM(G)1313 2137 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2640 y FN([)p FM(W)13 b(J)c FN(\()p FO(z)p FN(\)]\()2385 2639 y(^)2384 2640 y FO(z)23 b Fy(\000)c FO(z)p FN(\))2624 2543 y Fu(\015)2624 2593 y(\015)2624 2643 y(\015)2671 2697 y Fx(2)2731 2640 y FN(+)h FM(h:o:t:)15 b(;)826 2827 y FN(namely)-8 b(,)1311 2730 y Fu(\015)1311 2780 y(\015)1311 2829 y(\015)1375 2826 y FN(^)1373 2827 y FO(a)20 b Fy(\000)g FO(a)1601 2730 y Fu(\015)1601 2780 y(\015)1601 2829 y(\015)1647 2883 y FL(W)1810 2827 y Fy(\025)83 b FM(\033)2016 2841 y FL(min)2149 2730 y Fu(\015)2149 2780 y(\015)2149 2829 y(\015)2211 2826 y FN(^)2211 2827 y FO(z)20 b Fy(\000)g FO(z)2429 2730 y Fu(\015)2430 2780 y(\015)2430 2829 y(\015)2476 2883 y Fx(2)2535 2827 y FN(+)g FM(h:o:t:;)1068 3039 y FN(or)1361 2941 y Fu(\015)1361 2991 y(\015)1361 3041 y(\015)1423 3038 y FN(^)1422 3039 y FO(z)h Fy(\000)f FO(z)1641 2941 y Fu(\015)1642 2991 y(\015)1642 3041 y(\015)1688 3095 y Fx(2)1810 3039 y Fy(\024)1964 2920 y Fu(\022)2105 2977 y FN(1)p 2035 3018 186 4 v 2035 3101 a FM(\033)2087 3115 y FL(min)2230 2920 y Fu(\023)2322 2941 y(\015)2322 2991 y(\015)2322 3041 y(\015)2386 3038 y FN(^)2383 3039 y FO(a)g Fy(\000)g FO(a)2611 2941 y Fu(\015)2611 2991 y(\015)2611 3041 y(\015)2657 3095 y FL(W)2758 3039 y FN(+)g FM(h:o:t:)600 b FN(\(17\))94 3266 y(where)65 b FM(\033)444 3280 y FL(min)610 3266 y FN(,)34 b(the)f(smallest)h (singular)e(v)-5 b(alue)33 b(of)g FM(W)13 b(J)c FN(\()p FO(z)p FN(\),)35 b(is)d(strictly)i(p)s(ositiv)m(e)g(since)f FM(W)45 b FN(and)32 b FM(J)9 b FN(\()p FO(z)p FN(\))34 b(are)94 3379 y(of)d(full)e(rank.)40 b(The)30 b(distance)g Fy(k)1202 3378 y FN(^)1201 3379 y FO(z)21 b Fy(\000)e FO(z)c Fy(k)1464 3393 y Fx(2)1534 3379 y FN(is)30 b(the)g(forw)m(ard)g (error)f(and)h(the)g(w)m(eigh)m(ted)h(distance)g Fy(k)3465 3378 y FN(^)3462 3379 y FO(a)19 b Fy(\000)h FO(a)15 b Fy(k)3734 3393 y FL(W)94 3492 y FN(measures)36 b(the)h(bac)m(kw)m(ard)f (error.)57 b(Therefore,)38 b(the)e(sensitivit)m(y)i(of)e(the)g(ro)s(ot) g(v)m(ector)i(is)e(asymptotically)94 3605 y(b)s(ounded)29 b(b)m(y)666 3569 y Fx(1)p 603 3584 160 4 v 603 3636 a FL(\033)643 3646 y Fs(min)804 3605 y FN(times)j(the)f(size)h(of)f(the)g (m)m(ultiplicit)m(y-preserving)i(p)s(erturbation.)42 b(In)31 b(this)g(sense,)g(the)94 3718 y(m)m(ultiple)h(ro)s(ots)e(are)h (not)f(in\014nitely)h(sensitiv)m(e.)94 3961 y FO(De\014nition)36 b(3.3)117 b FC(L)-5 b(et)35 b FM(p)g FC(b)-5 b(e)36 b(a)g(p)-5 b(olynomial)38 b(and)f FO(z)f FC(b)-5 b(e)35 b(its)h(p)-5 b(ejor)g(ative)37 b(r)-5 b(o)g(ot)37 b(c)-5 b(orr)g(esp)g(onding)39 b(to)d(a)g(given)94 4074 y(multiplicity)29 b(structur)-5 b(e)29 b FM(`)f FC(and)h(weight)f FM(W)13 b FC(.)40 b(L)-5 b(et)28 b FM(G)1851 4089 y FL(`)1913 4074 y FC(b)-5 b(e)27 b(the)i(c)-5 b(o)g(e\016cient)28 b(op)-5 b(er)g(ator)32 b(asso)-5 b(ciate)g(d)30 b(with)f FM(`)p FC(,)f FM(J)37 b FC(b)-5 b(e)94 4186 y(its)28 b(Jac)-5 b(obian,)30 b(and)e FM(\033)842 4200 y FL(min)1003 4186 y FC(b)-5 b(e)27 b(the)h(smal)5 b(lest)29 b(singular)f(value)g(of)55 b FM(W)13 b(J)c FN(\()p FO(z)p FN(\))p FC(.)41 b(Then)27 b(the)h Fr(c)-5 b(ondition)32 b(numb)-5 b(er)94 4299 y(of)66 b FO(z)f(with)34 b(resp)s(ect)h(to)g(the)g(m)m(ultiplicit)m(y)g (structure)e FM(`)f FO(and)j(w)m(eigh)m(t)g FM(W)45 b FC(is)32 b(de\014ne)-5 b(d)34 b(as)1641 4520 y FM(\024)1693 4535 y FL(`;w)1799 4520 y FN(\()p FO(z)p FN(\))26 b(=)2117 4458 y(1)p 2047 4499 186 4 v 2047 4582 a FM(\033)2099 4596 y FL(min)2242 4520 y FM(:)94 4778 y FO(Remark:)53 b FN(The)35 b(condition)i(n)m(um)m(b)s(er)e FM(\024)1524 4793 y FL(`;w)1629 4778 y FN(\()p FO(z)p FN(\))i(is)f(structure)g(dep)s (enden)m(t.)56 b(The)36 b(arra)m(y)g FM(`)f FN(=)f([)p FM(`)3443 4792 y Fx(1)3483 4778 y FM(;)15 b Fy(\001)g(\001)g(\001)h FM(;)f(`)3722 4792 y FL(m)3789 4778 y FN(])94 4891 y(ma)m(y)24 b(or)f(ma)m(y)g(not)g(b)s(e)g(the)g FC(actual)h FN(m)m(ultiplicit)m(y)g (structure.)38 b(A)23 b(p)s(olynomial)g(has)g(di\013eren)m(t)g (condition)h(n)m(um-)94 5004 y(b)s(ers)29 b(corresp)s(onding)f(to)j (di\013eren)m(t)f(p)s(ejorativ)m(e)g(ro)s(ots)g(on)g(v)-5 b(arious)29 b(p)s(ejorativ)m(e)i(manifolds.)40 b(F)-8 b(or)30 b(example,)94 5117 y(see)h(T)-8 b(able)31 b(4)g(in)f Fy(x)p FN(3.6.2.)94 5324 y(W)-8 b(e)41 b(no)m(w)f(estimate)h(the)f (error)f(on)h(p)s(ejorativ)m(e)g(ro)s(ots)g(of)g(p)s(olynomials)g(with) f(inexact)i(co)s(e\016cien)m(ts.)70 b(In)94 5437 y(this)41 b(case,)j(the)c(giv)m(en)h(p)s(olynomial)48 b(^)-52 b FM(p)39 b FN(is)i(assumed)e(to)i(b)s(e)f(arbitrarily)g(p)s(erturb)s(ed) e(from)h FM(p)h FN(with)g(b)s(oth)1909 5686 y(12)p eop %%Page: 13 15 13 14 bop 94 99 a FN(p)s(olynomials)36 b(near)e(a)h(p)s(ejorativ)m(e)h (manifold)f(\005)1769 114 y FL(`)1802 99 y FN(.)54 b(In)34 b(exact)j(sense,)f(neither)f(p)s(olynomial)g(p)s(ossesses)f(the)94 211 y(structure)41 b FM(`)p FN(.)74 b(The)40 b(nearb)m(y)i(p)s (ejorativ)m(e)g(manifold)f(causes)h(b)s(oth)e(p)s(olynomials)i(b)s (eing)f(ill-conditioned)94 324 y(in)36 b(con)m(v)m(en)m(tional)j (sense.)58 b(Consequen)m(tly)-8 b(,)39 b(the)d(exact)h(ro)s(ots)g(of)44 b(^)-53 b FM(p)36 b FN(can)g(b)s(e)g(far)f(from)h(those)h(of)f FM(p)g FN(ev)m(en)g(if)94 437 y(t)m(w)m(o)i(p)s(olynomials)f(are)g (close)h(to)g(eac)m(h)f(other.)60 b(Ho)m(w)m(ev)m(er,)41 b(the)36 b(follo)m(wing)i(theorem)f(ensures)f(that)h(their)94 550 y(p)s(ejorativ)m(e)32 b(ro)s(ots,)f(not)f(exact)i(ro)s(ots,)f(ma)m (y)g(still)g(b)s(e)f(insensitiv)m(e)i(to)f(p)s(erturbation.)94 786 y FO(Theorem)k(3.3)83 b FC(F)-7 b(or)38 b(a)f(\014xe)-5 b(d)38 b FM(`)33 b FN(=)f([)p FM(`)1462 800 y Fx(1)1502 786 y FM(;)15 b Fy(\001)g(\001)g(\001)h FM(;)f(`)1741 800 y FL(m)1808 786 y FN(])p FC(,)38 b(let)f(the)g(p)-5 b(olynomial)85 b FN(^)-53 b FM(p)32 b Fy(\030)2873 762 y FN(^)2867 786 y FO(b)73 b FC(b)-5 b(e)37 b(an)g(appr)-5 b(oximation)94 899 y(to)73 b FM(p)30 b Fy(\030)h FO(b)36 b FC(with)g(p)-5 b(ejor)g(ative)38 b(r)-5 b(o)g(ots)38 b FO(z)e FC(and)1618 898 y FN(^)1618 899 y FO(z)g FC(r)-5 b(esp)g(e)g(ctively)37 b(that)g(ar)-5 b(e)37 b(c)-5 b(orr)g(esp)g (onding)39 b(to)d(the)g(multiplicity)94 1012 y(structur)-5 b(e)39 b FM(`)e FC(and)i(a)f(weight)g FM(W)13 b FC(.)56 b(Assume)37 b(the)h(c)-5 b(omp)g(onents)41 b(of)75 b FO(z)38 b FC(ar)-5 b(e)38 b(distinct.)58 b(If)75 b Fy(k)15 b FO(b)24 b Fy(\000)3440 988 y FN(^)3434 1012 y FO(b)15 b Fy(k)3552 1026 y FL(W)3670 1012 y FC(and)94 1125 y Fy(k)g FM(G)225 1140 y FL(`)260 1125 y FN(\()p FO(z)p FN(\))21 b Fy(\000)f FO(b)15 b Fy(k)606 1139 y FL(W)720 1125 y FC(ar)-5 b(e)33 b(su\016ciently)g(smal)5 b(l,)33 b(then)710 1247 y Fu(\015)710 1297 y(\015)710 1347 y(\015)771 1345 y FO(z)21 b Fy(\000)929 1344 y FN(^)929 1345 y FO(z)990 1247 y Fu(\015)990 1297 y(\015)990 1347 y(\015)1036 1401 y Fx(2)1101 1345 y Fy(\024)k FN(2)c Fy(\001)f FM(\024)1360 1360 y FL(`;w)1466 1345 y FN(\()p FO(z)p FN(\))h Fy(\001)1648 1251 y Fu(\020)1698 1247 y(\015)1698 1297 y(\015)1698 1347 y(\015)1759 1345 y FM(G)1830 1360 y FL(`)1864 1345 y FN(\()p FO(z)p FN(\))g Fy(\000)f FO(b)2165 1247 y Fu(\015)2165 1297 y(\015)2165 1347 y(\015)2211 1401 y FL(W)2312 1345 y FN(+)2403 1247 y Fu(\015)2403 1297 y(\015)2403 1347 y(\015)2464 1345 y FO(b)h Fy(\000)2640 1321 y FN(^)2633 1345 y FO(b)2707 1247 y Fu(\015)2707 1297 y(\015)2707 1347 y(\015)2753 1401 y FL(W)2833 1251 y Fu(\021)2903 1345 y FN(+)f FM(h:o:t:)455 b FN(\(18\))94 1601 y FO(Pro)s(of.)83 b FN(F)-8 b(rom)31 b(\(17\),)373 1723 y Fu(\015)373 1773 y(\015)373 1823 y(\015)434 1821 y FO(z)21 b Fy(\000)592 1820 y FN(^)592 1821 y FO(z)653 1723 y Fu(\015)653 1773 y(\015)653 1823 y(\015)699 1877 y Fx(2)822 1821 y Fy(\024)83 b FM(\024)1028 1836 y FL(`;w)1133 1821 y FN(\()p FO(z)p FN(\))1264 1723 y Fu(\015)1265 1773 y(\015)1265 1823 y(\015)1327 1821 y FM(G)1398 1836 y FL(`)1431 1821 y FN(\()p FO(z)p FN(\))21 b Fy(\000)f FM(G)1730 1836 y FL(`)1764 1821 y FN(\()1800 1820 y(^)1799 1821 y FO(z)q FN(\))1896 1723 y Fu(\015)1896 1773 y(\015)1896 1823 y(\015)1942 1877 y FL(W)2043 1821 y FN(+)g FM(h:o:t:)822 2007 y Fy(\024)83 b FM(\024)1028 2022 y FL(`;w)1133 2007 y FN(\()p FO(z)p FN(\))1264 1913 y Fu(\020)1330 1910 y(\015)1330 1960 y(\015)1330 2009 y(\015)1391 2007 y FM(G)1462 2022 y FL(`)1496 2007 y FN(\()p FO(z)p FN(\))21 b Fy(\000)f FO(b)1797 1910 y Fu(\015)1797 1960 y(\015)1797 2009 y(\015)1843 2063 y FL(W)1944 2007 y FN(+)2035 1910 y Fu(\015)2035 1960 y(\015)2035 2009 y(\015)2096 2007 y FO(b)h Fy(\000)2272 1983 y FN(^)2266 2007 y FO(b)2339 1910 y Fu(\015)2339 1960 y(\015)2339 2009 y(\015)2385 2063 y FL(W)2486 2007 y FN(+)2577 1910 y Fu(\015)2577 1960 y(\015)2577 2009 y(\015)2638 2007 y FM(G)2709 2022 y FL(`)2742 2007 y FN(\()2778 2006 y(^)2777 2007 y FO(z)q FN(\))g Fy(\000)2977 1983 y FN(^)2971 2007 y FO(b)3044 1910 y Fu(\015)3044 1960 y(\015)3044 2009 y(\015)3090 2063 y FL(W)3170 1913 y Fu(\021)3240 2007 y FN(+)f FM(h:o:t:)94 2250 y FN(Since)31 b Fy(k)15 b FM(G)463 2265 y FL(`)497 2250 y FN(\()533 2249 y(^)532 2250 y FO(z)q FN(\))20 b Fy(\000)732 2226 y FN(^)725 2250 y FO(b)15 b Fy(k)843 2264 y FL(W)955 2250 y FN(is)30 b(a)h(lo)s(cal)h(minim)m(um,)d(w)m(e)i (ha)m(v)m(e)659 2373 y Fu(\015)659 2422 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2307 3507 mt 2379 3507 L 2343 3471 mt 2343 3543 L 2717 3661 mt 2789 3661 L 2753 3625 mt 2753 3697 L 2634 3513 mt 2706 3513 L 2670 3477 mt 2670 3549 L 2238 3601 mt 2310 3601 L 2274 3565 mt 2274 3637 L 2233 3730 mt 2305 3730 L 2269 3694 mt 2269 3766 L 2355 3484 mt 2427 3484 L 2391 3448 mt 2391 3520 L 2489 3888 mt 2561 3888 L 2525 3852 mt 2525 3924 L 2637 3829 mt 2709 3829 L 2673 3793 mt 2673 3865 L 2649 3816 mt 2721 3816 L 2685 3780 mt 2685 3852 L 2239 3733 mt 2311 3733 L 2275 3697 mt 2275 3769 L 2473 3884 mt 2545 3884 L 2509 3848 mt 2509 3920 L 2681 3581 mt 2753 3581 L 2717 3545 mt 2717 3617 L 2684 3751 mt 2756 3751 L 2720 3715 mt 2720 3787 L 2517 3475 mt 2589 3475 L 2553 3439 mt 2553 3511 L 2695 3696 mt 2767 3696 L 2731 3660 mt 2731 3732 L 2252 3734 mt 2324 3734 L 2288 3698 mt 2288 3770 L 2643 3549 mt 2715 3549 L 2679 3513 mt 2679 3585 L 2470 3872 mt 2542 3872 L 2506 3836 mt 2506 3908 L 2572 3850 mt 2644 3850 L 2608 3814 mt 2608 3886 L 2247 3701 mt 2319 3701 L 2283 3665 mt 2283 3737 L 2541 3490 mt 2613 3490 L 2577 3454 mt 2577 3526 L 2591 3513 mt 2663 3513 L 2627 3477 mt 2627 3549 L 2351 3511 mt 2423 3511 L 2387 3475 mt 2387 3547 L 2502 3863 mt 2574 3863 L 2538 3827 mt 2538 3899 L 2264 3733 mt 2336 3733 L 2300 3697 mt 2300 3769 L 2259 3627 mt 2331 3627 L 2295 3591 mt 2295 3663 L 2468 3862 mt 2540 3862 L 2504 3826 mt 2504 3898 L 2678 3637 mt 2750 3637 L 2714 3601 mt 2714 3673 L 2345 3520 mt 2417 3520 L 2381 3484 mt 2381 3556 L 2636 3784 mt 2708 3784 L 2672 3748 mt 2672 3820 L 2502 3494 mt 2574 3494 L 2538 3458 mt 2538 3530 L 2296 3574 mt 2368 3574 L 2332 3538 mt 2332 3610 L 2276 3607 mt 2348 3607 L 2312 3571 mt 2312 3643 L 2446 3493 mt 2518 3493 L 2482 3457 mt 2482 3529 L 2661 3743 mt 2733 3743 L 2697 3707 mt 2697 3779 L 2276 3732 mt 2348 3732 L 2312 3696 mt 2312 3768 L 2266 3657 mt 2338 3657 L 2302 3621 mt 2302 3693 L 2467 3852 mt 2539 3852 L 2503 3816 mt 2503 3888 L 2316 3558 mt 2388 3558 L 2352 3522 mt 2352 3594 L 2421 3501 mt 2493 3501 L 2457 3465 mt 2457 3537 L 2600 3811 mt 2672 3811 L 2636 3775 mt 2636 3847 L 2650 3595 mt 2722 3595 L 2686 3559 mt 2686 3631 L 2521 3845 mt 2593 3845 L 2557 3809 mt 2557 3881 L 2644 3761 mt 2716 3761 L 2680 3725 mt 2680 3797 L 2563 3829 mt 2635 3829 L 2599 3793 mt 2599 3865 L 2286 3730 mt 2358 3730 L 2322 3694 mt 2322 3766 L 2466 3844 mt 2538 3844 L 2502 3808 mt 2502 3880 L 2282 3705 mt 2354 3705 L 2318 3669 mt 2318 3741 L 2586 3539 mt 2658 3539 L 2622 3503 mt 2622 3575 L 2438 3508 mt 2510 3508 L 2474 3472 mt 2474 3544 L 2662 3678 mt 2734 3678 L 2698 3642 mt 2698 3714 L 2282 3684 mt 2354 3684 L 2318 3648 mt 2318 3720 L 2387 3526 mt 2459 3526 L 2423 3490 mt 2423 3562 L 2296 3728 mt 2368 3728 L 2332 3692 mt 2332 3764 L 2529 3519 mt 2601 3519 L 2565 3483 mt 2565 3555 L 2631 3595 mt 2703 3595 L 2667 3559 mt 2667 3631 L 2602 3561 mt 2674 3561 L 2638 3525 mt 2638 3597 L 2465 3836 mt 2537 3836 L 2501 3800 mt 2501 3872 L 2654 3680 mt 2726 3680 L 2690 3644 mt 2690 3716 L 2289 3657 mt 2361 3657 L 2325 3621 mt 2325 3693 L 2619 3765 mt 2691 3765 L 2655 3729 mt 2655 3801 L 2491 3830 mt 2563 3830 L 2527 3794 mt 2527 3866 L 2323 3586 mt 2395 3586 L 2359 3550 mt 2359 3622 L 2305 3727 mt 2377 3727 L 2341 3691 mt 2341 3763 L 2537 3817 mt 2609 3817 L 2573 3781 mt 2573 3853 L 2643 3705 mt 2715 3705 L 2679 3669 mt 2679 3741 L 2488 3520 mt 2560 3520 L 2524 3484 mt 2524 3556 L 2464 3828 mt 2536 3828 L 2500 3792 mt 2500 3864 L 2303 3710 mt 2375 3710 L 2339 3674 mt 2339 3746 L 2635 3629 mt 2707 3629 L 2671 3593 mt 2671 3665 L 2551 3542 mt 2623 3542 L 2587 3506 mt 2587 3578 L 2453 3525 mt 2525 3525 L 2489 3489 mt 2489 3561 L 2313 3725 mt 2385 3725 L 2349 3689 mt 2349 3761 L 2311 3629 mt 2383 3629 L 2347 3593 mt 2347 3665 L 2630 3715 mt 2702 3715 L 2666 3679 mt 2666 3751 L 2463 3821 mt 2535 3821 L 2499 3785 mt 2499 3857 L 2631 3639 mt 2703 3639 L 2667 3603 mt 2667 3675 L 2339 3584 mt 2411 3584 L 2375 3548 mt 2375 3620 L 2394 3544 mt 2466 3544 L 2430 3508 mt 2430 3580 L 2365 3563 mt 2437 3563 L 2401 3527 mt 2401 3599 L 2564 3791 mt 2636 3791 L 2600 3755 mt 2600 3827 L 2518 3810 mt 2590 3810 L 2554 3774 mt 2554 3846 L 2572 3782 mt 2644 3782 L 2608 3746 mt 2608 3818 L 2321 3723 mt 2393 3723 L 2357 3687 mt 2357 3759 L 2476 3814 mt 2548 3814 L 2512 3778 mt 2512 3850 L 2463 3814 mt 2535 3814 L 2499 3778 mt 2499 3850 L 2629 3660 mt 2701 3660 L 2665 3624 mt 2665 3696 L 2522 3542 mt 2594 3542 L 2558 3506 mt 2558 3578 L 2313 3681 mt 2385 3681 L 2349 3645 mt 2349 3717 L 2321 3629 mt 2393 3629 L 2357 3593 mt 2357 3665 L 2407 3547 mt 2479 3547 L 2443 3511 mt 2443 3583 L 2591 3760 mt 2663 3760 L 2627 3724 mt 2627 3796 L 2625 3679 mt 2697 3679 L 2661 3643 mt 2661 3715 L 2594 3591 mt 2666 3591 L 2630 3555 mt 2630 3627 L 2566 3567 mt 2638 3567 L 2602 3531 mt 2602 3603 L 2586 3584 mt 2658 3584 L 2622 3548 mt 2622 3620 L 2381 3563 mt 2453 3563 L 2417 3527 mt 2417 3599 L 2328 3721 mt 2400 3721 L 2364 3685 mt 2364 3757 L 2462 3807 mt 2534 3807 L 2498 3771 mt 2498 3843 L 2487 3541 mt 2559 3541 L 2523 3505 mt 2523 3577 L 2349 3594 mt 2421 3594 L 2385 3558 mt 2385 3630 L 2596 3603 mt 2668 3603 L 2632 3567 mt 2632 3639 L 2602 3737 mt 2674 3737 L 2638 3701 mt 2638 3773 L 2550 3566 mt 2622 3566 L 2586 3530 mt 2586 3602 L 2341 3614 mt 2413 3614 L 2377 3578 mt 2377 3650 L 2334 3719 mt 2406 3719 L 2370 3683 mt 2370 3755 L 2611 3646 mt 2683 3646 L 2647 3610 mt 2647 3682 L 2464 3546 mt 2536 3546 L 2500 3510 mt 2500 3582 L 2461 3801 mt 2533 3801 L 2497 3765 mt 2497 3837 L 2512 3553 mt 2584 3553 L 2548 3517 mt 2548 3589 L 2459 3548 mt 2531 3548 L 2495 3512 mt 2495 3584 L 2328 3657 mt 2400 3657 L 2364 3621 mt 2364 3693 L 2334 3639 mt 2406 3639 L 2370 3603 mt 2370 3675 L 2511 3792 mt 2583 3792 L 2547 3756 mt 2547 3828 L 2441 3553 mt 2513 3553 L 2477 3517 mt 2477 3589 L 2402 3566 mt 2474 3566 L 2438 3530 mt 2438 3602 L 2339 3718 mt 2411 3718 L 2375 3682 mt 2375 3754 L 2602 3716 mt 2674 3716 L 2638 3680 mt 2638 3752 L 2332 3684 mt 2404 3684 L 2368 3648 mt 2368 3720 L 2570 3592 mt 2642 3592 L 2606 3556 mt 2606 3628 L 2461 3795 mt 2533 3795 L 2497 3759 mt 2497 3831 L 2588 3616 mt 2660 3616 L 2624 3580 mt 2624 3652 L 2363 3602 mt 2435 3602 L 2399 3566 mt 2399 3638 L 2346 3717 mt 2418 3717 L 2382 3681 mt 2382 3753 L 2531 3779 mt 2603 3779 L 2567 3743 mt 2567 3815 L 2601 3695 mt 2673 3695 L 2637 3659 mt 2637 3731 L 2584 3735 mt 2656 3735 L 2620 3699 mt 2620 3771 L 2422 3565 mt 2494 3565 L 2458 3529 mt 2458 3601 L 2342 3697 mt 2414 3697 L 2378 3661 mt 2378 3733 L 2462 3789 mt 2534 3789 L 2498 3753 mt 2498 3825 L 2556 3760 mt 2628 3760 L 2592 3724 mt 2592 3796 L 2551 3763 mt 2623 3763 L 2587 3727 mt 2587 3799 L 2593 3640 mt 2665 3640 L 2629 3604 mt 2629 3676 L 2510 3781 mt 2582 3781 L 2546 3745 mt 2546 3817 L 2484 3786 mt 2556 3786 L 2520 3750 mt 2520 3822 L 2466 3787 mt 2538 3787 L 2502 3751 mt 2502 3823 L 2598 3673 mt 2670 3673 L 2634 3637 mt 2634 3709 L 2588 3715 mt 2660 3715 L 2624 3679 mt 2624 3751 L 2387 3587 mt 2459 3587 L 2423 3551 mt 2423 3623 L 2351 3715 mt 2423 3715 L 2387 3679 mt 2387 3751 L 2348 3705 mt 2420 3705 L 2384 3669 mt 2384 3741 L 2528 3577 mt 2600 3577 L 2564 3541 mt 2564 3613 L 2461 3782 mt 2533 3782 L 2497 3746 mt 2497 3818 L 2576 3618 mt 2648 3618 L 2612 3582 mt 2612 3654 L 2558 3748 mt 2630 3748 L 2594 3712 mt 2594 3784 L 2550 3592 mt 2622 3592 L 2586 3556 mt 2586 3628 L 2514 3573 mt 2586 3573 L 2550 3537 mt 2550 3609 L 2356 3714 mt 2428 3714 L 2392 3678 mt 2392 3750 L 2349 3675 mt 2421 3675 L 2385 3639 mt 2385 3711 L 2360 3628 mt 2432 3628 L 2396 3592 mt 2396 3664 L 2353 3646 mt 2425 3646 L 2389 3610 mt 2389 3682 L 2562 3740 mt 2634 3740 L 2598 3704 mt 2598 3776 L 2405 3585 mt 2477 3585 L 2441 3549 mt 2441 3621 L 2434 3573 mt 2506 3573 L 2470 3537 mt 2470 3609 L 2469 3568 mt 2541 3568 L 2505 3532 mt 2505 3604 L 2483 3570 mt 2555 3570 L 2519 3534 mt 2519 3606 L 2502 3772 mt 2574 3772 L 2538 3736 mt 2538 3808 L 2461 3776 mt 2533 3776 L 2497 3740 mt 2497 3812 L 2466 3776 mt 2538 3776 L 2502 3740 mt 2502 3812 L 2361 3713 mt 2433 3713 L 2397 3677 mt 2397 3749 L 2583 3653 mt 2655 3653 L 2619 3617 mt 2619 3689 L 2586 3670 mt 2658 3670 L 2622 3634 mt 2622 3706 L 2583 3693 mt 2655 3693 L 2619 3657 mt 2619 3729 L 2372 3618 mt 2444 3618 L 2408 3582 mt 2408 3654 L 2527 3761 mt 2599 3761 L 2563 3725 mt 2563 3797 L 2578 3649 mt 2650 3649 L 2614 3613 mt 2614 3685 L 2461 3771 mt 2533 3771 L 2497 3735 mt 2497 3807 L 2503 3580 mt 2575 3580 L 2539 3544 mt 2539 3616 L 2539 3598 mt 2611 3598 L 2575 3562 mt 2575 3634 L 2366 3711 mt 2438 3711 L 2402 3675 mt 2402 3747 L 2563 3724 mt 2635 3724 L 2599 3688 mt 2599 3760 L 2359 3664 mt 2431 3664 L 2395 3628 mt 2395 3700 L 2453 3578 mt 2525 3578 L 2489 3542 mt 2489 3614 L 2392 3604 mt 2464 3604 L 2428 3568 mt 2428 3640 L 2386 3614 mt 2458 3614 L 2422 3578 mt 2422 3650 L 2461 3767 mt 2533 3767 L 2497 3731 mt 2497 3803 L 2572 3698 mt 2644 3698 L 2608 3662 mt 2608 3734 L 2534 3599 mt 2606 3599 L 2570 3563 mt 2570 3635 L 2560 3625 mt 2632 3625 L 2596 3589 mt 2596 3661 L 2370 3710 mt 2442 3710 L 2406 3674 mt 2406 3746 L 2519 3592 mt 2591 3592 L 2555 3556 mt 2555 3628 L 2377 3628 mt 2449 3628 L 2413 3592 mt 2413 3664 L 2364 3668 mt 2436 3668 L 2400 3632 mt 2400 3704 L 2513 3757 mt 2585 3757 L 2549 3721 mt 2549 3793 L 2415 3595 mt 2487 3595 L 2451 3559 mt 2451 3631 L 2548 3614 mt 2620 3614 L 2584 3578 mt 2584 3650 L 2482 3584 mt 2554 3584 L 2518 3548 mt 2518 3620 L 2404 3604 mt 2476 3604 L 2440 3568 mt 2440 3640 L 2461 3763 mt 2533 3763 L 2497 3727 mt 2497 3799 L 2375 3709 mt 2447 3709 L 2411 3673 mt 2411 3745 L 2370 3654 mt 2442 3654 L 2406 3618 mt 2406 3690 L 2491 3760 mt 2563 3760 L 2527 3724 mt 2527 3796 L 2558 3716 mt 2630 3716 L 2594 3680 mt 2594 3752 L 2454 3586 mt 2526 3586 L 2490 3550 mt 2490 3622 L 2370 3690 mt 2442 3690 L 2406 3654 mt 2406 3726 L 2370 3683 mt 2442 3683 L 2406 3647 mt 2406 3719 L 2569 3671 mt 2641 3671 L 2605 3635 mt 2605 3707 L 2433 3592 mt 2505 3592 L 2469 3556 mt 2469 3628 L 2492 3589 mt 2564 3589 L 2528 3553 mt 2528 3625 L 2548 3725 mt 2620 3725 L 2584 3689 mt 2584 3761 L 2526 3744 mt 2598 3744 L 2562 3708 mt 2562 3780 L 2461 3759 mt 2533 3759 L 2497 3723 mt 2497 3795 L 2554 3630 mt 2626 3630 L 2590 3594 mt 2590 3666 L 2562 3650 mt 2634 3650 L 2598 3614 mt 2598 3686 L 2564 3693 mt 2636 3693 L 2600 3657 mt 2600 3729 L 2379 3707 mt 2451 3707 L 2415 3671 mt 2415 3743 L 2546 3623 mt 2618 3623 L 2582 3587 mt 2582 3659 L 2482 3756 mt 2554 3756 L 2518 3720 mt 2518 3792 L 2559 3648 mt 2631 3648 L 2595 3612 mt 2595 3684 L 2378 3649 mt 2450 3649 L 2414 3613 mt 2414 3685 L 2536 3731 mt 2608 3731 L 2572 3695 mt 2572 3767 L 2462 3756 mt 2534 3756 L 2498 3720 mt 2498 3792 L 2436 3597 mt 2508 3597 L 2472 3561 mt 2472 3633 L 2388 3633 mt 2460 3633 L 2424 3597 mt 2424 3669 L 2383 3705 mt 2455 3705 L 2419 3669 mt 2419 3741 L 2519 3742 mt 2591 3742 L 2555 3706 mt 2555 3778 L 2507 3599 mt 2579 3599 L 2543 3563 mt 2543 3635 L 2383 3703 mt 2455 3703 L 2419 3667 mt 2419 3739 L 2560 3677 mt 2632 3677 L 2596 3641 mt 2596 3713 L 2463 3753 mt 2535 3753 L 2499 3717 mt 2499 3789 L 2470 3595 mt 2542 3595 L 2506 3559 mt 2506 3631 L 2449 3597 mt 2521 3597 L 2485 3561 mt 2485 3633 L 2553 3699 mt 2625 3699 L 2589 3663 mt 2589 3735 L 2406 3618 mt 2478 3618 L 2442 3582 mt 2442 3654 L 2505 3744 mt 2577 3744 L 2541 3708 mt 2541 3780 L 2498 3600 mt 2570 3600 L 2534 3564 mt 2534 3636 L 2549 3706 mt 2621 3706 L 2585 3670 mt 2585 3742 L 2462 3750 mt 2534 3750 L 2498 3714 mt 2498 3786 L 2388 3704 mt 2460 3704 L 2424 3668 mt 2424 3740 L 2476 3749 mt 2548 3749 L 2512 3713 mt 2512 3785 L 2421 3609 mt 2493 3609 L 2457 3573 mt 2457 3645 L 2530 3619 mt 2602 3619 L 2566 3583 mt 2566 3655 L 2540 3716 mt 2612 3716 L 2576 3680 mt 2576 3752 L 2384 3663 mt 2456 3663 L 2420 3627 mt 2420 3699 L 2387 3694 mt 2459 3694 L 2423 3658 mt 2423 3730 L 2389 3646 mt 2461 3646 L 2425 3610 mt 2425 3682 L 2384 3671 mt 2456 3671 L 2420 3635 mt 2420 3707 L 2554 3666 mt 2626 3666 L 2590 3630 mt 2590 3702 L 2462 3747 mt 2534 3747 L 2498 3711 mt 2498 3783 L 2392 3703 mt 2464 3703 L 2428 3667 mt 2428 3739 L 2497 3741 mt 2569 3741 L 2533 3705 mt 2533 3777 L 2548 3655 mt 2620 3655 L 2584 3619 mt 2584 3691 L 2514 3613 mt 2586 3613 L 2550 3577 mt 2550 3649 L 2480 3601 mt 2552 3601 L 2516 3565 mt 2516 3637 L 2388 3680 mt 2460 3680 L 2424 3644 mt 2424 3716 L 2502 3608 mt 2574 3608 L 2538 3572 mt 2538 3644 L 2462 3743 mt 2534 3743 L 2498 3707 mt 2498 3779 L 2395 3702 mt 2467 3702 L 2431 3666 mt 2431 3738 L 2415 3621 mt 2487 3621 L 2451 3585 mt 2451 3657 L 2405 3630 mt 2477 3630 L 2441 3594 mt 2441 3666 L 2544 3696 mt 2616 3696 L 2580 3660 mt 2580 3732 L 2525 3721 mt 2597 3721 L 2561 3685 mt 2561 3757 L 2547 3680 mt 2619 3680 L 2583 3644 mt 2583 3716 L 2539 3643 mt 2611 3643 L 2575 3607 mt 2575 3679 L 2529 3628 mt 2601 3628 L 2565 3592 mt 2565 3664 L 2513 3729 mt 2585 3729 L 2549 3693 mt 2549 3765 L 2434 3612 mt 2506 3612 L 2470 3576 mt 2470 3648 L 2466 3605 mt 2538 3605 L 2502 3569 mt 2502 3641 L 2535 3638 mt 2607 3638 L 2571 3602 mt 2571 3674 L 2462 3740 mt 2534 3740 L 2498 3704 mt 2498 3776 L 2398 3700 mt 2470 3700 L 2434 3664 mt 2434 3736 L 2394 3658 mt 2466 3658 L 2430 3622 mt 2430 3694 L 2405 3637 mt 2477 3637 L 2441 3601 mt 2441 3673 L 2473 3739 mt 2545 3739 L 2509 3703 mt 2509 3775 L 2487 3737 mt 2559 3737 L 2523 3701 mt 2523 3773 L 2400 3699 mt 2472 3699 L 2436 3663 mt 2436 3735 L 2450 3609 mt 2522 3609 L 2486 3573 mt 2486 3645 L 2459 3608 mt 2531 3608 L 2495 3572 mt 2495 3644 L 2463 3738 mt 2535 3738 L 2499 3702 mt 2499 3774 L 2398 3692 mt 2470 3692 L 2434 3656 mt 2434 3728 L 2418 3626 mt 2490 3626 L 2454 3590 mt 2454 3662 L 2402 3645 mt 2474 3645 L 2438 3609 mt 2438 3681 L 2527 3712 mt 2599 3712 L 2563 3676 mt 2563 3748 L 2523 3629 mt 2595 3629 L 2559 3593 mt 2559 3665 L 2542 3669 mt 2614 3669 L 2578 3633 mt 2578 3705 L 2519 3719 mt 2591 3719 L 2555 3683 mt 2555 3755 L 2535 3648 mt 2607 3648 L 2571 3612 mt 2571 3684 L 2480 3610 mt 2552 3610 L 2516 3574 mt 2516 3646 L 2538 3657 mt 2610 3657 L 2574 3621 mt 2574 3693 L 2533 3701 mt 2605 3701 L 2569 3665 mt 2569 3737 L 2440 3615 mt 2512 3615 L 2476 3579 mt 2476 3651 L 2513 3624 mt 2585 3624 L 2549 3588 mt 2549 3660 L 2495 3615 mt 2567 3615 L 2531 3579 mt 2531 3651 L 2403 3699 mt 2475 3699 L 2439 3663 mt 2439 3735 L 2463 3735 mt 2535 3735 L 2499 3699 mt 2499 3771 L 2398 3669 mt 2470 3669 L 2434 3633 mt 2434 3705 L 2508 3623 mt 2580 3623 L 2544 3587 mt 2544 3659 L 2477 3733 mt 2549 3733 L 2513 3697 mt 2513 3769 L 2497 3728 mt 2569 3728 L 2533 3692 mt 2533 3764 L 2471 3613 mt 2543 3613 L 2507 3577 mt 2507 3649 L 2401 3683 mt 2473 3683 L 2437 3647 mt 2437 3719 L 2407 3698 mt 2479 3698 L 2443 3662 mt 2443 3734 L 2463 3732 mt 2535 3732 L 2499 3696 mt 2499 3768 L 2426 3627 mt 2498 3627 L 2462 3591 mt 2462 3663 L 2536 3674 mt 2608 3674 L 2572 3638 mt 2572 3710 L 2535 3683 mt 2607 3683 L 2571 3647 mt 2571 3719 L 2508 3721 mt 2580 3721 L 2544 3685 mt 2544 3757 L 2482 3730 mt 2554 3730 L 2518 3694 mt 2518 3766 L 2534 3669 mt 2606 3669 L 2570 3633 mt 2570 3705 L 2404 3661 mt 2476 3661 L 2440 3625 mt 2440 3697 L 2409 3696 mt 2481 3696 L 2445 3660 mt 2445 3732 L 2463 3730 mt 2535 3730 L 2499 3694 mt 2499 3766 L 2499 3723 mt 2571 3723 L 2535 3687 mt 2535 3759 L 2530 3692 mt 2602 3692 L 2566 3656 mt 2566 3728 L 2407 3654 mt 2479 3654 L 2443 3618 mt 2443 3690 L 2494 3620 mt 2566 3620 L 2530 3584 mt 2530 3656 L 2419 3637 mt 2491 3637 L 2455 3601 mt 2455 3673 L 2435 3625 mt 2507 3625 L 2471 3589 mt 2471 3661 L 2451 3619 mt 2523 3619 L 2487 3583 mt 2487 3655 L 2518 3708 mt 2590 3708 L 2554 3672 mt 2554 3744 L 2412 3695 mt 2484 3695 L 2448 3659 mt 2448 3731 L 2408 3685 mt 2480 3685 L 2444 3649 mt 2444 3721 L 2406 3674 mt 2478 3674 L 2442 3638 mt 2442 3710 L 2464 3727 mt 2536 3727 L 2500 3691 mt 2500 3763 L 2522 3646 mt 2594 3646 L 2558 3610 mt 2558 3682 L 2518 3641 mt 2590 3641 L 2554 3605 mt 2554 3677 L 2416 3646 mt 2488 3646 L 2452 3610 mt 2452 3682 L 2477 3620 mt 2549 3620 L 2513 3584 mt 2513 3656 L 2468 3620 mt 2540 3620 L 2504 3584 mt 2504 3656 L 2521 3700 mt 2593 3700 L 2557 3664 mt 2557 3736 L 2511 3711 mt 2583 3711 L 2547 3675 mt 2547 3747 L 2528 3661 mt 2600 3661 L 2564 3625 mt 2564 3697 L 2464 3725 mt 2536 3725 L 2500 3689 mt 2500 3761 L 2414 3694 mt 2486 3694 L 2450 3658 mt 2450 3730 L 2509 3634 mt 2581 3634 L 2545 3598 mt 2545 3670 L 2486 3723 mt 2558 3723 L 2522 3687 mt 2522 3759 L 2525 3689 mt 2597 3689 L 2561 3653 mt 2561 3725 L 2414 3691 mt 2486 3691 L 2450 3655 mt 2450 3727 L 2525 3657 mt 2597 3657 L 2561 3621 mt 2561 3693 L 2428 3636 mt 2500 3636 L 2464 3600 mt 2464 3672 L 2470 3724 mt 2542 3724 L 2506 3688 mt 2506 3760 L 2479 3723 mt 2551 3723 L 2515 3687 mt 2515 3759 L 2494 3627 mt 2566 3627 L 2530 3591 mt 2530 3663 L 2465 3723 mt 2537 3723 L 2501 3687 mt 2501 3759 L 2462 3623 mt 2534 3623 L 2498 3587 mt 2498 3659 L 2417 3693 mt 2489 3693 L 2453 3657 mt 2453 3729 L 2412 3677 mt 2484 3677 L 2448 3641 mt 2448 3713 L 2521 3692 mt 2593 3692 L 2557 3656 mt 2557 3728 L 2444 3628 mt 2516 3628 L 2480 3592 mt 2480 3664 L 2472 3623 mt 2544 3623 L 2508 3587 mt 2508 3659 L 2418 3651 mt 2490 3651 L 2454 3615 mt 2454 3687 L 2417 3655 mt 2489 3655 L 2453 3619 mt 2453 3691 L 2434 3634 mt 2506 3634 L 2470 3598 mt 2470 3670 L 2510 3640 mt 2582 3640 L 2546 3604 mt 2546 3676 L 2414 3668 mt 2486 3668 L 2450 3632 mt 2450 3704 L 2415 3665 mt 2487 3665 L 2451 3629 mt 2451 3701 L 2487 3627 mt 2559 3627 L 2523 3591 mt 2523 3663 L 2465 3721 mt 2537 3721 L 2501 3685 mt 2501 3757 L 2523 3676 mt 2595 3676 L 2559 3640 mt 2559 3712 L 2420 3692 mt 2492 3692 L 2456 3656 mt 2456 3728 L 2499 3712 mt 2571 3712 L 2535 3676 mt 2535 3748 L 2490 3716 mt 2562 3716 L 2526 3680 mt 2526 3752 L 2499 3634 mt 2571 3634 L 2535 3598 mt 2535 3670 L 2418 3685 mt 2490 3685 L 2454 3649 mt 2454 3721 L 2518 3656 mt 2590 3656 L 2554 3620 mt 2554 3692 L 2420 3690 mt 2492 3690 L 2456 3654 mt 2456 3726 L 2454 3628 mt 2526 3628 L 2490 3592 mt 2490 3664 L 2514 3697 mt 2586 3697 L 2550 3661 mt 2550 3733 L 2465 3719 mt 2537 3719 L 2501 3683 mt 2501 3755 L 2493 3632 mt 2565 3632 L 2529 3596 mt 2529 3668 L 2471 3718 mt 2543 3718 L 2507 3682 mt 2507 3754 L 2519 3683 mt 2591 3683 L 2555 3647 mt 2555 3719 L 2422 3691 mt 2494 3691 L 2458 3655 mt 2458 3727 L 2512 3649 mt 2584 3649 L 2548 3613 mt 2548 3685 L 2430 3644 mt 2502 3644 L 2466 3608 mt 2466 3680 L 2441 3636 mt 2513 3636 L 2477 3600 mt 2477 3672 L 2503 3706 mt 2575 3706 L 2539 3670 mt 2539 3742 L 2465 3717 mt 2537 3717 L 2501 3681 mt 2501 3753 L 2502 3640 mt 2574 3640 L 2538 3604 mt 2538 3676 L 2518 3677 mt 2590 3677 L 2554 3641 mt 2554 3713 L 2518 3670 mt 2590 3670 L 2554 3634 mt 2554 3706 L 2508 3647 mt 2580 3647 L 2544 3611 mt 2544 3683 L 2426 3652 mt 2498 3652 L 2462 3616 mt 2462 3688 L 2477 3631 mt 2549 3631 L 2513 3595 mt 2513 3667 L 2449 3634 mt 2521 3634 L 2485 3598 mt 2485 3670 L 2424 3690 mt 2496 3690 L 2460 3654 mt 2460 3726 L 2517 3668 mt 2589 3668 L 2553 3632 mt 2553 3704 L 2472 3631 mt 2544 3631 L 2508 3595 mt 2508 3667 L 2498 3707 mt 2570 3707 L 2534 3671 mt 2534 3743 L 2465 3715 mt 2537 3715 L 2501 3679 mt 2501 3751 L 2506 3699 mt 2578 3699 L 2542 3663 mt 2542 3735 L 2423 3661 mt 2495 3661 L 2459 3625 mt 2459 3697 L 2485 3634 mt 2557 3634 L 2521 3598 mt 2521 3670 L 2426 3689 mt 2498 3689 L 2462 3653 mt 2462 3725 L 2511 3656 mt 2583 3656 L 2547 3620 mt 2547 3692 L 2464 3633 mt 2536 3633 L 2500 3597 mt 2500 3669 L 2423 3670 mt 2495 3670 L 2459 3634 mt 2459 3706 L 2423 3678 mt 2495 3678 L 2459 3642 mt 2459 3714 L 2458 3634 mt 2530 3634 L 2494 3598 mt 2494 3670 L 2466 3713 mt 2538 3713 L 2502 3677 mt 2502 3749 L 2483 3711 mt 2555 3711 L 2519 3675 mt 2519 3747 L 2488 3709 mt 2560 3709 L 2524 3673 mt 2524 3745 L 2509 3691 mt 2581 3691 L 2545 3655 mt 2545 3727 L 2428 3688 mt 2500 3688 L 2464 3652 mt 2464 3724 L 2433 3650 mt 2505 3650 L 2469 3614 mt 2469 3686 L 2512 3663 mt 2584 3663 L 2548 3627 mt 2548 3699 L 2438 3645 mt 2510 3645 L 2474 3609 mt 2474 3681 L 2491 3639 mt 2563 3639 L 2527 3603 mt 2527 3675 L 2455 3636 mt 2527 3636 L 2491 3600 mt 2491 3672 L 2427 3662 mt 2499 3662 L 2463 3626 mt 2463 3698 L 2426 3674 mt 2498 3674 L 2462 3638 mt 2462 3710 L 2430 3655 mt 2502 3655 L 2466 3619 mt 2466 3691 L 2466 3711 mt 2538 3711 L 2502 3675 mt 2502 3747 L 2430 3687 mt 2502 3687 L 2466 3651 mt 2466 3723 L 2442 3644 mt 2514 3644 L 2478 3608 mt 2478 3680 L 2477 3636 mt 2549 3636 L 2513 3600 mt 2513 3672 L 2475 3710 mt 2547 3710 L 2511 3674 mt 2511 3746 L 2467 3710 mt 2539 3710 L 2503 3674 mt 2503 3746 L 2495 3644 mt 2567 3644 L 2531 3608 mt 2531 3680 L 2511 3671 mt 2583 3671 L 2547 3635 mt 2547 3707 L 2481 3708 mt 2553 3708 L 2517 3672 mt 2517 3744 L 2510 3683 mt 2582 3683 L 2546 3647 mt 2546 3719 L 2495 3702 mt 2567 3702 L 2531 3666 mt 2531 3738 L 2446 3643 mt 2518 3643 L 2482 3607 mt 2482 3679 L 2502 3696 mt 2574 3696 L 2538 3660 mt 2538 3732 L 2502 3650 mt 2574 3650 L 2538 3614 mt 2538 3686 L 2508 3660 mt 2580 3660 L 2544 3624 mt 2544 3696 L 2431 3687 mt 2503 3687 L 2467 3651 mt 2467 3723 L 2470 3637 mt 2542 3637 L 2506 3601 mt 2506 3673 L 2429 3679 mt 2501 3679 L 2465 3643 mt 2465 3715 L 2467 3708 mt 2539 3708 L 2503 3672 mt 2503 3744 L 2469 3708 mt 2541 3708 L 2505 3672 mt 2505 3744 L 2486 3641 mt 2558 3641 L 2522 3605 mt 2522 3677 L 2496 3647 mt 2568 3647 L 2532 3611 mt 2532 3683 L 2507 3685 mt 2579 3685 L 2543 3649 mt 2543 3721 L 2508 3677 mt 2580 3677 L 2544 3641 mt 2544 3713 L 2465 3639 mt 2537 3639 L 2501 3603 mt 2501 3675 L 2437 3653 mt 2509 3653 L 2473 3617 mt 2473 3689 L 2432 3685 mt 2504 3685 L 2468 3649 mt 2468 3721 L 2503 3691 mt 2575 3691 L 2539 3655 mt 2539 3727 L 2433 3686 mt 2505 3686 L 2469 3650 mt 2469 3722 L 2501 3655 mt 2573 3655 L 2537 3619 mt 2537 3691 L 2467 3706 mt 2539 3706 L 2503 3670 mt 2503 3742 L 2452 3643 mt 2524 3643 L 2488 3607 mt 2488 3679 L 2496 3697 mt 2568 3697 L 2532 3661 mt 2532 3733 L 2480 3642 mt 2552 3642 L 2516 3606 mt 2516 3678 L 2505 3665 mt 2577 3665 L 2541 3629 mt 2541 3701 L 2440 3652 mt 2512 3652 L 2476 3616 mt 2476 3688 L 2435 3685 mt 2507 3685 L 2471 3649 mt 2471 3721 L 2433 3670 mt 2505 3670 L 2469 3634 mt 2469 3706 L 2433 3667 mt 2505 3667 L 2469 3631 mt 2469 3703 L 2450 3645 mt 2522 3645 L 2486 3609 mt 2486 3681 L 2433 3678 mt 2505 3678 L 2469 3642 mt 2469 3714 L 2437 3658 mt 2509 3658 L 2473 3622 mt 2473 3694 L 2474 3705 mt 2546 3705 L 2510 3669 mt 2510 3741 L 2490 3699 mt 2562 3699 L 2526 3663 mt 2526 3735 L 2496 3694 mt 2568 3694 L 2532 3658 mt 2532 3730 L 2505 3672 mt 2577 3672 L 2541 3636 mt 2541 3708 L 2467 3705 mt 2539 3705 L 2503 3669 mt 2503 3741 L 2434 3675 mt 2506 3675 L 2470 3639 mt 2470 3711 L 2484 3701 mt 2556 3701 L 2520 3665 mt 2520 3737 L 2468 3704 mt 2540 3704 L 2504 3668 mt 2504 3740 L 2437 3684 mt 2509 3684 L 2473 3648 mt 2473 3720 L 2436 3681 mt 2508 3681 L 2472 3645 mt 2472 3717 L 2472 3642 mt 2544 3642 L 2508 3606 mt 2508 3678 L 2502 3664 mt 2574 3664 L 2538 3628 mt 2538 3700 L 2501 3685 mt 2573 3685 L 2537 3649 mt 2537 3721 L 2460 3644 mt 2532 3644 L 2496 3608 mt 2496 3680 L 2499 3688 mt 2571 3688 L 2535 3652 mt 2535 3724 L 2458 3645 mt 2530 3645 L 2494 3609 mt 2494 3681 L 2497 3656 mt 2569 3656 L 2533 3620 mt 2533 3692 L 2483 3646 mt 2555 3646 L 2519 3610 mt 2519 3682 L 2502 3678 mt 2574 3678 L 2538 3642 mt 2538 3714 L 2467 3703 mt 2539 3703 L 2503 3667 mt 2503 3739 L 2438 3683 mt 2510 3683 L 2474 3647 mt 2474 3719 L 2444 3654 mt 2516 3654 L 2480 3618 mt 2480 3690 L 2438 3664 mt 2510 3664 L 2474 3628 mt 2474 3700 L 2492 3653 mt 2564 3653 L 2528 3617 mt 2528 3689 L 2437 3670 mt 2509 3670 L 2473 3634 mt 2473 3706 L 2439 3683 mt 2511 3683 L 2475 3647 mt 2475 3719 L 2487 3650 mt 2559 3650 L 2523 3614 mt 2523 3686 L 2477 3700 mt 2549 3700 L 2513 3664 mt 2513 3736 L 2468 3701 mt 2540 3701 L 2504 3665 mt 2504 3737 L 2440 3661 mt 2512 3661 L 2476 3625 mt 2476 3697 L 2474 3645 mt 2546 3645 L 2510 3609 mt 2510 3681 L 2440 3682 mt 2512 3682 L 2476 3646 mt 2476 3718 L 2500 3669 mt 2572 3669 L 2536 3633 mt 2536 3705 L 2500 3674 mt 2572 3674 L 2536 3638 mt 2536 3710 L 2487 3695 mt 2559 3695 L 2523 3659 mt 2523 3731 L 2496 3660 mt 2568 3660 L 2532 3624 mt 2532 3696 L 2467 3646 mt 2539 3646 L 2503 3610 mt 2503 3682 L 2492 3656 mt 2564 3656 L 2528 3620 mt 2528 3692 L 2493 3689 mt 2565 3689 L 2529 3653 mt 2529 3725 L 2481 3697 mt 2553 3697 L 2517 3661 mt 2517 3733 L 2468 3700 mt 2540 3700 L 2504 3664 mt 2504 3736 L 2469 3700 mt 2541 3700 L 2505 3664 mt 2505 3736 L 2489 3692 mt 2561 3692 L 2525 3656 mt 2525 3728 L 2455 3649 mt 2527 3649 L 2491 3613 mt 2491 3685 L 2485 3651 mt 2557 3651 L 2521 3615 mt 2521 3687 L 2442 3682 mt 2514 3682 L 2478 3646 mt 2478 3718 L 2443 3660 mt 2515 3660 L 2479 3624 mt 2479 3696 L 2497 3677 mt 2569 3677 L 2533 3641 mt 2533 3713 L 2495 3683 mt 2567 3683 L 2531 3647 mt 2531 3719 L 2473 3698 mt 2545 3698 L 2509 3662 mt 2509 3734 L 2468 3698 mt 2540 3698 L 2504 3662 mt 2504 3734 L 2446 3657 mt 2518 3657 L 2482 3621 mt 2482 3693 L 2478 3649 mt 2550 3649 L 2514 3613 mt 2514 3685 L 2497 3668 mt 2569 3668 L 2533 3632 mt 2533 3704 L 2443 3681 mt 2515 3681 L 2479 3645 mt 2479 3717 L 2494 3662 mt 2566 3662 L 2530 3626 mt 2530 3698 L 2472 3648 mt 2544 3648 L 2508 3612 mt 2508 3684 L 2481 3651 mt 2553 3651 L 2517 3615 mt 2517 3687 L 2442 3678 mt 2514 3678 L 2478 3642 mt 2478 3714 L 2452 3653 mt 2524 3653 L 2488 3617 mt 2488 3689 L 2457 3651 mt 2529 3651 L 2493 3615 mt 2493 3687 L 2485 3692 mt 2557 3692 L 2521 3656 mt 2521 3728 L 2468 3697 mt 2540 3697 L 2504 3661 mt 2504 3733 L 2487 3656 mt 2559 3656 L 2523 3620 mt 2523 3692 L 2444 3681 mt 2516 3681 L 2480 3645 mt 2480 3717 L 2456 3652 mt 2528 3652 L 2492 3616 mt 2492 3688 L 2442 3673 mt 2514 3673 L 2478 3637 mt 2478 3709 L 2450 3656 mt 2522 3656 L 2486 3620 mt 2486 3692 L 2494 3680 mt 2566 3680 L 2530 3644 mt 2530 3716 L 2469 3696 mt 2541 3696 L 2505 3660 mt 2505 3732 L 2490 3686 mt 2562 3686 L 2526 3650 mt 2526 3722 L 2495 3673 mt 2567 3673 L 2531 3637 mt 2531 3709 L 2444 3667 mt 2516 3667 L 2480 3631 mt 2480 3703 L 2463 3650 mt 2535 3650 L 2499 3614 mt 2499 3686 L 2445 3680 mt 2517 3680 L 2481 3644 mt 2481 3716 L 2444 3674 mt 2516 3674 L 2480 3638 mt 2480 3710 L 2467 3650 mt 2539 3650 L 2503 3614 mt 2503 3686 L 2449 3658 mt 2521 3658 L 2485 3622 mt 2485 3694 L 2493 3666 mt 2565 3666 L 2529 3630 mt 2529 3702 L 2490 3660 mt 2562 3660 L 2526 3624 mt 2526 3696 L 2478 3693 mt 2550 3693 L 2514 3657 mt 2514 3729 L 2469 3695 mt 2541 3695 L 2505 3659 mt 2505 3731 L 2475 3694 mt 2547 3694 L 2511 3658 mt 2511 3730 L 2446 3665 mt 2518 3665 L 2482 3629 mt 2482 3701 L 2446 3680 mt 2518 3680 L 2482 3644 mt 2482 3716 L 2482 3655 mt 2554 3655 L 2518 3619 mt 2518 3691 L 2469 3694 mt 2541 3694 L 2505 3658 mt 2505 3730 L 2490 3682 mt 2562 3682 L 2526 3646 mt 2526 3718 L 2492 3669 mt 2564 3669 L 2528 3633 mt 2528 3705 L 2484 3658 mt 2556 3658 L 2520 3622 mt 2520 3694 L 2472 3652 mt 2544 3652 L 2508 3616 mt 2508 3688 L 2468 3652 mt 2540 3652 L 2504 3616 mt 2504 3688 L 2447 3679 mt 2519 3679 L 2483 3643 mt 2483 3715 L 2492 3676 mt 2564 3676 L 2528 3640 mt 2528 3712 L 2446 3674 mt 2518 3674 L 2482 3638 mt 2482 3710 L 2471 3693 mt 2543 3693 L 2507 3657 mt 2507 3729 L 2469 3693 mt 2541 3693 L 2505 3657 mt 2505 3729 L 2483 3689 mt 2555 3689 L 2519 3653 mt 2519 3725 L 2482 3689 mt 2554 3689 L 2518 3653 mt 2518 3725 L 2474 3653 mt 2546 3653 L 2510 3617 mt 2510 3689 L 2490 3667 mt 2562 3667 L 2526 3631 mt 2526 3703 L 2448 3679 mt 2520 3679 L 2484 3643 mt 2484 3715 L 2457 3656 mt 2529 3656 L 2493 3620 mt 2493 3692 L 2447 3669 mt 2519 3669 L 2483 3633 mt 2483 3705 L 2469 3692 mt 2541 3692 L 2505 3656 mt 2505 3728 L 2477 3655 mt 2549 3655 L 2513 3619 mt 2513 3691 L 2488 3681 mt 2560 3681 L 2524 3645 mt 2524 3717 L 2467 3654 mt 2539 3654 L 2503 3618 mt 2503 3690 L 2486 3683 mt 2558 3683 L 2522 3647 mt 2522 3719 L 2473 3691 mt 2545 3691 L 2509 3655 mt 2509 3727 L 2448 3677 mt 2520 3677 L 2484 3641 mt 2484 3713 L 2449 3678 mt 2521 3678 L 2485 3642 mt 2485 3714 L 2451 3663 mt 2523 3663 L 2487 3627 mt 2487 3699 L 2450 3664 mt 2522 3664 L 2486 3628 mt 2486 3700 L 2460 3656 mt 2532 3656 L 2496 3620 mt 2496 3692 L 2455 3659 mt 2527 3659 L 2491 3623 mt 2491 3695 L 2479 3657 mt 2551 3657 L 2515 3621 mt 2515 3693 L 2469 3691 mt 2541 3691 L 2505 3655 mt 2505 3727 L 2449 3672 mt 2521 3672 L 2485 3636 mt 2485 3708 L 2450 3678 mt 2522 3678 L 2486 3642 mt 2486 3714 L 2489 3673 mt 2561 3673 L 2525 3637 mt 2525 3709 L 2484 3661 mt 2556 3661 L 2520 3625 mt 2520 3697 L 2485 3663 mt 2557 3663 L 2521 3627 mt 2521 3699 L 2483 3684 mt 2555 3684 L 2519 3648 mt 2519 3720 L 2488 3675 mt 2560 3675 L 2524 3639 mt 2524 3711 L 2475 3689 mt 2547 3689 L 2511 3653 mt 2511 3725 L 2470 3690 mt 2542 3690 L 2506 3654 mt 2506 3726 L 2450 3670 mt 2522 3670 L 2486 3634 mt 2486 3706 L 2460 3658 mt 2532 3658 L 2496 3622 mt 2496 3694 L 2471 3656 mt 2543 3656 L 2507 3620 mt 2507 3692 L 2485 3664 mt 2557 3664 L 2521 3628 mt 2521 3700 L 2477 3688 mt 2549 3688 L 2513 3652 mt 2513 3724 L 2451 3677 mt 2523 3677 L 2487 3641 mt 2487 3713 L 2486 3667 mt 2558 3667 L 2522 3631 mt 2522 3703 L 2485 3681 mt 2557 3681 L 2521 3645 mt 2521 3717 L 2470 3689 mt 2542 3689 L 2506 3653 mt 2506 3725 L 2486 3670 mt 2558 3670 L 2522 3634 mt 2522 3706 L 2479 3660 mt 2551 3660 L 2515 3624 mt 2515 3696 L 2465 3657 mt 2537 3657 L 2501 3621 mt 2501 3693 L 2462 3658 mt 2534 3658 L 2498 3622 mt 2498 3694 L 2452 3677 mt 2524 3677 L 2488 3641 mt 2488 3713 L 2470 3688 mt 2542 3688 L 2506 3652 mt 2506 3724 L 2451 3675 mt 2523 3675 L 2487 3639 mt 2487 3711 L 2453 3667 mt 2525 3667 L 2489 3631 mt 2489 3703 L 2480 3684 mt 2552 3684 L 2516 3648 mt 2516 3720 L 2481 3662 mt 2553 3662 L 2517 3626 mt 2517 3698 L 2456 3663 mt 2528 3663 L 2492 3627 mt 2492 3699 L 2456 3662 mt 2528 3662 L 2492 3626 mt 2492 3698 L 2486 3674 mt 2558 3674 L 2522 3638 mt 2522 3710 L 2470 3687 mt 2542 3687 L 2506 3651 mt 2506 3723 L 2473 3687 mt 2545 3687 L 2509 3651 mt 2509 3723 L 2453 3677 mt 2525 3677 L 2489 3641 mt 2489 3713 L 2485 3677 mt 2557 3677 L 2521 3641 mt 2521 3713 L 2470 3658 mt 2542 3658 L 2506 3622 mt 2506 3694 L 2475 3660 mt 2547 3660 L 2511 3624 mt 2511 3696 L 2477 3685 mt 2549 3685 L 2513 3649 mt 2513 3721 L 2470 3686 mt 2542 3686 L 2506 3650 mt 2506 3722 L 2485 3671 mt 2557 3671 L 2521 3635 mt 2521 3707 L 2453 3676 mt 2525 3676 L 2489 3640 mt 2489 3712 L 2455 3665 mt 2527 3665 L 2491 3629 mt 2491 3701 L 2481 3682 mt 2553 3682 L 2517 3646 mt 2517 3718 L 2483 3678 mt 2555 3678 L 2519 3642 mt 2519 3714 L 2473 3660 mt 2545 3660 L 2509 3624 mt 2509 3696 L 2466 3659 mt 2538 3659 L 2502 3623 mt 2502 3695 L 2454 3670 mt 2526 3670 L 2490 3634 mt 2490 3706 L 2454 3676 mt 2526 3676 L 2490 3640 mt 2490 3712 L 2470 3686 mt 2542 3686 L 2506 3650 mt 2506 3722 L 2459 3662 mt 2531 3662 L 2495 3626 mt 2495 3698 L 2476 3684 mt 2548 3684 L 2512 3648 mt 2512 3720 L 2478 3662 mt 2550 3662 L 2514 3626 mt 2514 3698 L 2462 3661 mt 2534 3661 L 2498 3625 mt 2498 3697 L 2454 3673 mt 2526 3673 L 2490 3637 mt 2490 3709 L 2455 3676 mt 2527 3676 L 2491 3640 mt 2491 3712 L 2481 3666 mt 2553 3666 L 2517 3630 mt 2517 3702 L 2471 3685 mt 2543 3685 L 2507 3649 mt 2507 3721 L 2482 3669 mt 2554 3669 L 2518 3633 mt 2518 3705 L 2481 3679 mt 2553 3679 L 2517 3643 mt 2517 3715 L 2483 3673 mt 2555 3673 L 2519 3637 mt 2519 3709 L 2481 3666 mt 2553 3666 L 2517 3630 mt 2517 3702 L 2456 3675 mt 2528 3675 L 2492 3639 mt 2492 3711 L 2457 3667 mt 2529 3667 L 2493 3631 mt 2493 3703 L 2472 3661 mt 2544 3661 L 2508 3625 mt 2508 3697 L 2471 3684 mt 2543 3684 L 2507 3648 mt 2507 3720 L 2459 3665 mt 2531 3665 L 2495 3629 mt 2495 3701 L 2465 3662 mt 2537 3662 L 2501 3626 mt 2501 3698 L 2482 3675 mt 2554 3675 L 2518 3639 mt 2518 3711 L 2471 3684 mt 2543 3684 L 2507 3648 mt 2507 3720 L 2456 3671 mt 2528 3671 L 2492 3635 mt 2492 3707 L 2456 3675 mt 2528 3675 L 2492 3639 mt 2492 3711 L 2473 3683 mt 2545 3683 L 2509 3647 mt 2509 3719 L 2472 3683 mt 2544 3683 L 2508 3647 mt 2508 3719 L 2456 3674 mt 2528 3674 L 2492 3638 mt 2492 3710 L 2473 3662 mt 2545 3662 L 2509 3626 mt 2509 3698 L 2468 3662 mt 2540 3662 L 2504 3626 mt 2504 3698 L 2475 3663 mt 2547 3663 L 2511 3627 mt 2511 3699 L 2471 3683 mt 2543 3683 L 2507 3647 mt 2507 3719 L 2457 3675 mt 2529 3675 L 2493 3639 mt 2493 3711 L 2463 3663 mt 2535 3663 L 2499 3627 mt 2499 3699 L 2477 3665 mt 2549 3665 L 2513 3629 mt 2513 3701 L 2469 3662 mt 2541 3662 L 2505 3626 mt 2505 3698 L 2461 3665 mt 2533 3665 L 2497 3629 mt 2497 3701 L 2481 3672 mt 2553 3672 L 2517 3636 mt 2517 3708 L 2471 3682 mt 2543 3682 L 2507 3646 mt 2507 3718 L 2458 3674 mt 2530 3674 L 2494 3638 mt 2494 3710 L 2477 3680 mt 2549 3680 L 2513 3644 mt 2513 3716 L 2458 3670 mt 2530 3670 L 2494 3634 mt 2494 3706 L 2479 3669 mt 2551 3669 L 2515 3633 mt 2515 3705 L 2479 3676 mt 2551 3676 L 2515 3640 mt 2515 3712 L 2471 3682 mt 2543 3682 L 2507 3646 mt 2507 3718 L 2475 3680 mt 2547 3680 L 2511 3644 mt 2511 3716 L 2477 3679 mt 2549 3679 L 2513 3643 mt 2513 3715 L 2459 3668 mt 2531 3668 L 2495 3632 mt 2495 3704 L 2478 3678 mt 2550 3678 L 2514 3642 mt 2514 3714 L 2458 3674 mt 2530 3674 L 2494 3638 mt 2494 3710 L 2477 3667 mt 2549 3667 L 2513 3631 mt 2513 3703 L 2471 3681 mt 2543 3681 L 2507 3645 mt 2507 3717 L 2459 3674 mt 2531 3674 L 2495 3638 mt 2495 3710 L 2479 3672 mt 2551 3672 L 2515 3636 mt 2515 3708 L 2474 3680 mt 2546 3680 L 2510 3644 mt 2510 3716 L 2466 3664 mt 2538 3664 L 2502 3628 mt 2502 3700 L 2471 3680 mt 2543 3680 L 2507 3644 mt 2507 3716 L 2460 3674 mt 2532 3674 L 2496 3638 mt 2496 3710 L 2459 3672 mt 2531 3672 L 2495 3636 mt 2495 3708 L 2462 3667 mt 2534 3667 L 2498 3631 mt 2498 3703 L 2461 3668 mt 2533 3668 L 2497 3632 mt 2497 3704 L 2473 3665 mt 2545 3665 L 2509 3629 mt 2509 3701 L 2471 3664 mt 2543 3664 L 2507 3628 mt 2507 3700 L 2460 3673 mt 2532 3673 L 2496 3637 mt 2496 3709 L 2475 3666 mt 2547 3666 L 2511 3630 mt 2511 3702 L 2478 3674 mt 2550 3674 L 2514 3638 mt 2514 3710 L 2471 3680 mt 2543 3680 L 2507 3644 mt 2507 3716 L 2477 3669 mt 2549 3669 L 2513 3633 mt 2513 3705 L 2461 3673 mt 2533 3673 L 2497 3637 mt 2497 3709 L 2461 3671 mt 2533 3671 L 2497 3635 mt 2497 3707 L 2478 3674 mt 2550 3674 L 2514 3638 mt 2514 3710 L 2475 3678 mt 2547 3678 L 2511 3642 mt 2511 3714 L 2467 3665 mt 2539 3665 L 2503 3629 mt 2503 3701 L 2461 3673 mt 2533 3673 L 2497 3637 mt 2497 3709 L 2465 3666 mt 2537 3666 L 2501 3630 mt 2501 3702 L 2471 3679 mt 2543 3679 L 2507 3643 mt 2507 3715 L 2466 3666 mt 2538 3666 L 2502 3630 mt 2502 3702 L 2462 3673 mt 2534 3673 L 2498 3637 mt 2498 3709 L 2470 3666 mt 2542 3666 L 2506 3630 mt 2506 3702 L 2476 3676 mt 2548 3676 L 2512 3640 mt 2512 3712 L 2471 3679 mt 2543 3679 L 2507 3643 mt 2507 3715 L 2477 3672 mt 2549 3672 L 2513 3636 mt 2513 3708 L 2462 3670 mt 2534 3670 L 2498 3634 mt 2498 3706 L 2463 3669 mt 2535 3669 L 2499 3633 mt 2499 3705 L 2462 3673 mt 2534 3673 L 2498 3637 mt 2498 3709 L 2472 3678 mt 2544 3678 L 2508 3642 mt 2508 3714 L 2476 3670 mt 2548 3670 L 2512 3634 mt 2512 3706 L 2464 3668 mt 2536 3668 L 2500 3632 mt 2500 3704 L 2472 3666 mt 2544 3666 L 2508 3630 mt 2508 3702 L 2474 3668 mt 2546 3668 L 2510 3632 mt 2510 3704 L 2471 3678 mt 2543 3678 L 2507 3642 mt 2507 3714 L 2472 3678 mt 2544 3678 L 2508 3642 mt 2508 3714 L 2471 3678 mt 2543 3678 L 2507 3642 mt 2507 3714 L 2474 3676 mt 2546 3676 L 2510 3640 mt 2510 3712 L 2463 3672 mt 2535 3672 L 2499 3636 mt 2499 3708 L 2476 3672 mt 2548 3672 L 2512 3636 mt 2512 3708 L 2466 3668 mt 2538 3668 L 2502 3632 mt 2502 3704 L 2471 3678 mt 2543 3678 L 2507 3642 mt 2507 3714 L 2475 3669 mt 2547 3669 L 2511 3633 mt 2511 3705 L 2470 3667 mt 2542 3667 L 2506 3631 mt 2506 3703 L 2473 3668 mt 2545 3668 L 2509 3632 mt 2509 3704 L 2463 3672 mt 2535 3672 L 2499 3636 mt 2499 3708 L 2468 3667 mt 2540 3667 L 2504 3631 mt 2504 3703 L 2465 3669 mt 2537 3669 L 2501 3633 mt 2501 3705 L 2471 3677 mt 2543 3677 L 2507 3641 mt 2507 3713 L 2474 3675 mt 2546 3675 L 2510 3639 mt 2510 3711 L 2475 3674 mt 2547 3674 L 2511 3638 mt 2511 3710 L 2464 3672 mt 2536 3672 L 2500 3636 mt 2500 3708 L 2464 3671 mt 2536 3671 L 2500 3635 mt 2500 3707 L 2471 3677 mt 2543 3677 L 2507 3641 mt 2507 3713 L 2474 3670 mt 2546 3670 L 2510 3634 mt 2510 3706 L 2464 3672 mt 2536 3672 L 2500 3636 mt 2500 3708 L 2473 3670 mt 2545 3670 L 2509 3634 mt 2509 3706 L 2471 3676 mt 2543 3676 L 2507 3640 mt 2507 3712 L 2474 3672 mt 2546 3672 L 2510 3636 mt 2510 3708 L 2470 3668 mt 2542 3668 L 2506 3632 mt 2506 3704 L 2468 3668 mt 2540 3668 L 2504 3632 mt 2504 3704 L 2464 3672 mt 2536 3672 L 2500 3636 mt 2500 3708 L 2472 3669 mt 2544 3669 L 2508 3633 mt 2508 3705 L 2471 3676 mt 2543 3676 L 2507 3640 mt 2507 3712 L 2472 3675 mt 2544 3675 L 2508 3639 mt 2508 3711 L 2465 3672 mt 2537 3672 L 2501 3636 mt 2501 3708 L 2466 3670 mt 2538 3670 L 2502 3634 mt 2502 3706 L 2465 3672 mt 2537 3672 L 2501 3636 mt 2501 3708 L 2471 3675 mt 2543 3675 L 2507 3639 mt 2507 3711 L 2473 3674 mt 2545 3674 L 2509 3638 mt 2509 3710 L 2471 3675 mt 2543 3675 L 2507 3639 mt 2507 3711 L 2473 3673 mt 2545 3673 L 2509 3637 mt 2509 3709 L 2473 3674 mt 2545 3674 L 2509 3638 mt 2509 3710 L 2466 3671 mt 2538 3671 L 2502 3635 mt 2502 3707 L 2471 3675 mt 2543 3675 L 2507 3639 mt 2507 3711 L 2467 3670 mt 2539 3670 L 2503 3634 mt 2503 3706 L 2466 3671 mt 2538 3671 L 2502 3635 mt 2502 3707 L 2471 3675 mt 2543 3675 L 2507 3639 mt 2507 3711 L 2472 3671 mt 2544 3671 L 2508 3635 mt 2508 3707 L 2470 3670 mt 2542 3670 L 2506 3634 mt 2506 3706 L 2469 3669 mt 2541 3669 L 2505 3633 mt 2505 3705 L 2471 3671 mt 2543 3671 L 2507 3635 mt 2507 3707 L 2466 3671 mt 2538 3671 L 2502 3635 mt 2502 3707 L 2467 3671 mt 2539 3671 L 2503 3635 mt 2503 3707 L 2471 3674 mt 2543 3674 L 2507 3638 mt 2507 3710 L 2472 3672 mt 2544 3672 L 2508 3636 mt 2508 3708 L 2471 3674 mt 2543 3674 L 2507 3638 mt 2507 3710 L 2467 3671 mt 2539 3671 L 2503 3635 mt 2503 3707 L 2468 3670 mt 2540 3670 L 2504 3634 mt 2504 3706 L 2471 3674 mt 2543 3674 L 2507 3638 mt 2507 3710 L 2467 3671 mt 2539 3671 L 2503 3635 mt 2503 3707 L 2471 3674 mt 2543 3674 L 2507 3638 mt 2507 3710 L 2471 3673 mt 2543 3673 L 2507 3637 mt 2507 3709 L 2470 3671 mt 2542 3671 L 2506 3635 mt 2506 3707 L 2467 3671 mt 2539 3671 L 2503 3635 mt 2503 3707 L 2468 3671 mt 2540 3671 L 2504 3635 mt 2504 3707 L 2471 3673 mt 2543 3673 L 2507 3637 mt 2507 3709 L 2468 3671 mt 2540 3671 L 2504 3635 mt 2504 3707 L 2471 3673 mt 2543 3673 L 2507 3637 mt 2507 3709 L 2471 3672 mt 2543 3672 L 2507 3636 mt 2507 3708 L 2470 3672 mt 2542 3672 L 2506 3636 mt 2506 3708 L 2470 3673 mt 2542 3673 L 2506 3637 mt 2506 3709 L 2468 3671 mt 2540 3671 L 2504 3635 mt 2504 3707 L 2470 3672 mt 2542 3672 L 2506 3636 mt 2506 3708 L 2470 3671 mt 2542 3671 L 2506 3635 mt 2506 3707 L 2470 3672 mt 2542 3672 L 2506 3636 mt 2506 3708 L 2468 3671 mt 2540 3671 L 2504 3635 mt 2504 3707 L 2469 3671 mt 2541 3671 L 2505 3635 mt 2505 3707 L 2470 3672 mt 2542 3672 L 2506 3636 mt 2506 3708 L 2469 3671 mt 2541 3671 L 2505 3635 mt 2505 3707 L 2470 3672 mt 2542 3672 L 2506 3636 mt 2506 3708 L 2469 3671 mt 2541 3671 L 2505 3635 mt 2505 3707 L 2470 3672 mt 2542 3672 L 2506 3636 mt 2506 3708 L 2469 3672 mt 2541 3672 L 2505 3636 mt 2505 3708 L gs 898 388 5357 4224 rc gr 3310 4940 mt (real part) s 565 2961 mt -90 rotate (imaginary part) s 90 rotate end eplot %%EndObject graph 1 epage end showpage %%Trailer %%EOF %%EndDocument @endspecial 1714 2003 a FN(Figure)37 b(7:)52 b FA(Result)30 b(for)h(the)f(degree)h(1000)h(p)r(olynomial)f(b)n(y)e(Matlab)1714 2094 y(function)d Fz(roots)94 768 y FN(The)31 b(m)m(ultiplicities)i(of) e(the)g(ro)s(ots)g(are)g(100,)94 881 y(200,)36 b(300)f(and)e(400.)52 b(These)33 b(m)m(ultiplicities)94 994 y(are)21 b(\\h)m(uge"compared)h (to)f(other)g(n)m(umerical)94 1107 y(examples,)66 b(usually)57 b(with)g(m)m(ultiplicities)94 1220 y(less)37 b(than)f(ten,)j(used)c(in) h(the)h(ro)s(ot-\014nding)94 1332 y(literature.)103 b(In)50 b(addition)h(to)g(suc)m(h)g(high)94 1445 y(m)m(ultiplicities,)28 b(w)m(e)d(p)s(erturb)d(the)i(sixth)g(dig-)94 1558 y(its)44 b(of)f(all)h(co)s(e\016cien)m(ts)g(of)86 b FM(g)k FN(b)m(y)43 b(m)m(ulti-)94 1671 y(plying)30 b(\(1)19 b Fy(\006)f FN(10)647 1638 y FE(\000)p Fx(6)742 1671 y FN(\))30 b(on)f(eac)m(h)i (one)f(of)f(them.)94 1784 y(Using)36 b(an)m(y)f(con)m(v)m(en)m(tional)k (approac)m(h,)d(this)94 1897 y(p)s(erturbation)g(will)h(result)f(in)h (a)g(total)h(loss)94 2010 y(of)27 b(forw)m(ard)g(accuracy)-8 b(,)29 b(ev)m(en)f(if)e(m)m(ultipreci-)94 2123 y(sion)46 b(is)f(used.)85 b(The)44 b(co)s(de)i FI(PejR)m(oot)e FN(of)94 2236 y(Algorithm)36 b(I)e(tak)m(es)i(a)f(few)g(seconds)f (under)f(Matlab)j(to)f(calculate)i(all)f(ro)s(ots)f(up)e(to)i(7)g (digits)h(accuracy)-8 b(.)94 2349 y(T)g(aking)23 b(the)g(condition)f(n) m(um)m(b)s(er)f(0.58)j(in)m(to)f(accoun)m(t,)j(this)c(accuracy)h(is)f (optimal.)39 b(On)22 b(the)g(same)g(mac)m(hine,)94 2462 y(Matlab)36 b(function)e Ff(roots)f FN(tak)m(es)j(ab)s(out)e(15)h(min)m (utes)f(to)h(pro)s(duce)e(1000)k(incorrect)e(ro)s(ots,)g(as)g(sho)m(wn) f(in)94 2574 y(Figure)d(7.)616 2771 y FM(z)658 2785 y Fx(1)p 1040 2794 4 79 v 1493 2771 a FM(z)1535 2785 y Fx(2)p 1918 2794 V 2371 2771 a FM(z)2413 2785 y Fx(3)p 2795 2794 V 3248 2771 a FM(z)3290 2785 y Fx(4)p 199 2798 3511 4 v 249 2853 a(.289)242 b(+.601i)p 1040 2876 4 79 v 276 w(.100)276 b(+.702i)p 1918 2876 V 276 w(.702)h(+.498i)p 2795 2876 V 276 w(.301)f(+.399i)249 2932 y(.309)242 b(+.602i)p 1040 2955 V 276 w(.097)276 b(+.698i)p 1918 2955 V 276 w(.698)h(+.499i)p 2795 2955 V 276 w(.299)f(+.400i)249 3011 y(.293)242 b(+.596i)p 1040 3034 V 276 w(.101)276 b(+.7003i)p 1918 3034 V 241 w(.7002)242 b(+.5005i)p 2795 3034 V 241 w(.3007)f(+.4003i)249 3089 y(.3003)207 b(+.5994i)p 1040 3113 V 241 w(.09994)f(+.70008i)p 1918 3113 V 206 w(.69996)h(+.50003i)p 2795 3113 V 206 w(.29996)f(+.40007i)249 3168 y(.300005)137 b(+.600006)p 1040 3192 V 191 w(.099998)171 b(+.6999992i)p 1918 3192 V 136 w(.69999992)102 b(+.4999993i)p 2795 3192 V 136 w(.2999992)136 b(+.3999992i)249 3247 y(.3000002)102 b(+.60000005i)p 1040 3271 V 101 w(.09999995)f (+.69999998i)p 1918 3271 V 101 w(.69999997)h(+.49999998i)p 2795 3271 V 101 w(.29999997)f(+.400000002i)94 3594 y FQ(4)135 b(Algorithm)52 b(I)t(I:)f(the)h(m)l(ultiplicit)l(y)h (structure)e(and)g(initial)i(ro)t(ot)e(es-)296 3743 y(timates)94 3946 y FN(While)31 b(Algorithm)f(I)f(can)g(b)s(e)g(used)f(on)h(an)m(y)h (particular)g(p)s(ejorativ)m(e)g(manifold,)g(of)f(course,)h(the)f (\\correct")94 4059 y(m)m(ultiplicit)m(y)36 b(structure)d(is)g (preferred)f(if)h(it)h(is)f(attainable.)52 b(W)-8 b(e)34 b(presen)m(t)g(Algorithm)g(I)s(I)e(that)i(calculates)94 4172 y(the)i(m)m(ultiplicit)m(y)g(structure)f(of)g(a)g(giv)m(en)h(p)s (olynomial)f(as)g(w)m(ell)h(as)f(the)g(initial)i(ro)s(ot)e(appro)m (ximation)h(for)94 4285 y(Algorithm)c(I.)94 4528 y FP(4.1)113 b(Remarks)38 b(on)g(the)f(univ)-6 b(ariate)38 b(GCD)g(computation)94 4700 y FN(F)-8 b(or)34 b(a)g(giv)m(en)g(p)s(olynomial)f FM(p)g FN(with)g FM(u)d FN(=)g FM(GC)7 b(D)17 b FN(\()q FM(p;)e(p)1926 4667 y FE(0)1949 4700 y FN(\),)34 b(Lagrange)h(p)s(oin)m (ted)e(out)g(in)g(1769)i(that)67 b FM(v)33 b FN(=)c FM(p=u)94 4813 y FN(has)36 b(the)f(same)h(distinct)g(ro)s(ots)f(as)h FM(p)p FN(,)g(and)f(all)h(ro)s(ots)g(of)f FM(v)k FN(are)c(simple.)56 b(If)35 b FM(v)j FN(is)e(obtainable,)i(its)d(simple)94 4926 y(ro)s(ots)f(can)f(b)s(e)g(calculated)i(using)e(standard)f(ro)s (ot-\014nders.)48 b(Based)34 b(on)f(this)g(observ)-5 b(ation,)35 b(the)e(follo)m(wing)94 5038 y(pro)s(cess)f(that)h(is)f (describ)s(ed)f(b)m(y)i(Gauss)f(in)g(1863)i(is)e(a)g(natural)h(approac) m(h)f(to)h(attain)h(total)g(factorization)1909 5686 y(18)p eop %%Page: 19 21 19 20 bop 94 99 a FN(of)31 b(the)g(p)s(olynomial)f FM(p)p FN(.)818 215 y Fu(8)818 290 y(>)818 315 y(>)818 340 y(>)818 365 y(>)818 390 y(>)818 415 y(>)818 440 y(>)818 465 y(<)818 614 y(>)818 639 y(>)818 664 y(>)818 689 y(>)818 714 y(>)818 739 y(>)818 763 y(>)818 788 y(:)934 298 y FM(u)986 312 y Fx(0)1050 298 y FN(=)25 b FM(p)934 411 y FN(for)30 b FM(j)h FN(=)25 b(1)p FM(;)15 b FN(2)p FM(;)g Fy(\001)g(\001)g(\001)r FN(,)61 b(while)31 b FM(deg)s FN(\()p FM(u)2060 425 y FL(j)t FE(\000)p Fx(1)2188 411 y FN(\))25 b FM(>)g FN(0)31 b(do)1136 545 y(calculate)h FM(u)1566 559 y FL(j)1628 545 y FN(=)25 b FM(GC)7 b(D)1960 451 y 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FM(S)2289 4774 y FL(j)2325 4760 y FN(\()p FM(f)10 b FN(\),)31 b FM(j)h FN(=)27 b(1)p FM(;)15 b FN(2)p FM(;)g Fy(\001)g(\001)g(\001)64 b FN(and)31 b(stop)g(at)h(the)f(\014rst)94 4873 y(rank)g(de\014cien)m (t)h(matrix)g FM(S)1020 4888 y FL(k)1062 4873 y FN(\()p FM(f)10 b FN(\).)43 b(With)32 b(this)f FM(S)1725 4888 y FL(k)1767 4873 y FN(\()p FM(f)10 b FN(\),)32 b(not)f(only)h(the)f (degrees)h(of)f(the)g(GCD)h(triplet)g FM(u;)15 b(v)s(;)g(w)94 4986 y FN(are)41 b(a)m(v)-5 b(ailable,)46 b(w)m(e)41 b(also)h(obtain)f(co)s(e\016cien)m(ts)h(of)f FM(v)j FN(and)c FM(w)j FN(automatically)g(from)d(the)h(resulting)g(righ)m(t)94 5098 y(singular)e(v)m(ector.)66 b(In)38 b(com)m(bination)h(with)f(a)h (least)h(squares)e(division)g(in)g Fy(x)p FN(4.2.3)j(of)d(the)h (unstable)f(long)94 5211 y(division,)d(w)m(e)f(can)g(generate)h(an)f (appro)m(ximation)h(to)f(the)g(GCD)g(triplet,)h(and)e(obtain)h(an)g (initial)h(iterate)94 5324 y(that)g(is)f(not)g(clearly)h(indicated)g (in)f(the)g(approac)m(h)g(of)g(Corless)g(et)h(al.)52 b(Consequen)m(tly)-8 b(,)36 b(a)e(blac)m(kb)s(o)m(x-t)m(yp)s(e)94 5437 y(soft)m(w)m(are)e(computing)f FM(GC)7 b(D)s FN(\()p FM(f)e(;)15 b(f)1302 5404 y FE(0)1324 5437 y FN(\))31 b(is)f(dev)m(elop)s(ed)h(for)f(the)h(pro)s(cess)f(\(20\).)1909 5686 y(19)p eop %%Page: 20 22 20 21 bop 94 99 a FN(The)28 b(discussion)f(on)h(GCD)g(computation)h(is) f(limited)h(to)f FM(GC)7 b(D)s FN(\()p FM(f)e(;)15 b(f)2492 66 y FE(0)2515 99 y FN(\))28 b(b)s(ecause)g(our)f(ob)5 b(jectiv)m(e)30 b(is)e(mainly)94 211 y(ro)s(ot-\014nding)34 b(in)f(this)h(pap)s(er.)51 b(With)34 b(minor)g(mo)s(di\014cations,)h (our)e(GCD-\014nder)h(can)g(easily)h(b)s(e)e(adapted)94 324 y(to)f(the)e(general)i(GCD)e(problem)g(of)h(arbitrary)f(p)s (olynomial)g(pairs.)94 591 y FP(4.2)113 b(Calculating)38 b(the)g(greatest)f(common)h(divisor)94 763 y FN(Algorithm)32 b(I)s(I)d(is)i(based)f(on)g(the)g(follo)m(wing)i(GCD-\014nder)e(for)g (an)g(arbitrary)g(p)s(olynomial)h FM(f)10 b FN(:)196 994 y FO(STEP)34 b(1.)41 b FN(Find)30 b(the)h(degree)g FM(m)f FN(of)g FM(GC)7 b(D)s FN(\()p FM(f)e(;)15 b(f)1893 961 y FE(0)1916 994 y FN(\).)196 1107 y FO(STEP)34 b(2.)41 b FN(Set)31 b(up)e(the)i(system)f(\(21\))i(in)e(accordance)i(with)e (the)h(degree)g FM(m)p FN(.)196 1220 y FO(STEP)j(3.)41 b FN(Find)30 b(an)g(initial)i(appro)m(ximation)f(to)g FM(u)p FN(,)g FM(v)j FN(and)29 b FM(w)k FN(for)d(the)h(GCD)g(system)f (\(21\).)196 1333 y FO(STEP)k(4.)41 b FN(Use)31 b(the)f(Gauss-Newton)i (iteration)g(to)f(re\014ne)e(the)i(GCD)g(triplet)g(\()p FM(u;)15 b(v)s(;)g(w)r FN(\).)94 1545 y(W)-8 b(e)32 b(shall)f(describ)s (e)e(eac)m(h)j(step)e(in)g(detail.)94 1809 y FO(4.2.1)106 b(Finding)35 b(the)g(degrees)g(of)g(the)f(GCD)h(triplet)94 1981 y FN(Let)28 b FM(f)37 b FN(b)s(e)27 b(a)h(p)s(olynomial)f(of)h (degree)g FM(n)p FN(.)39 b(By)28 b(Lemma)g(2.2,)h(the)f(degree)g(of)g FM(u)d FN(=)g FM(GC)7 b(D)s FN(\()p FM(f)e(;)15 b(f)3236 1948 y FE(0)3258 1981 y FN(\))28 b(is)f FM(m)e FN(=)g FM(n)14 b Fy(\000)g FM(k)94 2094 y FN(if)34 b(and)f(only)h(if)f(the)h (the)g FM(k)s FN(-th)g(Sylv)m(ester)g(discriminan)m(t)g(matrix)f(is)h (the)g(\014rst)f(one)g(b)s(eing)h(rank-de\014cien)m(t.)94 2206 y(Therefore,)i FM(m)c FN(=)f FM(deg)s FN(\()p FM(u)p FN(\))37 b(can)d(b)s(e)g(iden)m(ti\014ed)h(b)m(y)f(calculating)j(the)d (sequence)h(of)f(the)h(smallest)h(singular)94 2319 y(v)-5 b(alues)37 b FM(&)404 2333 y FL(j)478 2319 y FN(of)f FM(S)643 2333 y FL(j)680 2319 y FN(\()p FM(f)10 b FN(\),)38 b FM(j)j FN(=)36 b(1)p FM(;)15 b FN(2)p FM(;)g Fy(\001)g(\001)g(\001)s FN(,)38 b(un)m(til)f(reac)m(hing)h FM(&)2016 2334 y FL(k)2095 2319 y FN(that)f(is)g FC(appr)-5 b(oximately)40 b FN(zero.)61 b(Since)36 b(only)h(one)94 2432 y(singular)43 b(pair)g(\(i.e.)81 b(the)43 b(singular)g(v)-5 b(alue)44 b(and)e(the)i(righ)m(t)f(singular) g(v)m(ector\))j(is)d(needed,)j(the)e(in)m(v)m(erse)94 2545 y(iteration)39 b(describ)s(ed)d(in)h(Lemma)g(2.4)h(is)f(suitable)h (for)f(this)f(purp)s(ose.)59 b(Moreo)m(v)m(er,)42 b(w)m(e)37 b(can)g(reduce)g(the)94 2658 y(computing)27 b(cost)h(ev)m(en)f(further) e(b)m(y)i(recycling)h(and)e(up)s(dating)f(the)i(QR)f(decomp)s(osition)i (of)e FM(S)3353 2672 y FL(j)3390 2658 y FN(\()p FM(f)10 b FN(\)'s)26 b(along)94 2771 y(the)31 b(w)m(a)m(y)-8 b(.)42 b(More)31 b(sp)s(eci\014cally)-8 b(,)32 b(let)500 2975 y FM(f)10 b FN(\()p FM(x)p FN(\))25 b(=)g FM(a)846 2989 y Fx(0)886 2975 y FM(x)938 2938 y FL(n)1005 2975 y FN(+)20 b FM(a)1144 2989 y Fx(1)1183 2975 y FM(x)1235 2938 y FL(n)p FE(\000)p Fx(1)1392 2975 y FN(+)g Fy(\001)15 b(\001)g(\001)22 b FN(+)d FM(a)1748 2989 y FL(n)1796 2975 y FM(;)91 b(f)1967 2938 y FE(0)1989 2975 y FN(\()p FM(x)p FN(\))26 b(=)f FM(b)2272 2989 y Fx(0)2312 2975 y FM(x)2364 2938 y FL(n)p FE(\000)p Fx(1)2521 2975 y FN(+)20 b FM(b)2651 2989 y Fx(1)2690 2975 y FM(x)2742 2938 y FL(n)p FE(\000)p Fx(2)2900 2975 y FN(+)f Fy(\001)c(\001)g(\001) 22 b FN(+)e FM(b)3247 2989 y FL(n)p FE(\000)p Fx(1)3384 2975 y FM(:)94 3184 y FN(W)-8 b(e)32 b(rotate)g(the)f(columns)f(of)g FM(S)1189 3198 y FL(j)1225 3184 y FN(\()p FM(f)10 b FN(\))31 b(to)g(form)1722 3161 y(^)1707 3184 y FM(S)1763 3198 y FL(j)1799 3184 y FN(\()p FM(f)10 b FN(\))30 b(in)g(suc)m(h)g(a)h(w)m (a)m(y)558 3339 y Fs(j)s Fw(+1)393 3393 y Fu(z)p 430 3393 143 10 v 143 w(}|)p 647 3393 V 143 w({)1160 3339 y Fs(j)969 3393 y Fu(z)p 1006 3393 131 10 v 131 w(}|)p 1211 3393 V 131 w({)205 3448 y(0)205 3595 y(B)205 3644 y(B)205 3694 y(B)205 3744 y(B)205 3794 y(B)205 3844 y(B)205 3893 y(B)205 3943 y(B)205 3993 y(B)205 4043 y(B)205 4093 y(B)205 4143 y(B)205 4192 y(B)205 4245 y(@)320 3512 y Fq(b)353 3520 y Fw(0)966 3512 y Fq(a)1007 3520 y Fw(0)320 3663 y Fq(b)353 3671 y Fw(1)561 3605 y FA(.)591 3630 y(.)620 3655 y(.)966 3663 y Fq(a)1007 3671 y Fw(1)1136 3605 y FA(.)1166 3630 y(.)1196 3655 y(.)320 3749 y(.)320 3782 y(.)320 3815 y(.)561 3757 y(.)591 3782 y(.)620 3807 y(.)729 3815 y Fq(b)762 3823 y Fw(0)966 3749 y FA(.)966 3782 y(.)966 3815 y(.)1136 3757 y(.)1166 3782 y(.)1196 3807 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Fq(b)3469 3990 y Fw(0)2537 4134 y Fq(a)2578 4142 y Fs(n)2786 4076 y FA(.)2816 4101 y(.)2846 4126 y(.)2954 4068 y(.)2954 4101 y(.)2954 4134 y(.)3191 4068 y(.)3191 4101 y(.)3191 4134 y(.)224 b Fq(b)3469 4142 y Fw(1)2954 4286 y Fq(b)2987 4294 y Fs(n)p FB(\000)p Fw(1)3191 4286 y Fq(a)3232 4294 y Fs(n)p FB(\000)p Fw(1)3436 4219 y FA(.)3436 4252 y(.)3436 4286 y(.)3191 4377 y Fq(a)3232 4385 y Fs(n)3436 4377 y Fq(b)3469 4385 y Fs(n)p FB(\000)p Fw(1)3631 3448 y Fu(1)3631 3595 y(C)3631 3644 y(C)3631 3694 y(C)3631 3744 y(C)3631 3794 y(C)3631 3844 y(C)3631 3893 y(C)3631 3943 y(C)3631 3993 y(C)3631 4043 y(C)3631 4093 y(C)3631 4143 y(C)3631 4192 y(C)3631 4245 y(A)94 4565 y FN(that)39 b(the)f(o)s(dd)f(and)h(ev)m(en)h(columns)e(of)1538 4542 y(^)1523 4565 y FM(S)1579 4579 y FL(j)1615 4565 y FN(\()p FM(f)10 b FN(\))38 b(consist)h(of)f(the)g(co)s(e\016cien)m(ts)i(of)e FM(f)2994 4532 y FE(0)3055 4565 y FN(and)f FM(f)47 b FN(resp)s(ectiv)m(ely)-8 b(.)94 4678 y(Consequen)m(tly)g(,)37 b(the)d(matrix)1156 4655 y(^)1141 4678 y FM(S)1197 4692 y FL(j)t Fx(+1)1323 4678 y FN(\()p FM(f)10 b FN(\))34 b(is)h(simply)e(formed)h(b)m(y)g(adding)g(a)h(zero)g(ro)m(w)f(at)h(the) f(b)s(ottom)h(and)94 4791 y(t)m(w)m(o)k(columns)e(in)f(the)i(righ)m(t)f (on)1285 4768 y(^)1269 4791 y FM(S)1325 4805 y FL(j)1362 4791 y FN(\()p FM(f)10 b FN(\).)61 b(Up)s(dating)36 b(the)i(QR)e (decomp)s(osition)i(of)f(eac)m(h)3264 4768 y(^)3249 4791 y FM(S)3305 4805 y FL(j)3341 4791 y FN(\()p FM(f)10 b FN(\))37 b(requires)94 4904 y(only)31 b FM(O)s FN(\()p FM(n)p FN(\))f(additional)i(\015ops.)40 b(The)30 b(in)m(v)m(erse)h (iteration)h(\(1\))f(requires)f FM(O)s FN(\()p FM(j)2693 4871 y Fx(2)2733 4904 y FN(\))h(\015ops)f(at)h(eac)m(h)g FM(S)3383 4918 y FL(j)3420 4904 y FN(\()p FM(f)10 b FN(\).)94 5111 y(Let)32 b FM(\022)g FN(b)s(e)e(a)h(giv)m(en)h FC(zer)-5 b(o)34 b(singular)f(value)g(thr)-5 b(eshold)p FN(.)45 b(W)-8 b(e)31 b(shall)g(discuss)f(more)h(ab)s(out)f(this)h(parameter)g (in)94 5224 y Fy(x)p FN(4.4.)41 b(With)26 b(successiv)m(e)h(QR)d(up)s (dating)h(and)f(the)i(in)m(v)m(erse)g(iteration,)i(the)e(pro)s(cess)f (of)g(\014nding)f(the)i(degrees)94 5337 y(of)31 b(the)g(GCD)f(triplet)h (\()p FM(u;)15 b(v)s(;)g(w)r FN(\))33 b(can)e(b)s(e)e(summarized)h(as)h (follo)m(ws.)1909 5686 y(20)p eop %%Page: 21 23 21 22 bop 297 99 a FN(Calculate)32 b(the)e(QR)g(decomp)s(osition)h(of)g (the)f(\()p FM(n)21 b FN(+)e(1\))i Fy(\002)f FN(3)31 b(matrix)2664 76 y(^)2649 99 y FM(S)2705 113 y Fx(1)2744 99 y FN(\()p FM(f)10 b FN(\))25 b(=)g FM(Q)3062 113 y Fx(1)3102 99 y FM(R)3171 113 y Fx(1)297 211 y Ff(For)60 b FM(j)31 b FN(=)24 b(1)p FM(;)15 b FN(2)p FM(;)g Fy(\001)g(\001)g (\001)64 b Ff(do)469 331 y FN(use)30 b(the)g(in)m(v)m(erse)i(iteration) g(\(1\))f(to)g(\014nd)e(the)i(smallest)g(singular)f(v)-5 b(alue)31 b FM(&)3010 345 y FL(j)3077 331 y FN(of)3196 308 y(^)3181 331 y FM(S)3237 345 y FL(j)3273 331 y FN(\()p FM(f)10 b FN(\))30 b(and)641 444 y(the)g(corresp)s(onding)g(righ)m(t)h (singular)f(v)m(ector)i FO(y)2274 458 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b(calculated)h(in)e Ff(STEP)47 b(1)p FN(.)39 b(W)-8 b(e)29 b(no)m(w)e(form)m(ulate)h(the)g (GCD)94 1471 y(system)j(\(21\))h(of)e Ff(STEP)47 b(2)30 b FN(in)g(v)m(ector)i(form)e(with)g(unkno)m(wn)f(v)m(ectors)j FO(u)p FN(,)e FO(v)i FN(and)e FO(w)q FN(:)640 1594 y Fu(2)640 1740 y(6)640 1794 y(4)737 1675 y FM(u)789 1689 y Fx(0)737 1788 y FM(conv)s FN(\()p FO(u)p FM(;)15 b FO(v)q FN(\))737 1901 y FM(conv)s FN(\()p FO(u)p FM(;)g FO(w)q FN(\))1209 1594 y Fu(3)1209 1740 y(7)1209 1794 y(5)1290 1788 y FN(=)1386 1594 y Fu(2)1386 1740 y(6)1386 1794 y(4)1483 1675 y FN(1)1483 1788 y FO(f)1483 1901 y(f)1525 1868 y FE(0)1590 1594 y Fu(3)1590 1740 y(7)1590 1794 y(5)1660 1788 y FM(;)177 b FN(for)2047 1594 y Fu(0)2047 1740 y(B)2047 1794 y(@)2170 1675 y FO(u)2171 1788 y(v)2161 1901 y(w)2279 1594 y Fu(1)2279 1740 y(C)2279 1794 y(A)2377 1788 y Fy(2)25 b FO(C)2539 1750 y FL(m)p Fx(+1)2716 1788 y Fy(\002)19 b FO(C)2882 1750 y FL(k)r Fx(+1)3035 1788 y Fy(\002)h FO(C)3202 1750 y FL(k)3244 1788 y FM(:)384 b 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b(either)h(explicitly)h(with)e(an)g(equation)h(or)g (implicitly)g(b)m(y)g(eliminating)g FM(u)2935 5225 y Fx(0)3000 5211 y FN(from)f(the)h(v)-5 b(ariables)26 b(to)94 5324 y(ensure)i(the)g(regularit)m(y)h(of)f(the)g(system)g(\(22\).)42 b(In)27 b(Remark)h(1)g(of)g([5)q(,)h Fy(x)p FN(4.3])g(with)f(a)g (heuristic)g(explanation,)94 5437 y(Chin,)e(Corless)f(and)f(Corless)i (realized)g(that)g(the)f(restriction)h FM(u)2281 5451 y Fx(0)2346 5437 y FN(=)f(1)g(ma)m(y)h(mak)m(e)g(their)f (Divisor-Quotien)m(t)1909 5686 y(21)p eop %%Page: 22 24 22 23 bop 94 99 a FN(Iteration)28 b(con)m(v)m(erge,)h(but)c(abandoned)g (it)i(since)f(their)g(\\test)h(sho)m(w)m(ed)g(that)f(the)g(o)m(v)m (erall)j(p)s(erformance)c(w)m(as)94 211 y(w)m(orse)39 b(when)f(this)h(constrain)m(t)g(w)m(as)g(in)g(place")h([5,)h Fy(x)p FN(5.1.4].)68 b(Because)40 b(w)m(e)f(use)f(a)h(di\013eren)m(t)h (re\014nemen)m(t)94 324 y(approac)m(h)k(in)g(our)f(GCD-\014nder,)j (preserving)e(this)f(constrain)m(t,)49 b(and)43 b(thereb)m(y)h(the)f (regularit)m(y)i(of)f(the)94 437 y(GCD)32 b(system)e(\(22\),)j(ma)m(y)e (b)s(e)f(the)h(v)m(ery)g(reason)g(for)g(our)f(metho)s(d)g(to)i(obtain)f (more)g(robust)f(test)h(results.)94 550 y(Without)d(this)e(regularit)m (y)-8 b(,)29 b(the)e(lo)s(cal)g(con)m(v)m(ergence)i(of)d(the)h (Gauss-Newton)g(iteration)h(w)m(e)f(use)f(w)m(ould)g(not)94 663 y(b)s(e)k(guaran)m(teed.)94 871 y FO(Theorem)35 b(4.1)75 b FC(L)-5 b(et)65 b FN(~)-51 b FM(u)25 b FN(=)g FM(GC)7 b(D)s FN(\()p FM(f)e(;)15 b(f)1503 838 y FE(0)1525 871 y FN(\))59 b FC(with)j FN(~)-48 b FM(v)61 b FC(and)77 b FN(~)-63 b FM(w)60 b FC(satisfying)30 b(\(22\),)h(and)f(let)58 b FM(W)70 b FC(b)-5 b(e)29 b(a)h(weight)94 1000 y(matrix.)64 b(Then)40 b(ther)-5 b(e)40 b(exists)80 b FM(")38 b(>)f FN(0)79 b FC(such)40 b(that)h(for)f(al)5 b(l)79 b FO(u)2322 1014 y Fx(0)2362 1000 y FC(,)40 b FO(v)2485 1014 y Fx(0)2525 1000 y FC(,)h FO(w)2670 1014 y Fx(0)2748 1000 y FC(satisfying)3160 902 y Fu(\015)3160 952 y(\015)3160 1002 y(\015)3221 1000 y FO(u)3279 1014 y Fx(0)3343 1000 y Fy(\000)3446 999 y FN(~)3439 1000 y FO(u)3513 902 y Fu(\015)3513 952 y(\015)3513 1002 y(\015)3559 1056 y Fx(2)3636 1000 y FM(<)c(")p FC(,)94 1064 y Fu(\015)94 1114 y(\015)94 1164 y(\015)156 1162 y FO(v)211 1176 y Fx(0)271 1162 y Fy(\000)367 1161 y FN(~)361 1162 y FO(v)433 1064 y Fu(\015)433 1114 y(\015)433 1164 y(\015)479 1218 y Fx(2)544 1162 y FM(<)25 b(")65 b FC(and)924 1064 y Fu(\015)924 1114 y(\015)924 1164 y(\015)985 1162 y FO(w)1061 1176 y Fx(0)1120 1162 y Fy(\000)1227 1161 y FN(~)1211 1162 y FO(w)1303 1064 y Fu(\015)1303 1114 y(\015)1303 1164 y(\015)1350 1218 y Fx(2)1414 1162 y FM(<)25 b(")p FC(,)33 b(the)g(Gauss-Newton)h(iter)-5 b(ation)303 1293 y Fu(2)303 1439 y(6)303 1492 y(4)409 1373 y FO(u)467 1387 y FL(j)t Fx(+1)410 1486 y FO(v)465 1500 y FL(j)t Fx(+1)400 1599 y FO(w)476 1613 y FL(j)t Fx(+1)644 1293 y Fu(3)644 1439 y(7)644 1492 y(5)725 1486 y FN(=)821 1293 y Fu(2)821 1439 y(6)821 1492 y(4)926 1373 y FO(u)984 1387 y FL(j)928 1486 y FO(v)983 1500 y FL(j)917 1599 y FO(w)993 1613 y FL(j)1071 1293 y Fu(3)1071 1439 y(7)1071 1492 y(5)1146 1486 y Fy(\000)20 b FM(J)9 b FN(\()p FO(u)1389 1500 y FL(j)1426 1486 y FM(;)15 b FO(v)1521 1500 y FL(j)1559 1486 y FM(;)g FO(w)1675 1500 y FL(j)1711 1486 y FN(\))1746 1448 y Fx(+)1746 1514 y FL(W)1842 1293 y Fu(2)1842 1439 y(6)1842 1492 y(4)1954 1373 y FO(e)2002 1340 y FE(>)2002 1396 y Fx(1)2061 1373 y FO(u)2119 1387 y FL(j)2540 1373 y Fy(\000)83 b FN(1)1954 1486 y FM(conv)s FN(\()p FO(u)2232 1500 y FL(j)2269 1486 y FM(;)15 b FO(v)2364 1500 y FL(j)2401 1486 y FN(\))104 b Fy(\000)83 b FO(f)1954 1599 y FM(conv)s FN(\()p FO(u)2232 1613 y FL(j)2269 1599 y FM(;)15 b FO(w)2385 1613 y FL(j)2422 1599 y FN(\))83 b Fy(\000)g FO(f)2736 1566 y FE(0)2816 1293 y Fu(3)2816 1439 y(7)2816 1492 y(5)2886 1486 y FM(;)91 b(j)31 b FN(=)25 b(0)p FM(;)15 b FN(1)p FM(;)g Fy(\001)g(\001)g(\001) 212 b FN(\(25\))94 1789 y FC(c)-5 b(onver)g(ges)35 b(to)f FN([)642 1788 y(~)636 1789 y FO(u)p FM(;)740 1788 y FN(~)734 1789 y FO(v)r FM(;)847 1788 y FN(~)831 1789 y FO(w)r FN(])934 1756 y FE(>)1026 1789 y FC(quadr)-5 b(atic)g(al)5 b(ly.)47 b(Her)-5 b(e)34 b FM(J)9 b FN(\()p Fy(\001)p FN(\))1972 1750 y Fx(+)1972 1816 y FL(W)2080 1789 y FN(=)27 b([)p FM(J)9 b FN(\()p Fy(\001)p FN(\))2357 1756 y FL(H)2426 1789 y FM(W)2525 1756 y Fx(2)2563 1789 y FM(J)g FN(\()p Fy(\001)p FN(\)])2742 1756 y FE(\000)p Fx(1)2838 1789 y FM(J)g FN(\()p Fy(\001)p FN(\))2992 1756 y FL(H)3061 1789 y FM(W)3160 1756 y Fx(2)3232 1789 y FC(is)33 b(the)h(weighte)-5 b(d)94 1901 y(pseudo-inverse)34 b(of)f(the)g(Jac)-5 b(obian)33 b FM(J)9 b FN(\()p Fy(\001)p FN(\))34 b FC(as)f(de\014ne)-5 b(d)34 b(in)e(\(23\).)94 2110 y FO(Pro)s(of.)83 b FN(A)30 b(straigh)m(tforw)m(ard)h(v)m(eri\014cation)h(b)m(y)f(using)e(Lemma)i (2.6)g(and)f(Lemma)h(4.1.)393 b(Q.E.D.)94 2345 y FO(4.2.3)106 b(Setting)34 b(up)h(the)g(initial)g(iterate)94 2517 y FN(W)-8 b(e)30 b(no)m(w)d(need)h(initial)h(iterates)57 b FO(u)1312 2531 y Fx(0)1352 2517 y FM(;)15 b FO(v)1447 2531 y Fx(0)1487 2517 y FM(;)g FO(w)1603 2531 y Fx(0)1670 2517 y FN(for)28 b(the)g(Gauss-Newton)g(iteration)i(\(25\).)41 b(In)28 b Ff(STEP)46 b(1)p FN(,)28 b(when)94 2630 y(the)g(singular)e(v) -5 b(alue)28 b FM(&)853 2645 y FL(k)922 2630 y FN(is)f(calculated,)j (the)d(asso)s(ciated)i(singular)d(v)m(ector)j FO(y)2710 2645 y FL(k)2780 2630 y FN(consists)e(of)g FO(v)3266 2644 y Fx(0)3333 2630 y FN(and)f FO(w)3582 2644 y Fx(0)3648 2630 y FN(that)94 2743 y(are)33 b(appro)m(ximations)g(to)g FO(v)h FN(and)d FO(w)i FN(in)f(\(22\))i(resp)s(ectiv)m(ely)f(\(see)h (Lemma)e(2.3\).)47 b(Because)34 b(of)e(the)h(column)94 2856 y(rotation)f(in)f Fy(x)p FN(4.2.1,)i(the)d(o)s(dd)g(and)g(ev)m(en) h(en)m(tries)g(of)g FO(y)2003 2871 y FL(k)2076 2856 y FN(form)f FO(v)2346 2870 y Fx(0)2416 2856 y FN(and)g FO(w)2669 2870 y Fx(0)2738 2856 y FN(resp)s(ectiv)m(ely)-8 b(.)43 b(F)-8 b(or)31 b(the)g(initial)94 2969 y(appro)m(ximation)h FO(u)759 2983 y Fx(0)798 2969 y FN(,)f(notice)h(that)f(in)f(theory)g (the)h(long)g(division)f(yields,)1480 3145 y FM(f)10 b FN(\()p FM(x)p FN(\))26 b(=)f FM(v)1823 3159 y Fx(0)1862 3145 y FN(\()p FM(x)p FN(\))p FM(q)s FN(\()p FM(x)p FN(\))c(+)f FM(r)s FN(\()p FM(x)p FN(\))1225 b(\(26\))94 3321 y(with)31 b FM(u)354 3335 y Fx(0)393 3321 y FN(\()p FM(x)p FN(\))26 b(=)f FM(q)s FN(\()p FM(x)p FN(\))31 b(and)e FM(r)s FN(\()p FM(x)p FN(\))d(=)f(0.)41 b(The)30 b(pro)s(cess)g(itself)h(ma)m(y)g(not) g(b)s(e)e(n)m(umerically)i(stable.)94 3528 y(In)c(the)g(con)m(text)i (of)e(Corless-Gianni-T)-8 b(rager-W)g(att)31 b(metho)s(d,)c(Corless)g (et)h(al.)40 b([6)q(,)27 b(Lemma)g(3])h(prop)s(ose)e(the)94 3641 y(use)33 b(of)g(least)h(squares)f(metho)s(d)f(to)i(minimize)f Fy(k)p FN(\001)p FO(f)10 b Fy(k)1946 3655 y Fx(2)2015 3641 y FN(=)29 b Fy(k)15 b FM(C)2240 3655 y FL(n)p FE(\000)p FL(m)2405 3641 y FN(\()p FM(d)p FN(\))p FO(h)23 b Fy(\000)f FO(f)j Fy(k)2798 3655 y Fx(2)2871 3641 y FN(whenev)m(er)32 b(a)i(candidate)f FM(d)94 3754 y FN(of)d(degree)h FM(m)f 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FN(of)k FM(GC)7 b(D)s FN(\()p FM(f)e(;)15 b(f)605 4173 y FE(0)627 4206 y FN(\))36 b(from)f(giv)m(en)i FM(v)1205 4220 y Fx(0)1281 4206 y FN(and)e FM(w)1528 4220 y Fx(0)1603 4206 y FN(in)h(our)f(metho)s(d,)i (and)e(justify)h(it)g(via)g(a)h(condition)f(theory)g(of)94 4319 y(linear)31 b(systems.)94 4526 y(By)g(Lemma)g(2.3,)g(the)g(long)g (division)f(\(26\))i(with)e FM(r)s FN(\()p FM(x)p FN(\))c(=)f(0)30 b(is)h(equiv)-5 b(alen)m(t)32 b(to)f(solving)g(the)f(linear)h(system) 1674 4702 y FM(C)1739 4716 y FL(m)1805 4702 y FN(\()p FM(v)1884 4716 y Fx(0)1924 4702 y FN(\))15 b FO(u)2032 4716 y Fx(0)2097 4702 y FN(=)25 b FO(f)1428 b FN(\(27\))94 4894 y(for)22 b(a)g(least)h(squares)f(solution)g FO(u)1203 4908 y Fx(0)1264 4894 y FN(that)h(minimizes)1869 4797 y Fu(\015)1869 4846 y(\015)1869 4896 y(\015)1930 4894 y FM(conv)s FN(\()p FO(u)2208 4908 y Fx(0)2248 4894 y FM(;)15 b FO(v)2343 4908 y Fx(0)2383 4894 y FN(\))s Fy(\000)s FO(f)2553 4797 y Fu(\015)2553 4846 y(\015)2553 4896 y(\015)2599 4950 y Fx(2)2638 4894 y FN(.)38 b(This)21 b(\\least)j(squares)d (division")94 5027 y(is)37 b(more)f(accurate)h(than)f(the)g(long)h (division)f(\(26\).)60 b(In)35 b(fact,)j(the)f(long)f(division)g (\(26\))i(is)e(equiv)-5 b(alen)m(t)38 b(to)94 5139 y(solving)32 b(the)e(\()p FM(n)20 b FN(+)g(1\))h Fy(\002)f FN(\()p FM(n)g FN(+)g(1\))31 b(lo)m(w)m(er)h(triangular)f(linear)f(system)551 5381 y FM(L)613 5395 y FL(m)680 5381 y FN(\()p FM(v)759 5395 y Fx(0)798 5381 y FN(\))848 5237 y Fu( )956 5324 y FO(q)962 5437 y(r)1053 5237 y Fu(!)1144 5381 y FN(=)25 b FO(f)10 b FM(;)106 b FN(with)91 b FM(L)1743 5395 y FL(m)1809 5381 y FN(\()p FM(v)1888 5395 y Fx(0)1928 5381 y FN(\))26 b(=)2084 5237 y Fu( )2192 5381 y FM(C)2257 5395 y FL(m)2323 5381 y FN(\()p FM(v)2402 5395 y Fx(0)2442 5381 y FN(\))2561 5234 y Fu(\014)2561 5284 y(\014)2561 5333 y(\014)2561 5383 y(\014)2561 5433 y(\014)2631 5323 y FN(0)2676 5342 y Fx(\()p FL(m)p Fx(+1\))p FE(\002)p Fx(\()p FL(n)p FE(\000)p FL(m)p Fx(\))2630 5437 y FM(I)2670 5456 y Fx(\()p FL(n)p FE(\000)p FL(m)p Fx(\))p FE(\002)p Fx(\()p FL(n)p FE(\000)p FL(m)p Fx(\))3252 5237 y Fu(!)3333 5381 y FM(:)295 b FN(\(28\))1909 5686 y(22)p eop %%Page: 23 25 23 24 bop 94 99 a FN(The)35 b(follo)m(wing)h(theorem)g(indicates)g (that)f(solving)h(\(27\))g(for)f FO(u)2327 113 y Fx(0)2401 99 y FN(ma)m(y)h(b)s(e)e(more)h(preferable)g(than)g(using)94 211 y(the)c(long)g(division)f(\(26\).)94 450 y FO(Theorem)35 b(4.2)46 b FC(L)-5 b(et)31 b FM(\024)p FN(\()p FM(A)p FN(\))h FC(denote)g(the)f(c)-5 b(ondition)32 b(numb)-5 b(er)31 b(of)g(an)g(arbitr)-5 b(ary)33 b(matrix)e FM(A)g FC(with)h(r)-5 b(esp)g(e)g(ct)32 b(to)94 563 y(the)h(matrix)h(2-norm.) 43 b(Then)98 b FM(\024)15 b FN(\()q FM(C)1366 577 y FL(m)1433 563 y FN(\()p FM(v)s FN(\)\))26 b Fy(\024)f FM(\024)p FN(\()p FM(L)1856 577 y FL(m)1924 563 y FN(\()p FM(v)s FN(\)\))66 b FC(for)33 b(any)g(p)-5 b(olynomial)36 b FM(v)g FC(and)e FM(m)25 b(>)g FN(0)p FC(.)94 793 y FO(Pro)s(of.)78 b FN(F)-8 b(or)28 b(an)m(y)g(matrix)f FM(A)p FN(,)i FM(\024)p FN(\()p FM(A)p FN(\))d(=)1519 749 y FL(\033)1559 757 y Fs(max)1686 749 y Fx(\()p FL(A)p Fx(\))p 1519 772 276 4 v 1523 825 a FL(\033)1563 835 y Fs(min)1683 825 y Fx(\()p FL(A)p Fx(\))1804 793 y FN(,)i(where)54 b FM(\033)2196 807 y FL(max)2340 793 y FN(\()p FM(A)p FN(\))26 b(=)60 b(max)2600 857 y FE(k)p Ft(x)p FE(k)2713 866 y Fw(2)2748 857 y Fx(=1)2854 696 y Fu(\015)2854 745 y(\015)2854 795 y(\015)2915 793 y FM(A)p FO(x)3053 696 y Fu(\015)3053 745 y(\015)3053 795 y(\015)3099 849 y Fx(2)3193 793 y FN(and)54 b FM(\033)3446 807 y FL(min)3580 793 y FN(\()p FM(A)p FN(\))26 b(=)138 987 y(min)94 1051 y FE(k)p Ft(x)p FE(k)207 1060 y Fw(2)243 1051 y Fx(=1)348 890 y Fu(\015)348 940 y(\015)348 989 y(\015)409 987 y FM(A)p FO(x)547 890 y Fu(\015)548 940 y(\015)548 989 y(\015)594 1043 y Fx(2)664 987 y FN(are)k(the)h(largest)h(and)d(smallest)j(singular)e(v)-5 b(alues)31 b(of)f FM(A)h FN(resp)s(ectiv)m(ely)-8 b(.)42 b(Therefore)509 1251 y FM(\033)561 1265 y FL(max)705 1251 y FN(\()p FM(C)805 1265 y FL(m)872 1251 y FN(\()p FM(v)s FN(\)\))26 b(=)61 b(max)1146 1315 y FE(k)p Ft(u)p FE(k)1261 1324 y Fw(2)1297 1315 y Fx(=1)1402 1154 y Fu(\015)1402 1204 y(\015)1402 1254 y(\015)1463 1251 y FM(C)1528 1265 y FL(m)1595 1251 y FN(\()p FM(v)s FN(\))15 b FO(u)1800 1154 y Fu(\015)1801 1204 y(\015)1801 1254 y(\015)1847 1308 y Fx(2)1912 1251 y FN(=)131 b(max)2008 1315 y FE(k)p Ft(q)p FE(k)2121 1324 y Fw(2)2156 1315 y Fx(=1)p FL(;)p Ft(r)p Fx(=0)2405 1154 y Fu(\015)2405 1204 y(\015)2405 1254 y(\015)2466 1251 y FM(C)2531 1265 y FL(m)2598 1251 y FN(\()p FM(v)s FN(\))15 b FO(q)21 b FN(+)f FO(r)2955 1154 y Fu(\015)2955 1204 y(\015)2955 1254 y(\015)3001 1308 y Fx(2)592 1520 y FN(=)189 b(max)746 1584 y FE(k)p Ft(q)p FE(k)859 1593 y Fw(2)894 1584 y Fx(=1)p FL(;)p Ft(r)p Fx(=0)1143 1373 y Fu(\015)1143 1423 y(\015)1143 1473 y(\015)1143 1523 y(\015)1143 1572 y(\015)1189 1520 y FM(L)1251 1534 y FL(m)1317 1520 y FN(\()p FM(v)s FN(\))1449 1377 y Fu( )1558 1464 y FO(q)1564 1577 y(r)1654 1377 y Fu(!)1720 1373 y(\015)1720 1423 y(\015)1720 1473 y(\015)1720 1523 y(\015)1720 1572 y(\015)1766 1626 y Fx(2)1881 1520 y Fy(\024)107 b FN(max)2028 1584 y FE(k)p FL(y)r FE(k)2135 1593 y Fw(2)2170 1584 y Fx(=1)2276 1423 y Fu(\015)2276 1473 y(\015)2276 1523 y(\015)2337 1520 y FM(L)2399 1534 y FL(m)2465 1520 y FN(\()p FM(v)s FN(\))p FO(y)2655 1423 y Fu(\015)2655 1473 y(\015)2655 1523 y(\015)2701 1577 y Fx(2)2766 1520 y FN(=)25 b FM(\033)2914 1534 y FL(max)3058 1520 y FN(\()p FM(L)3155 1534 y FL(m)3221 1520 y FN(\()p FM(v)s FN(\)\))p FM(:)94 1773 y FN(Similarly)-8 b(,)62 b FM(\033)581 1787 y FL(min)714 1773 y FN(\()p FM(C)814 1787 y FL(m)881 1773 y FN(\()p FM(v)s FN(\)\))27 b Fy(\025)e FM(\033)1208 1787 y FL(min)1341 1773 y FN(\()p FM(L)1438 1787 y FL(m)1505 1773 y FN(\()p FM(v)s FN(\)\),)32 b(and)e(consequen)m (tly)-8 b(,)32 b FM(\024)p FN(\()p FM(C)2599 1787 y FL(m)2666 1773 y FN(\()p FM(v)s FN(\)\))27 b Fy(\024)e FM(\024)p FN(\()p FM(L)3090 1787 y FL(m)3157 1773 y FN(\()p FM(v)s FN(\)\).)203 b(Q.E.D.)94 1980 y(The)23 b(magnitude)g(gap)g(b)s(et)m(w)m (een)g(the)g(condition)g(n)m(um)m(b)s(ers)e FM(\024)p FN(\()p FM(C)2274 1994 y FL(m)2342 1980 y FN(\()p FM(v)s FN(\)\))j(and)e FM(\024)p FN(\()p FM(L)2836 1994 y FL(m)2903 1980 y FN(\()p FM(v)s FN(\)\))i(can)f(b)s(e)f(tremendous)94 2093 y(for)k(seemingly)g(harmless)f FM(v)k FN(and)c(mo)s(derate)g FM(m)p FN(.)39 b(Actually)-8 b(,)29 b FM(L)2241 2107 y FL(m)2307 2093 y FN(\()p FM(v)s FN(\))d(can)g(b)s(e)f(pathetically)j (ill-conditioned,)94 2206 y(making)e(the)g(long)g(division)f(\(26\))i (virtually)f(a)g(singular)f(pro)s(cess,)h(while)g FM(C)2682 2220 y FL(m)2748 2206 y FN(\()p FM(v)s FN(\))h(is)e(still)h(w)m(ell)h (conditioned.)94 2319 y(F)-8 b(or)45 b(example,)j(consider)c(a)g (simple)g(p)s(olynomial)g FM(v)s FN(\()p FM(x)p FN(\))49 b(=)e FM(x)30 b FN(+)e(25.)83 b(When)43 b FM(m)h FN(increases,)k FM(\024)p FN(\()p FM(L)3594 2333 y FL(m)3661 2319 y FN(\()p FM(v)s FN(\)\))94 2432 y(gro)m(ws)38 b(exp)s(onen)m(tially)g(but)e 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2758 4673 V 102 w(11.)p 3278 4673 V 628 4752 V 711 4728 a(30.17981700)p 1132 4752 V 1612 4752 V 2251 4752 V 2758 4752 V 3278 4752 V 628 4831 V 746 4807 a(8.33455813)p 1132 4831 V 1612 4831 V 2251 4831 V 2758 4831 V 3278 4831 V 630 4834 2650 4 v 325 5017 a FN(T)-8 b(able)31 b(6:)41 b(A)31 b(n)m(umerical)g(comparison)f (b)s(et)m(w)m(een)h(long)g(division)f(and)g(least)i(squares)e(division) 94 5170 y(Extracting)35 b FO(v)606 5184 y Fx(0)678 5170 y FN(and)e FO(w)934 5184 y Fx(0)1006 5170 y FN(from)f(the)i(singular)f (v)m(ector)h(and)f(solving)h(\(27\))g(for)f FO(u)2888 5184 y Fx(0)2928 5170 y FN(,)g(w)m(e)h(shall)f(use)g(them)g(as)94 5283 y(the)26 b(initial)g(iterates)g(for)f(the)g(Gauss-Newton)h (iteration)h(\(25\))f(that)g(re\014nes)e(the)h(GCD)g(triplet.)40 b(Moreo)m(v)m(er,)94 5396 y(the)29 b(linear)g(system)g(\(27\))g(is)g (banded,)f(with)g(bandwidth)f(b)s(eing)h(one)g(plus)g(the)g(n)m(um)m(b) s(er)g(of)g(distinct)h(ro)s(ots.)94 5509 y(Therefore,)i(the)f(cost)i (of)e(solving)h(\(27\))h(is)f(insigni\014can)m(t)g(in)f(the)g(o)m(v)m (erall)j(complexit)m(y)-8 b(.)1909 5686 y(23)p eop %%Page: 24 26 24 25 bop 94 99 a FO(4.2.4)106 b(Re\014ning)36 b(the)e(GCD)h(with)f (the)h(Gauss-Newton)f(iteration)94 270 y FN(The)c(Gauss-Newton)i (iteration)f(is)g(exp)s(ected)g(to)g(reduce)f(the)g(residual)606 338 y Fu(\015)606 388 y(\015)606 438 y(\015)606 487 y(\015)606 537 y(\015)667 341 y( )774 429 y FM(conv)s FN(\()p FO(u)1052 443 y FL(j)1090 429 y FM(;)15 b FO(v)1185 443 y FL(j)1222 429 y FN(\))774 542 y FM(conv)s FN(\()p FO(u)1052 556 y FL(j)1090 542 y FM(;)g FO(w)1206 556 y FL(j)1242 542 y FN(\))1319 341 y Fu(!)1405 485 y Fy(\000)1496 341 y Fu( )1603 429 y FO(f)1603 542 y(f)1645 509 y FE(0)1710 341 y Fu(!)1775 338 y(\015)1775 388 y(\015)1775 438 y(\015)1775 487 y(\015)1775 537 y(\015)1821 591 y FL(W)1927 485 y FN(=)2023 338 y Fu(\015)2023 388 y(\015)2023 438 y(\015)2023 487 y(\015)2023 537 y(\015)2085 485 y FM(W)2198 341 y Fu( )2305 429 y FM(conv)s FN(\()p 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(system)919 1340 y Fu(h)983 1434 y FM(W)13 b(J)c FN(\()p FO(u)1234 1448 y FL(j)1271 1434 y FM(;)15 b FO(v)1366 1448 y FL(j)1403 1434 y FM(;)g FO(w)1519 1448 y FL(j)1555 1434 y FN(\))1616 1340 y Fu(i)1680 1434 y FO(z)26 b FN(=)f FM(W)1961 1240 y Fu(2)1961 1386 y(6)1961 1439 y(4)2073 1321 y FO(e)2121 1288 y FE(>)2121 1344 y Fx(1)2180 1321 y FO(u)2238 1335 y FL(j)2660 1321 y Fy(\000)82 b FN(1)2073 1434 y FM(conv)s FN(\()p FO(u)2351 1448 y FL(j)2389 1434 y FM(;)15 b FO(v)2484 1448 y FL(j)2521 1434 y FN(\))104 b Fy(\000)82 b FO(f)2073 1547 y FM(conv)s FN(\()p FO(u)2351 1561 y FL(j)2389 1547 y FM(;)15 b FO(w)2505 1561 y FL(j)2541 1547 y FN(\))84 b Fy(\000)e FO(f)2855 1514 y FE(0)2935 1240 y Fu(3)2935 1386 y(7)2935 1439 y(5)1965 2639 y @beginspecial 195 @llx 201 @lly 431 @urx 590 @ury 647 @rwi 1187 @rhi @setspecial %%BeginDocument: sparse.eps %!PS-Adobe-2.0 EPSF-1.2 %%Creator: MATLAB, The Mathworks, Inc. %%Title: E:\AAA\illcond\sparse.eps %%CreationDate: 12/27/2002 14:42:23 %%DocumentNeededFonts: Helvetica %%DocumentProcessColors: Cyan Magenta Yellow Black %%Pages: 1 %%BoundingBox: 195 201 431 590 %%EndComments %%BeginProlog % MathWorks dictionary /MathWorks 160 dict begin % definition operators /bdef {bind def} bind def /ldef {load def} bind def /xdef {exch def} bdef /xstore {exch store} bdef % operator abbreviations /c /clip ldef /cc /concat ldef /cp /closepath ldef /gr /grestore ldef /gs /gsave ldef /mt /moveto ldef /np /newpath ldef /cm /currentmatrix ldef /sm /setmatrix ldef /rm /rmoveto ldef /rl /rlineto ldef /s /show ldef /sc {setcmykcolor} bdef /sr /setrgbcolor ldef /sg /setgray ldef /w /setlinewidth ldef /j /setlinejoin ldef /cap /setlinecap ldef /rc {rectclip} bdef /rf {rectfill} bdef % page state control /pgsv () def /bpage {/pgsv save def} bdef /epage {pgsv restore} bdef /bplot /gsave ldef /eplot {stroke grestore} bdef % orientation switch /portraitMode 0 def /landscapeMode 1 def /rotateMode 2 def % coordinate system mappings /dpi2point 0 def % font control /FontSize 0 def /FMS {/FontSize xstore findfont [FontSize 0 0 FontSize neg 0 0] makefont setfont} bdef /ISOLatin1Encoding where {pop /WindowsLatin1Encoding 256 array bdef ISOLatin1Encoding WindowsLatin1Encoding copy pop /.notdef/.notdef/quotesinglbase/florin/quotedblbase/ellipsis/dagger /daggerdbl/circumflex/perthousand/Scaron/guilsinglleft/OE/.notdef/.notdef /.notdef/.notdef/quoteleft/quoteright/quotedblleft/quotedblright/bullet /endash/emdash/tilde/trademark/scaron/guilsinglright/oe/.notdef/.notdef /Ydieresis WindowsLatin1Encoding 128 32 getinterval astore pop} {/WindowsLatin1Encoding StandardEncoding bdef} ifelse /reencode {exch dup where {pop load} {pop StandardEncoding} ifelse exch dup 3 1 roll findfont dup length dict begin { 1 index /FID ne {def}{pop pop} ifelse } forall /Encoding exch def currentdict end definefont pop} bdef /isroman {findfont /CharStrings get /Agrave known} bdef /FMSR {3 1 roll 1 index dup isroman {reencode} {pop pop} ifelse exch FMS} bdef /csm {1 dpi2point div -1 dpi2point div scale neg translate dup landscapeMode eq {pop -90 rotate} {rotateMode eq {90 rotate} if} ifelse} bdef % line types: solid, dotted, dashed, dotdash /SO { [] 0 setdash } bdef /DO { [.5 dpi2point mul 4 dpi2point mul] 0 setdash } bdef /DA { [6 dpi2point mul] 0 setdash } bdef /DD { [.5 dpi2point mul 4 dpi2point mul 6 dpi2point mul 4 dpi2point mul] 0 setdash } bdef % macros for lines and objects /L {lineto stroke} bdef /MP {3 1 roll moveto 1 sub {rlineto} repeat} bdef /AP {{rlineto} repeat} bdef /PDlw -1 def /W {/PDlw currentlinewidth def setlinewidth} def /PP {closepath eofill} bdef /DP {closepath stroke} bdef /MR {4 -2 roll moveto dup 0 exch rlineto exch 0 rlineto neg 0 exch rlineto closepath} bdef /FR {MR stroke} bdef /PR {MR fill} bdef /L1i {{currentfile picstr readhexstring pop} image} bdef /tMatrix matrix def /MakeOval {newpath tMatrix currentmatrix pop translate scale 0 0 1 0 360 arc tMatrix setmatrix} bdef /FO {MakeOval stroke} bdef /PO {MakeOval fill} bdef /PD {currentlinewidth 2 div 0 360 arc fill PDlw -1 eq not {PDlw w /PDlw -1 def} if} def /FA {newpath tMatrix currentmatrix pop translate scale 0 0 1 5 -2 roll arc tMatrix setmatrix stroke} bdef /PA {newpath tMatrix currentmatrix pop translate 0 0 moveto scale 0 0 1 5 -2 roll arc closepath tMatrix setmatrix fill} bdef /FAn {newpath tMatrix currentmatrix pop translate scale 0 0 1 5 -2 roll arcn tMatrix setmatrix stroke} bdef /PAn {newpath tMatrix currentmatrix pop translate 0 0 moveto scale 0 0 1 5 -2 roll arcn closepath tMatrix setmatrix fill} bdef /vradius 0 def /hradius 0 def /lry 0 def /lrx 0 def /uly 0 def /ulx 0 def /rad 0 def /MRR {/vradius xdef /hradius xdef /lry xdef /lrx xdef /uly xdef /ulx xdef newpath tMatrix currentmatrix pop ulx hradius add uly vradius add translate hradius vradius scale 0 0 1 180 270 arc tMatrix setmatrix lrx hradius sub uly vradius add translate hradius vradius scale 0 0 1 270 360 arc tMatrix setmatrix lrx hradius sub lry vradius sub translate hradius vradius scale 0 0 1 0 90 arc tMatrix setmatrix ulx hradius add lry vradius sub translate hradius vradius scale 0 0 1 90 180 arc tMatrix setmatrix closepath} bdef /FRR {MRR stroke } bdef /PRR {MRR fill } bdef /MlrRR {/lry xdef /lrx xdef /uly xdef /ulx xdef /rad lry uly sub 2 div def newpath tMatrix currentmatrix pop ulx rad add uly rad add translate rad rad scale 0 0 1 90 270 arc tMatrix setmatrix lrx rad sub lry rad sub translate rad rad scale 0 0 1 270 90 arc tMatrix setmatrix closepath} bdef /FlrRR {MlrRR stroke } bdef /PlrRR {MlrRR fill } bdef /MtbRR {/lry xdef /lrx xdef /uly xdef /ulx xdef /rad lrx ulx sub 2 div def newpath tMatrix currentmatrix pop ulx rad add uly rad add translate rad rad scale 0 0 1 180 360 arc tMatrix setmatrix lrx rad sub lry rad sub translate rad rad scale 0 0 1 0 180 arc tMatrix setmatrix closepath} bdef /FtbRR {MtbRR stroke } bdef /PtbRR {MtbRR fill } bdef /stri 6 array def /dtri 6 array def /smat 6 array def /dmat 6 array def /tmat1 6 array def /tmat2 6 array def /dif 3 array def /asub {/ind2 exch def /ind1 exch def dup dup ind1 get exch ind2 get sub exch } bdef /tri_to_matrix { 2 0 asub 3 1 asub 4 0 asub 5 1 asub dup 0 get exch 1 get 7 -1 roll astore } bdef /compute_transform { dmat dtri tri_to_matrix tmat1 invertmatrix smat stri tri_to_matrix tmat2 concatmatrix } bdef /ds {stri astore pop} bdef /dt {dtri astore pop} bdef /db {2 copy /cols xdef /rows xdef mul dup string currentfile exch readhexstring pop /bmap xdef pop pop} bdef /it {gs np dtri aload pop moveto lineto lineto cp c cols rows 8 compute_transform {bmap} image gr}bdef /il {newpath moveto lineto stroke}bdef currentdict end def %%EndProlog %%BeginSetup MathWorks begin 0 cap end %%EndSetup %%Page: 1 1 %%BeginPageSetup %%PageBoundingBox: 195 201 431 590 MathWorks begin bpage %%EndPageSetup %%BeginObject: obj1 bplot /dpi2point 12 def portraitMode 0204 7344 csm 2141 256 2837 4675 MR c np 85 dict begin %Colortable dictionary /c0 { 0 0 0 sr} bdef /c1 { 1 1 1 sr} bdef /c2 { 1 0 0 sr} bdef /c3 { 0 1 0 sr} bdef /c4 { 0 0 1 sr} bdef /c5 { 1 1 0 sr} bdef /c6 { 1 0 1 sr} bdef /c7 { 0 1 1 sr} bdef c0 1 j 1 sg 0 0 6913 5185 PR 6 w 0 -4225 2535 0 0 4225 2309 389 4 MP PP -2535 0 0 -4225 2535 0 0 4225 2309 389 5 MP stroke 4 w 6 w 0 sg 2309 4614 mt 4844 4614 L 2309 389 mt 4844 389 L 4844 389 mt 4844 4614 L 2309 389 mt 2309 4614 L 2309 4614 mt 4844 4614 L 2309 389 mt 2309 4614 L 2309 4614 mt 2309 4571 L 2309 389 mt 2309 431 L %%IncludeResource: font Helvetica /Helvetica /WindowsLatin1Encoding 120 FMSR 2276 4759 mt (0) s 2732 4614 mt 2732 4571 L 2732 389 mt 2732 431 L 2699 4759 mt (5) s 3154 4614 mt 3154 4571 L 3154 389 mt 3154 431 L 3088 4759 mt (10) s 3577 4614 mt 3577 4571 L 3577 389 mt 3577 431 L 3511 4759 mt (15) s 3999 4614 mt 3999 4571 L 3999 389 mt 3999 431 L 3933 4759 mt (20) s 4422 4614 mt 4422 4571 L 4422 389 mt 4422 431 L 4356 4759 mt (25) s 4844 4614 mt 4844 4571 L 4844 389 mt 4844 431 L 4778 4759 mt (30) s 2309 389 mt 2351 389 L 4844 389 mt 4802 389 L 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PD 56 W 4760 3769 PD 56 W 4760 3853 PD 56 W 4760 3938 PD 56 W 4760 4022 PD 56 W 4760 4107 PD 56 W 4760 4191 PD 56 W 4760 4276 PD 56 W 4760 4360 PD 56 W 4760 4445 PD 56 W 4760 4529 PD gs 2309 389 2536 4226 MR c np gr gr 3346 4902 mt (nz = 409) s end eplot %%EndObject epage end showpage %%Trailer %%EOF %%EndDocument @endspecial 2680 2074 a FL(QR)2596 2159 y Fe(\000)-29 b(!)2964 2639 y @beginspecial 195 @llx 201 @lly 431 @urx 590 @ury 647 @rwi 1187 @rhi @setspecial %%BeginDocument: sparser.eps %!PS-Adobe-2.0 EPSF-1.2 %%Creator: MATLAB, The Mathworks, Inc. %%Title: E:\AAA\illcond\sparser.eps %%CreationDate: 12/27/2002 14:42:54 %%DocumentNeededFonts: Helvetica %%DocumentProcessColors: Cyan Magenta Yellow Black %%Pages: 1 %%BoundingBox: 195 201 431 590 %%EndComments %%BeginProlog % MathWorks dictionary /MathWorks 160 dict begin % definition operators /bdef {bind def} bind def /ldef {load def} bind def /xdef {exch def} bdef /xstore {exch store} bdef % operator abbreviations /c /clip ldef /cc 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tMatrix setmatrix fill} bdef /FAn {newpath tMatrix currentmatrix pop translate scale 0 0 1 5 -2 roll arcn tMatrix setmatrix stroke} bdef /PAn {newpath tMatrix currentmatrix pop translate 0 0 moveto scale 0 0 1 5 -2 roll arcn closepath tMatrix setmatrix fill} bdef /vradius 0 def /hradius 0 def /lry 0 def /lrx 0 def /uly 0 def /ulx 0 def /rad 0 def /MRR {/vradius xdef /hradius xdef /lry xdef /lrx xdef /uly xdef /ulx xdef newpath tMatrix currentmatrix pop ulx hradius add uly vradius add translate hradius vradius scale 0 0 1 180 270 arc tMatrix setmatrix lrx hradius sub uly vradius add translate hradius vradius scale 0 0 1 270 360 arc tMatrix setmatrix lrx hradius sub lry vradius sub translate hradius vradius scale 0 0 1 0 90 arc tMatrix setmatrix ulx hradius add lry vradius sub translate hradius vradius scale 0 0 1 90 180 arc tMatrix setmatrix closepath} bdef /FRR {MRR stroke } bdef /PRR {MRR fill } bdef /MlrRR {/lry xdef /lrx xdef /uly xdef /ulx xdef /rad lry uly sub 2 div def newpath tMatrix currentmatrix pop ulx rad add uly rad add translate rad rad scale 0 0 1 90 270 arc tMatrix setmatrix lrx rad sub lry rad sub translate rad rad scale 0 0 1 270 90 arc tMatrix setmatrix closepath} bdef /FlrRR {MlrRR stroke } bdef /PlrRR {MlrRR fill } bdef /MtbRR {/lry xdef /lrx xdef /uly xdef /ulx xdef /rad lrx ulx sub 2 div def newpath tMatrix currentmatrix pop ulx rad add uly rad add translate rad rad scale 0 0 1 180 360 arc tMatrix setmatrix lrx rad sub lry rad sub translate rad rad scale 0 0 1 0 180 arc tMatrix setmatrix closepath} bdef /FtbRR {MtbRR stroke } bdef /PtbRR {MtbRR fill } bdef /stri 6 array def /dtri 6 array def /smat 6 array def /dmat 6 array def /tmat1 6 array def /tmat2 6 array def /dif 3 array def /asub {/ind2 exch def /ind1 exch def dup dup ind1 get exch ind2 get sub exch } bdef /tri_to_matrix { 2 0 asub 3 1 asub 4 0 asub 5 1 asub dup 0 get exch 1 get 7 -1 roll astore } bdef /compute_transform { dmat dtri tri_to_matrix tmat1 invertmatrix smat stri tri_to_matrix tmat2 concatmatrix } bdef /ds {stri astore pop} bdef /dt {dtri astore pop} bdef /db {2 copy /cols xdef /rows xdef mul dup string currentfile exch readhexstring pop /bmap xdef pop pop} bdef /it {gs np dtri aload pop moveto lineto lineto cp c cols rows 8 compute_transform {bmap} image gr}bdef /il {newpath moveto lineto stroke}bdef currentdict end def %%EndProlog %%BeginSetup MathWorks begin 0 cap end %%EndSetup %%Page: 1 1 %%BeginPageSetup %%PageBoundingBox: 195 201 431 590 MathWorks begin bpage %%EndPageSetup %%BeginObject: obj1 bplot /dpi2point 12 def portraitMode 0204 7344 csm 2141 256 2837 4675 MR c np 85 dict begin %Colortable dictionary /c0 { 0 0 0 sr} bdef /c1 { 1 1 1 sr} bdef /c2 { 1 0 0 sr} bdef /c3 { 0 1 0 sr} bdef /c4 { 0 0 1 sr} bdef /c5 { 1 1 0 sr} bdef /c6 { 1 0 1 sr} bdef /c7 { 0 1 1 sr} bdef c0 1 j 1 sg 0 0 6913 5185 PR 6 w 0 -4225 2535 0 0 4225 2309 389 4 MP PP -2535 0 0 -4225 2535 0 0 4225 2309 389 5 MP stroke 4 w 6 w 0 sg 2309 4614 mt 4844 4614 L 2309 389 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FO(v)q FM(;)g FO(w)q FN(\))26 b(in)m(to)f(accoun)m(t,)i(it)e(can)f(b)s(e)f(v)m (eri\014ed)i(that)f(the)h(sparse)e(QR)h(decomp)s(osition)h(costs)94 3167 y FM(O)s FN(\()p FM(mk)331 3134 y Fx(2)388 3167 y FN(+)17 b FM(m)556 3134 y Fx(2)595 3167 y FM(k)j FN(+)d FM(k)800 3134 y Fx(3)840 3167 y FN(\),)29 b(where,)g(as)g(b)s(efore,)g FM(k)j FN(is)d(the)g(n)m(um)m(b)s(er)e(of)i(distinct)g(ro)s(ots)g(and)g FM(m)c FN(=)g FM(n)16 b Fy(\000)h FM(k)s FN(.)40 b(If)28 b FM(k)k FN(is)94 3280 y FM(o)p FN(\()p FM(n)p FN(\),)f(then)f(the)h (complexit)m(y)h(is)f(reduced)e(to)i FM(O)s FN(\()p FM(k)s(n)1898 3247 y Fx(2)1938 3280 y FN(\).)94 3515 y FP(4.3)113 b(Computing)38 b(the)f(m)m(ultiplicit)m(y)h(structure)f(and)h(initial)h(ro)s(ot)e (estimates)94 3687 y FN(F)-8 b(or)32 b(a)g(giv)m(en)g(p)s(olynomial)g FM(p)p FN(,)f(the)g(pro)s(cedure)f(\(20\))j(generates)f(a)g(sequence)g (of)f(square-free)g(p)s(olynomials)94 3800 y FM(v)138 3814 y Fx(1)178 3800 y FM(;)15 b(v)262 3814 y Fx(2)302 3800 y FM(;)g 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(distinct)94 4222 y(ro)s(ots)h(of)f FM(p)p FN(,)g(while)g(ro)s(ots)g (of)h FM(v)1168 4236 y Fx(2)1241 4222 y FN(consists)f(of)g(all)h (distinct)g(ro)s(ots)f(of)g FM(p=v)2625 4236 y Fx(1)2664 4222 y FN(,)h(etc.)53 b(With)34 b(these)h(prop)s(erties,)94 4335 y(the)c(m)m(ultiplicit)m(y)h(structure)e(is)h(determined)f(b)m(y)g (the)h(degrees)g FM(d)2344 4349 y Fx(1)2383 4335 y FM(;)15 b(d)2470 4349 y Fx(2)2510 4335 y FM(;)g Fy(\001)g(\001)g(\001)i FM(;)e(d)2759 4349 y FL(s)2797 4335 y FN(.)40 b(F)-8 b(or)31 b(example,)h(consider)1356 4490 y FM(p)p FN(\()p FM(x)p FN(\))26 b(=)f(\()p FM(x)c Fy(\000)e FM(a)p FN(\)\()p FM(x)i Fy(\000)f FM(b)p FN(\))2200 4452 y Fx(3)2240 4490 y FN(\()p FM(x)h Fy(\000)e FM(c)p FN(\))2512 4452 y Fx(4)94 4645 y FN(for)31 b(an)m(y)f FM(a)p FN(,)h FM(b)f FN(and)g FM(c)p FN(.)41 b(W)-8 b(e)32 b(ha)m(v)m(e)1726 4878 y FM(v)1770 4892 y FL(j)1807 4878 y FN('s)372 b(degrees)31 b(of)g FM(v)2707 4892 y FL(j)2743 4878 y FN('s)309 b(ro)s(ots)p 546 4915 2817 4 v 596 4994 a FM(v)640 5008 y Fx(1)680 4994 y FN(\()p FM(x)p FN(\))25 b(=)324 b(\()p FM(x)20 b Fy(\000)g FM(a)p FN(\)\()p FM(x)h Fy(\000)f FM(b)p FN(\)\()p FM(x)h Fy(\000)f FM(c)p FN(\))321 b FM(d)2417 5008 y Fx(1)2507 4994 y FN(=)51 b(3)331 b FM(a)91 b(b)g(c)596 5107 y(v)640 5121 y Fx(2)680 5107 y FN(\()p FM(x)p FN(\))25 b(=)606 b(\()p FM(x)20 b Fy(\000)g FM(b)p FN(\)\()p FM(x)h Fy(\000)f FM(c)p FN(\))321 b FM(d)2417 5121 y Fx(2)2507 5107 y FN(=)51 b(2)470 b FM(b)91 b(c)596 5220 y(v)640 5234 y Fx(3)680 5220 y FN(\()p FM(x)p FN(\))25 b(=)606 b(\()p FM(x)20 b Fy(\000)g FM(b)p FN(\)\()p FM(x)h Fy(\000)f FM(c)p FN(\))321 b FM(d)2417 5234 y Fx(3)2507 5220 y FN(=)51 b(2)470 b FM(b)91 b(c)596 5333 y(v)640 5347 y Fx(4)680 5333 y FN(\()p FM(x)p FN(\))25 b(=)878 b(\()p FM(x)21 b Fy(\000)f FM(c)p FN(\))321 b FM(d)2417 5347 y Fx(4)2507 5333 y FN(=)51 b(1)600 b FM(c)p 546 5370 V 998 5449 a FN(m)m(ultiplicit)m(y)32 b(structure)e(of)h FM(p)p FN(:)945 b(1)91 b(3)h(4)1909 5686 y(24)p eop %%Page: 25 27 25 26 bop 94 99 a FN(Without)34 b(lo)s(cating)g(the)f(ro)s(ots)f FM(a)p FN(,)i FM(b)e FN(and)g FM(c)p FN(,)h(the)g(m)m(ultiplicit)m(y)i (structure)d([)p FM(`)2760 113 y Fx(1)2799 99 y FM(;)15 b(`)2877 113 y Fx(2)2917 99 y FM(;)g(`)2995 113 y Fx(3)3035 99 y FN(])29 b(=)f([1)p FM(;)15 b FN(3)p FM(;)g FN(4])35 b(is)e(deter-)94 211 y(mined)d(solely)i(from)e(the)g(degrees)h FM(d)1355 225 y Fx(1)1395 211 y FM(;)15 b Fy(\001)g(\001)g(\001)h FM(;)f(d)1643 225 y Fx(4)1684 211 y FN(:)906 415 y FM(`)944 429 y Fx(1)1009 415 y FN(=)25 b(1)166 b(since)91 b FM(d)1646 429 y Fx(1)2141 415 y Fy(\025)25 b FN(3)152 b(=)25 b(\()p FM(d)2612 429 y Fx(1)2672 415 y FN(+)20 b(1\))h Fy(\000)f FN(1)906 552 y FM(`)944 566 y Fx(2)1009 552 y FN(=)25 b(3)166 b(since)91 b FM(d)1646 566 y Fx(1)1686 552 y FM(;)15 b(d)1773 566 y Fx(2)1813 552 y FM(;)g(d)1900 566 y Fx(3)2143 552 y Fy(\025)24 b FN(2)152 b(=)25 b(\()p FM(d)2613 566 y Fx(1)2674 552 y FN(+)20 b(1\))h Fy(\000)f FN(2)906 690 y FM(`)944 704 y Fx(3)1009 690 y FN(=)25 b(4)166 b(since)91 b FM(d)1646 704 y Fx(1)1686 690 y FM(;)15 b(d)1773 704 y Fx(2)1813 690 y FM(;)g(d)1900 704 y Fx(3)1941 690 y FM(;)g(d)2028 704 y Fx(4)2143 690 y Fy(\025)25 b FN(1)152 b(=)25 b(\()p FM(d)2614 704 y Fx(1)2675 690 y FN(+)20 b(1\))h Fy(\000)e FN(3)94 1031 y(Generally)-8 b(,)33 b(w)m(e)e(ha)m(v)m(e)g(the)g(follo)m(wing)h (theorem)e(on)h(iden)m(tifying)g(the)f(m)m(ultiplicit)m(y)j(structure.) 94 1266 y FO(Theorem)i(4.3)46 b FC(F)-7 b(or)33 b(a)e(given)f(p)-5 b(olynomial)34 b FM(p)p FC(,)d(let)g FM(v)1929 1280 y Fx(1)1969 1266 y FM(;)15 b Fy(\001)g(\001)g(\001)h FM(;)f(v)2214 1280 y FL(s)2282 1266 y FC(b)-5 b(e)31 b(the)h(squar)-5 b(efr)g(e)g(e)32 b(factors)g(of)f FM(p)g FC(gener)-5 b(ate)g(d)94 1379 y(by)30 b(the)g(pr)-5 b(o)g(c)g(e)g(dur)g(e)32 b(\(20\))f(with)f(de)-5 b(gr)g(e)g(es)31 b FM(d)1497 1393 y Fx(1)1562 1379 y Fy(\025)25 b FM(d)1705 1393 y Fx(2)1770 1379 y Fy(\025)g(\001)15 b(\001)g(\001)26 b(\025)f FM(d)2140 1393 y FL(s)2207 1379 y FC(r)-5 b(esp)g(e)g(ctively.)42 b(L)-5 b(et)30 b FM(k)e FN(=)d FM(d)3090 1393 y Fx(1)3155 1379 y FN(=)g FM(deg)s FN(\()p FM(v)3465 1393 y Fx(1)3506 1379 y FN(\))p FC(.)41 b(Then)94 1492 y(the)33 b(multiplicity)h (structur)-5 b(e)33 b FM(`)g FC(of)g FM(p)f FC(c)-5 b(onsists)34 b(of)f(c)-5 b(omp)g(onents)964 1706 y FM(`)1002 1720 y FL(j)1063 1706 y FN(=)25 b(max)1344 1612 y Fu(n)1414 1706 y FM(t)1462 1608 y Fu(\014)1462 1658 y(\014)1462 1708 y(\014)1505 1706 y FM(d)1552 1720 y FL(t)1607 1706 y Fy(\025)g FN(\()p FM(d)1785 1720 y Fx(1)1845 1706 y FN(+)20 b(1\))h Fy(\000)f FM(j)2186 1612 y Fu(o)2257 1706 y FM(;)91 b(j)31 b FN(=)25 b(1)p FM(;)15 b FN(2)p FM(;)g Fy(\001)g(\001)g(\001)i FM(;)e(k)s(:)709 b FN(\(30\))94 2022 y FO(Pro)s(of.)83 b FN(A)30 b(straigh)m(tforw)m(ard)h(v)m (eri\014cation.)1877 b(Q.E.D.)94 2230 y(The)30 b(lo)s(cation)i(of)f (the)f(ro)s(ots)h(is)f(not)h(needed)f(in)g(deciding)h(the)f(structure.) 94 2437 y(The)41 b(initial)h(ro)s(ot)g(appro)m(ximation)g(is)f (determined)f(based)h(on)g(the)g(fact)h(that)g(an)e FM(l)r FN(-fold)i(ro)s(ot)f(of)g FM(p)p FN(\()p FM(x)p FN(\))94 2550 y(app)s(ears)32 b FM(l)i FN(times)e(as)g(a)h(simple)f(ro)s(ot)g (of)g(eac)m(h)i(p)s(olynomial)e(among)h FM(v)2532 2564 y Fx(1)2571 2550 y FM(;)15 b Fy(\001)g(\001)g(\001)i FM(;)e(v)2817 2565 y FL(l)2843 2550 y FN(.)46 b(After)32 b(calculating)i(ro)s(ots)94 2663 y(of)h(eac)m(h)g FM(v)454 2677 y FL(j)524 2663 y FN(with)e(a)i(standard)e(ro)s(ot-\014nder,)h(n)m (umerically)g(\\iden)m(tical")j(ro)s(ots)d(of)g FM(v)3018 2677 y FL(j)3055 2663 y FN('s)g(are)g(group)s(ed)e(in)i(a)94 2776 y(straigh)m(tforw)m(ard)26 b(manner,)f(according)g(to)h(the)f(m)m (ultiplicit)m(y)h(structure)e([)p FM(`)2666 2790 y Fx(1)2706 2776 y FM(;)15 b Fy(\001)g(\001)g(\001)h FM(;)f(`)2945 2791 y FL(k)2988 2776 y FN(])25 b(determined)f(b)m(y)h(\(30\),)94 2889 y(to)32 b(form)d(the)i(initial)h(ro)s(ot)e(appro)m(ximation)i(\()p FM(z)1718 2903 y Fx(1)1758 2889 y FM(;)15 b Fy(\001)g(\001)g(\001)h FM(;)f(z)2001 2904 y FL(k)2045 2889 y FN(\))30 b(that)h(is)g(needed)f (b)m(y)g(Algorithm)h(I.)94 3132 y FP(4.4)113 b(Con)m(trol)37 b(parameters)94 3303 y FN(W)-8 b(e)38 b(use)d(three)h(con)m(trol)i (parameters)e(for)g(the)g(recursiv)m(e)g(GCD)g(computation.)59 b(The)35 b(default)h(v)-5 b(alues)36 b(of)94 3416 y(those)23 b(parameters)f(giv)m(en)g(b)s(elo)m(w)g(are)g(selected)h(under)d(the)i (assumption)f(that)h(the)g(IEEE)f(standard)g(double)94 3529 y(precision)37 b(of)f(16)h(decimal)g(digits)g(is)f(used.)57 b(The)35 b(\014rst)h(con)m(trol)h(parameter)g(is)f(the)g FC(zer)-5 b(o)39 b(singular)f(value)94 3642 y(thr)-5 b(eshold)65 b FM(\022)34 b FN(for)c(iden)m(tifying)i(the)f(zero)h (singular)f(v)-5 b(alue.)43 b(The)31 b(default)g(c)m(hoice)i(is)e FM(\022)d FN(=)f(10)3260 3609 y FE(\000)p Fx(8)3355 3642 y FN(.)42 b(When)31 b(the)94 3766 y(smallest)i(singular)f(v)-5 b(alue)32 b FM(&)1060 3781 y FL(l)1117 3766 y FN(of)1237 3743 y(^)1222 3766 y FM(S)1278 3781 y FL(l)1304 3766 y FN(\()p FM(u)1391 3780 y FL(m)p FE(\000)p Fx(1)1548 3766 y FN(\))g(is)g(less)g(than)f FM(\022)2136 3668 y Fu(\015)2136 3718 y(\015)2136 3768 y(\015)2196 3766 y FO(u)2254 3780 y FL(m)p FE(\000)p Fx(1)2426 3668 y Fu(\015)2426 3718 y(\015)2426 3768 y(\015)2472 3822 y Fx(2)2512 3766 y FN(,)h(it)g(will)g(b)s(e)f FC(tentatively)i FN(considered)94 3898 y(as)c(a)f(zero)h(\(p)s(ending)e(con\014rmation)h(from)f(the)h (residual)g(information)g(pro)s(duced)f(b)m(y)g(the)i(Gauss-Newton)94 4011 y(iteration\).)57 b(Then)34 b(the)h(Gauss-Newton)h(iteration)g(is) f(initiated)i(to)e(further)f(reduce)h(the)g(residual)g(as)g(in)94 4124 y(\(29\))c(to)e(its)g(n)m(umerical)g(limit.)41 b(W)-8 b(e)30 b(use)e(the)h(second)g(con)m(trol)h(parameter,)g(the)f FC(initial)i(r)-5 b(esidual)33 b(toler)-5 b(anc)g(e)94 4237 y FM(\045)p FN(,)34 b(to)g(decide)g(if)f(the)g(re\014ned)f (residual)h(is)g(acceptable.)51 b(Our)32 b(default)i(c)m(hoice)h(is)e FM(\045)d FN(=)f(10)3198 4204 y FE(\000)p Fx(10)3329 4237 y FN(.)49 b(W)-8 b(e)34 b(accept)94 4350 y(the)d(GCD)g(triplet)g (\()p FM(u)853 4364 y FL(m)920 4350 y FM(;)15 b(v)1004 4364 y FL(m)1071 4350 y FM(;)g(w)1176 4364 y FL(m)1243 4350 y FN(\))31 b(when)e(the)h(residual)848 4618 y FM(\032)895 4632 y FL(m)987 4618 y FN(=)1083 4471 y Fu(\015)1083 4521 y(\015)1083 4571 y(\015)1083 4621 y(\015)1083 4670 y(\015)1144 4474 y( )1251 4562 y FM(conv)s FN(\()p FO(u)1529 4576 y FL(m)1597 4562 y FM(;)15 b FO(v)1692 4576 y FL(m)1759 4562 y FN(\))20 b Fy(\000)g FO(u)1963 4576 y FL(m)p FE(\000)p Fx(1)1251 4675 y FM(conv)s FN(\()p FO(u)1529 4689 y FL(m)1597 4675 y FM(;)15 b FO(w)1713 4689 y FL(m)1779 4675 y FN(\))21 b Fy(\000)f FO(u)1984 4642 y FE(0)1984 4698 y FL(m)p FE(\000)p Fx(1)2182 4474 y Fu(!)2248 4471 y(\015)2248 4521 y(\015)2248 4571 y(\015)2248 4621 y(\015)2248 4670 y(\015)2294 4724 y FL(W)2450 4618 y Fy(\024)76 b FM(\045)2659 4521 y Fu(\015)2659 4571 y(\015)2659 4621 y(\015)2720 4618 y FO(u)2778 4632 y FL(m)p FE(\000)p Fx(1)2950 4521 y Fu(\015)2950 4571 y(\015)2950 4621 y(\015)2996 4675 y Fx(2)3036 4618 y FM(:)592 b FN(\(31\))94 4891 y(Otherwise,)31 b(w)m(e)g(con)m(tin)m(ue)g(to)g(up)s(date)f FM(S)1520 4906 y FL(l)1546 4891 y FN(\()p FM(u)1633 4905 y FL(m)p FE(\000)p Fx(1)1790 4891 y FN(\))g(to)i FM(S)2023 4906 y FL(l)q Fx(+1)2139 4891 y FN(\()p FM(u)2226 4905 y FL(m)p FE(\000)p Fx(1)2383 4891 y FN(\))e(and)g(c)m(hec)m(k)i FM(&)2903 4906 y FL(l)q Fx(+1)3019 4891 y FM(;)15 b Fy(\001)g(\001)g (\001)r FN(.)94 5098 y(The)47 b(third)e(parameter)j(is)e(the)h FC(r)-5 b(esidual)49 b(toler)-5 b(anc)g(e)50 b(gr)-5 b(owth)49 b(factor)f FM(\036)p FN(.)89 b(Whenev)m(er)48 b(a)f(GCD)g(triplet)94 5211 y(\()p FM(u)181 5225 y FL(m)248 5211 y FM(;)15 b(v)332 5225 y FL(m)399 5211 y FM(;)g(w)504 5225 y FL(m)572 5211 y FN(\))34 b(and)g FM(\032)869 5225 y FL(m)970 5211 y FN(are)g(calculated,)k(The)c(error)g(in)g(\()p FM(u)2200 5225 y FL(m)2267 5211 y FM(;)15 b(v)2351 5225 y FL(m)2418 5211 y FM(;)g(w)2523 5225 y FL(m)2590 5211 y FN(\))34 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3999 V 3045 3999 V 3437 3999 V 468 4081 4 83 v 571 4057 a(correct)p 882 4081 V 151 w(parameters)p 1323 4081 V 187 w(co)r(de)p 1735 4081 V 187 w FL(x)1826 4066 y Fw(1)1880 4057 y Fx(=)c(0)p FL(:)2018 4043 y Fx(_)2010 4057 y(9)2053 4043 y(_)2045 4057 y(0)p 2207 4081 V 178 w FL(x)2298 4066 y Fw(2)2352 4057 y Fx(=)g(1)p FL(:)2490 4043 y Fx(_)2482 4057 y(8)2525 4043 y(_)2517 4057 y(1)p 2644 4081 V 143 w FL(x)2735 4066 y Fw(3)2789 4057 y Fx(=)g(2)p FL(:)2927 4043 y Fx(_)2919 4057 y(7)2962 4043 y(_)2954 4057 y(2)p 3045 4081 V 108 w(bac)n(kw)n(ard)p 3437 4081 V 468 4160 4 79 v 592 4136 a(digits)p 882 4160 V 286 w FL(\045)p Fx(,)j FL(\022)p 1323 4160 V 1735 4160 V 2207 4160 V 2644 4160 V 3045 4160 V 2009 w Fx(error)p 3437 4160 V 470 4163 2969 4 v 470 4180 V 468 4259 4 79 v 575 4235 a FL(k)e Fx(=)f(10)p 882 4259 V 166 w FL(\045)g Fx(=)g(1)p FL(e)15 b FE(\000)h Fx(9)p 1323 4259 V 111 w Fc(GcdR)n(oot)p 1735 4259 V 100 w Fx(0.90909090)p 2207 4259 V 137 w(1.8181818)p 2644 4259 V 137 w(2.7272727)p 3045 4259 V 140 w(1.7e-08)p 3437 4259 V 468 4338 V 882 4338 V 945 4314 a FL(\022)21 b Fx(=)f(1)p FL(e)c FE(\000)f Fx(7)p 1323 4338 V 123 w Fc(PejR)n(oot)p 1735 4338 V 112 w Fx(0.909090909)p 2207 4338 V 102 w(1.81818181)p 2644 4338 V 102 w(2.7272727)p 3045 4338 V 140 w(2.4e-10)p 3437 4338 V 470 4341 2969 4 v 468 4420 4 79 v 593 4396 a FL(k)21 b Fx(=)f(9)p 882 4420 V 183 w FL(\045)g Fx(=)g(1)p FL(e)15 b FE(\000)h Fx(8)p 1323 4420 V 111 w Fc(GcdR)n(oot)p 1735 4420 V 100 w Fx(0.909090)p 2207 4420 V 207 w(1.81818)p 2644 4420 V 207 w(2.72727)p 3045 4420 V 210 w(7.0e-06)p 3437 4420 V 468 4499 V 882 4499 V 945 4475 a FL(\022)21 b Fx(=)f(1)p FL(e)c FE(\000)f Fx(6)p 1323 4499 V 123 w Fc(PejR)n(oot)p 1735 4499 V 112 w Fx(0.9090909)p 2207 4499 V 172 w(1.8181818)p 2644 4499 V 137 w(2.727272)p 3045 4499 V 175 w(2.3e-09)p 3437 4499 V 470 4502 2969 4 v 468 4581 4 79 v 593 4557 a FL(k)21 b Fx(=)f(8)p 882 4581 V 183 w FL(\045)g Fx(=)g(1)p FL(e)15 b FE(\000)h Fx(7)p 1323 4581 V 111 w Fc(GcdR)n(oot)p 1735 4581 V 100 w Fx(0.90909)p 2207 4581 V 242 w(1.8182)p 2644 4581 V 242 w(2.727)p 3045 4581 V 280 w(1.3e-04)p 3437 4581 V 468 4660 V 882 4660 V 945 4636 a FL(\022)21 b Fx(=)f(1)p FL(e)c FE(\000)f Fx(5)p 1323 4660 V 123 w Fc(PejR)n(oot)p 1735 4660 V 112 w Fx(0.9090909)p 2207 4660 V 172 w(1.818181)p 2644 4660 V 172 w(2.72727)p 3045 4660 V 210 w(2.3e-08)p 3437 4660 V 470 4663 2969 4 v 468 4742 4 79 v 593 4718 a FL(k)21 b Fx(=)f(7)p 882 4742 V 183 w FL(\045)g Fx(=)g(1)p FL(e)15 b FE(\000)h Fx(6)p 1323 4742 V 111 w Fc(GcdR)n(oot)p 1735 4742 V 100 w Fx(0.9090)p 2207 4742 V 277 w(1.82)p 2644 4742 V 312 w(2.7)p 3045 4742 V 350 w(1.3e-02)p 3437 4742 V 468 4821 V 882 4821 V 945 4797 a FL(\022)21 b Fx(=)f(1)p FL(e)c FE(\000)f Fx(4)p 1323 4821 V 123 w Fc(PejR)n(oot)p 1735 4821 V 112 w Fx(0.90909)p 2207 4821 V 242 w(1.81818)p 2644 4821 V 207 w(2.7272)p 3045 4821 V 245 w(2.3e-07)p 3437 4821 V 470 4824 2969 4 v 468 4903 4 79 v 593 4879 a FL(k)21 b Fx(=)f(6)p 882 4903 V 218 w FE(\000)15 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y(As)39 b(sho)m(wn)g(in)f(this)h(test,)j(b)s(oth)c (metho)s(ds)g(allo)m(w)i(inexact)g(co)s(e\016cien)m(ts)h(to)f(certain)f (exten)m(t.)68 b(As)39 b(usual,)94 870 y(Algorithm)22 b(I)f(is)g(more)g(robust)f(than)g(Algorithm)i(I)s(I,)e(but)g(Algorithm) i(I)f(dep)s(ends)e(on)h(a)i(structure)e(iden)m(ti\014er.)94 1161 y FP(5.2)113 b(The)37 b(e\013ect)h(of)g(nearb)m(y)g(m)m(ultiple)g (ro)s(ots)94 1356 y FN(When)33 b(t)m(w)m(o)g(or)g(more)f(m)m(ultiple)i (ro)s(ots)e(are)h(nearb)m(y)-8 b(,)33 b(it)g(can)g(b)s(e)f(di\016cult)g (to)i(iden)m(tify)e(the)h(correct)h(m)m(ulti-)94 1469 y(plicit)m(y)e(structure.)41 b(W)-8 b(e)31 b(test)g(the)g(example)1166 1674 y FM(p)1212 1688 y FL(")1248 1674 y FN(\()p FM(x)p FN(\))26 b(=)f(\()p FM(x)c Fy(\000)f FN(1)g(+)g FM(")p FN(\))1924 1636 y Fx(20)2000 1674 y FN(\()p FM(x)g Fy(\000)g FN(1\))2278 1636 y Fx(20)2354 1674 y FN(\()p FM(x)g FN(+)g(0)p FM(:)p FN(5\))2702 1636 y Fx(5)94 1878 y FN(for)25 b(decreasing)g(ro)s (ot)g(gap)g FM(")h FN(=)f(0)p FM(:)p FN(1)p 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164 2450 3581 4 v 162 2529 4 79 v 214 2505 a Fx(ro)r(ot)24 b(gap)p 620 2529 V 1032 2529 V 1645 2529 V 2258 2529 V 2966 2529 V 2543 w(bac)n(kw)n(ard)p 3357 2529 V 101 w(condition)p 3743 2529 V 162 2608 V 308 2584 a FL(")p 620 2608 V 417 w Fx(co)r(de)p 1032 2608 V 187 w FL(x)1123 2593 y Fw(1)1177 2584 y Fx(=)c(1)c FE(\000)f FL(")p 1645 2608 V 291 w(x)1737 2593 y Fw(2)1790 2584 y Fx(=)20 b(1)p 2258 2608 V 410 w FL(x)2350 2593 y Fw(3)2404 2584 y Fx(=)f FE(\000)p Fx(0)p FL(:)p Fx(5)p 2966 2608 V 466 w(error)p 3357 2608 V 199 w(n)n(um)n(b)r(er)p 3743 2608 V 164 2611 3581 4 v 164 2628 V 162 2707 4 79 v 214 2683 a FL(")g Fx(=)h(0)p FL(:)p Fx(1)p 620 2707 V 241 w Fc(GcdR)n(oot)p 1032 2707 V 100 w Fx(0.89999999999)p 1645 2707 V 174 w(0.99999999999)p 2258 2707 V 173 w(-0.49999999999999)p 2966 2707 V 176 w(9.7e-10)p 3357 2707 V 3743 2707 V 162 2785 V 620 2785 V 683 2762 a Fc(PejR)n(oot)p 1032 2785 V 112 w Fx(0.9000000000000)p 1645 2785 V 104 w(0.9999999999999)p 2258 2785 V 103 w(-0.50000000000000)p 2966 2785 V 176 w(2.7e-13)p 3357 2785 V 370 w(.7)p 3743 2785 V 164 2789 3581 4 v 162 2868 4 79 v 214 2844 a FL(")f Fx(=)h(0)p FL(:)p Fx(01)p 620 2868 V 206 w Fc(GcdR)n(oot)p 1032 2868 V 100 w Fx(0.98999999)p 1645 2868 V 279 w(0.99999999)p 2258 2868 V 278 w(-0.50000000000000)p 2966 2868 V 176 w(3.2e-07)p 3357 2868 V 3743 2868 V 162 2946 V 620 2946 V 683 2923 a Fc(PejR)n(oot)p 1032 2946 V 112 w Fx(0.989999999999)p 1645 2946 V 139 w(1.000000000000)p 2258 2946 V 138 w(-0.49999999999999) p 2966 2946 V 176 w(1.0e-12)p 3357 2946 V 335 w(6.7)p 3743 2946 V 164 2950 3581 4 v 162 3029 4 79 v 214 3005 a FL(")f Fx(=)h(0)p FL(:)p Fx(001)p 620 3029 V 171 w Fc(GcdR)n(oot)p 1032 3029 V 100 w Fx(0.99900)p 1645 3029 V 384 w(1.00000)p 2258 3029 V 383 w(-0.49999999999999)p 2966 3029 V 176 w(1.9e-04)p 3357 3029 V 3743 3029 V 162 3108 V 620 3108 V 683 3084 a Fc(PejR)n(oot)p 1032 3108 V 112 w Fx(0.99899999999)p 1645 3108 V 174 w(1.00000000000)p 2258 3108 V 173 w(-0.500000000000000)p 2966 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b(Ross)33 b(Lipp)s(ert,)h(Hans)f(Stetter,)i(Joab)f(Winkler)g(and)f (anon)m(ymous)94 4727 y(referees)f(made)g(v)-5 b(aluable)32 b(suggestions)g(on)g(the)f(presen)m(tation.)45 b(One)31 b(of)h(the)f(referees)h(p)s(oin)m(ted)f(out)h(some)94 4839 y(imp)s(ortan)m(t)38 b(previous)f(w)m(ork)h(in)f([6)q(,)h(36)q(])f (Barry)h(Da)m(yton)h(found)d(an)i(error)f(in)g(the)h(early)g(v)m (ersion)g(of)g(the)94 4952 y(man)m(uscript.)94 5132 y FQ(References)136 5338 y FK([1])45 b FJ(D.)26 b(H.)h(Bailey)p FK(,)d Fa(A)h(Fortr)l(an-90)i(b)l(ase)l(d)f(multipr)l(e)l(cision)h (system)p FK(,)e(A)n(CM)e(T)-7 b(rans.)23 b(Math.)g(Soft)n(w)n(are,)g (21)g(\(1995\),)269 5437 y(pp.)28 b(379{387.)1909 5686 y FN(30)p eop %%Page: 31 33 31 32 bop 136 99 a FK([2])45 b FJ(D.)23 b(Bini)g(and)h(G.)e(Fiorentino) p FK(,)g Fa(Numeric)l(al)h(c)l(omputation)g(of)h(p)l(olynomial)h(r)l(o) l(ots)e(using)f(MPSolve)i({)g(version)269 198 y(2.0)p FK(.)38 b(man)n(uscript,)27 b(Soft)n(w)n(are)g(and)g(pap)r(er)g(a)n(v) -5 b(ailable)26 b(at)i Fp(ftp://ftp.dm.uni)o(pi)o(.it)o(/p)o(ub/)o(mp)o (so)o(lve)o(/)p FK(,)22 b(1999.)136 346 y([3])45 b FJ(L.)29 b(Br)n(ugnana)n(o)g(and)g(D.)g(Trigiante)p FK(,)e Fa(Polynomial)j(r)l (o)l(ots:)38 b(the)28 b(ultimate)g(answer?)p FK(,)f(Linear)d(Alg.)i (and)f(Its)269 445 y(Appl.,)j(225)f(\(1995\),)f(pp.)i(207{219.)136 593 y([4])45 b FJ(L.)29 b(Br)n(ugnano)p FK(,)c Fa(Numeric)l(al)k (implementation)g(of)f(a)h(new)e(algorithm)j(for)f(p)l(olynomials)h (with)e(multiple)h(r)l(o)l(ots)p FK(,)269 692 y(J.)e(Di\013erence)h (Eq.)f(and)h(Appl.,)g(1)f(\(1995\),)g(pp.)h(187{207.)136 840 y([5])45 b FJ(P.)24 b(Chin,)i(R.)f(M.)f(Corless,)i(and)e(G.)g(F.)h (Corless)p FK(,)e Fa(Optimization)i(str)l(ate)l(gies)f(for)g(the)h (appr)l(oximate)g(GCD)269 939 y(pr)l(oblem)p FK(,)k(in)f(ISSA)n(C)g (1998,)d(New)j(Y)-7 b(ork,)27 b(1998,)f(A)n(CM)i(Press,)e(pp.)i (228{235.)136 1087 y([6])45 b FJ(R.)32 b(M.)f(Corless,)g(P.)g(M.)g (Gianni,)g(B.)g(M.)g(Tra)n(ger,)h(and)g(S.)e(M.)i(W)-10 b(a)k(tt)p FK(,)28 b 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(o)l(ach)i(to)e(p)l(erturb)l(ation)g(the)l(ory)h(of)f(matric)l(es)g (and)h(and)f(matrix)g(p)l(encils.)h(p)l(art)f(II:)g(a)269 2967 y(str)l(ati\014c)l(ation-enhanc)l(e)l(d)f(stair)l(c)l(ase)h (algorithm)p FK(,)e(SIAM)f(J.)g(Matrix)f(Anal.)g(Appl.,)i(20)d (\(1999\),)h(pp.)h(667{699.)94 3114 y([13])45 b FJ(I.)31 b(Z.)f(Emiris,)h(A.)f(Galligo,)g(and)h(H.)f(Lombardi)p FK(,)e Fa(Certi\014e)l(d)i(appr)l(oximate)h(univariate)g(GCDs)p FK(,)d(J.)f(Pure)269 3214 y(Appl.)h(Algebra,)f(117/118)d(\(1997\),)j (pp.)h(229{251.)94 3361 y([14])45 b FJ(M.)30 b(R.)h(F)-10 b(armer)32 b(and)e(G.)g(Loizou)p FK(,)c Fa(A)n(n)j(algorithm)h(for)h (the)e(total,)h(or)g(p)l(artial,)h(factorization)g(of)f(a)g(p)l(olyno-) 269 3461 y(mial)p FK(,)f(Math.)f(Pro)r(c.)e(Cam)n(b.)h(Phil.)h(So)r (c.,)f(82)g(\(1977\),)f(pp.)i(427{437.)94 3608 y([15])p 269 3595 V 236 w(,)g Fa(L)l(o)l(c)l(ating)i(multiple)g(zer)l(os)g (inter)l(actively)p FK(,)f(Comp.)f(Math.)f(Appl.,)i(11)d(\(1985\),)h (pp.)h(595{603.)94 3756 y([16])45 b FJ(S.)32 b(F)n(or)-6 b(tune)p FK(,)29 b Fa(A)n(n)h(iter)l(ate)l(d)h(eigenvalue)h(algorithm)g (for)g(appr)l(oximating)g(r)l(o)l(ots)f(of)h(univariate)f(p)l (olynomials)p FK(,)269 3856 y(J.)c(Sym)n(b)r(olic)h(Comput.,)g(33)f (\(2002\),)f(pp.)i(627{646.)94 4003 y([17])45 b FJ(W.)27 b(Ga)n(utschi)p FK(,)d Fa(Questions)i(of)h(numeric)l(al)g(c)l(ondition) g(r)l(elate)l(d)g(to)g(p)l(olynomials)p FK(,)g(in)d(MAA)g(Studies)h(in) f(Math-)269 4103 y(ematics,)k(V)-7 b(ol.)28 b(24,)f(Studies)i(in)f (Numerical)g(Analysis,)f(G.)i(H.)f(Golub,)g(ed.,)h(USA,)g(1984,)d(The)i (Mathematical)269 4202 y(Asso)r(ciation)f(of)g(America,)g(pp.)h (140{177.)94 4350 y([18])45 b FJ(V.)33 b(Hribernig)i(and)e(H.)h(J.)f (Stetter)p FK(,)f Fa(Dete)l(ction)g(and)g(validation)j(of)e(clusters)e (of)i(p)l(olynomial)i(zer)l(os)p FK(,)c(J.)269 4449 y(Sym)n(b.)d (Comput.,)g(24)e(\(1997\),)h(pp.)h(667{681.)94 4597 y([19])45 b FJ(M.)30 b(Igarashi)g(and)g(T.)g(Ypma)p FK(,)d Fa(R)l(elationships)j (b)l(etwe)l(en)f(or)l(der)g(and)h(e\016ciency)g(of)g(a)f(class)g(of)h (metho)l(ds)g(for)269 4696 y(multiple)g(zer)l(os)g(of)h(p)l(olynomials) p FK(,)f(J.)d(Comput.)h(Appl.)g(Math.,)g(60)f(\(1995\),)f(pp.)i (101{113.)94 4844 y([20])45 b FJ(M.)32 b(A.)g(Jenkins)g(and)g(J.)g(F.)g (Tra)n(ub)p FK(,)e Fa(Principles)i(for)g(testing)e(p)l(olynomial)j(zer) l(o\014nding)e(pr)l(o)l(gr)l(ams)p FK(,)e(A)n(CM)269 4943 y(T)-7 b(rans.)27 b(Math.)h(Soft)n(w)n(are,)e(1)h(\(1975\),)f(pp.) i(26{34.)94 5091 y([21])45 b FJ(W.)39 b(Kahan)p FK(,)f Fa(Conserving)g(c)l(on\015uenc)l(e)f(curbs)g(il)t(l-c)l(ondition)p FK(.)63 b(T)-7 b(ec)n(hnical)35 b(Rep)r(ort)h(6,)h(Computer)f(Science,) 269 5190 y(Univ)n(ersit)n(y)27 b(of)g(California,)g(Berk)n(eley)-7 b(,)26 b(1972.)94 5338 y([22])45 b FJ(N.)23 b(K.)g(Karmarkar)g(and)g (Y.)f(N.)h(Lakshman)p FK(,)e Fa(On)h(appr)l(oximate)i(p)l(olynomial)h (gr)l(e)l(atest)e(c)l(ommon)g(divisors)p FK(,)269 5437 y(J.)k(Sym)n(b.)h(Comput.,)g(26)f(\(1998\),)f(pp.)i(653{666.)1909 5686 y FN(31)p eop %%Page: 32 34 32 33 bop 94 99 a FK([23])45 b FJ(P.)38 b(Kra)-7 b(v)g(anja)38 b(and)h(M.)f(V)-10 b(an)39 b(Barel)p FK(,)d Fa(Computing)h(Zer)l(os)g (of)g(A)n(nalytic)g(F)-6 b(unctions,)38 b(L)l(e)l(ctur)l(e)d(Notes)h (in)269 198 y(Mathematics,)c(1727)p FK(,)d(Springer-V)-7 b(erlag,)25 b(2000.)94 348 y([24])45 b FJ(R.)e(A.)f(Lipper)-6 b(t)44 b(and)f(A.)f(Edelman)p FK(,)f Fa(The)g(c)l(omputation)f(and)g (sensitivity)h(of)g(double)g(eigenvalues)p FK(,)h(in)269 447 y(Adv)-5 b(ances)27 b(in)g(computational)f(mathematics,)h(Lecture)g (Notes)g(in)g(Pure)f(and)h(Appl.)g(Math.)g(202,)f(New)h(Y)-7 b(ork,)269 547 y(1999,)26 b(Dekk)n(er,)h(pp.)h(353{393.)94 696 y([25])45 b FJ(T.)25 b(Miy)-6 b(ak)n(od)n(a)p FK(,)22 b Fa(Iter)l(ative)i(metho)l(ds)h(for)g(multiple)f(zer)l(os)g(of)h(a)f (p)l(olynomial)i(by)f(clustering)p FK(,)d(J.)f(Comput.)h(Appl.)269 796 y(Math.,)28 b(28)f(\(1989\),)f(pp.)i(315{326.)94 945 y([26])45 b FJ(V.)36 b(Y.)f(P)-7 b(an)p FK(,)33 b Fa(Numeric)l(al)h(c)l(omputation)h(of)g(a)f(p)l(olynomial)j(gc)l(d)d (and)h(extensions)p FK(.)50 b(Researc)n(h)31 b(Rep)r(ort)h(2996,)269 1045 y(,Institut)k(National)e(de)h(Rec)n(herc)n(he)f(en)h(Informatique) g(et)g(en)g(Automatique)g(\(INRIA\),)h(Sophia-An)n(tip)r(olis,)269 1145 y(F)-7 b(rance,)27 b(1996.)94 1294 y([27])p 269 1281 191 4 v 236 w(,)33 b Fa(Solving)h(p)l(olynomial)h(e)l(quations:)46 b(some)34 b(history)h(and)e(r)l(e)l(c)l(ent)g(pr)l(o)l(gr)l(ess)p FK(,)g(SIAM)f(Review,)g(39)f(\(1997\),)269 1394 y(pp.)d(187{220.)94 1543 y([28])45 b FJ(D.)35 b(R)n(upprecht)p FK(,)f Fa(A)n(n)e(algorithm) j(for)f(c)l(omputing)g(c)l(erti\014e)l(d)f(appr)l(oximate)i(GCD)f(of)g (n)f(univariate)i(p)l(olyno-)269 1643 y(mials)p FK(,)29 b(J.)e(Pure)g(and)g(Appl.)i(Alg.,)e(139)f(\(1999\),)h(pp.)h(255{284.)94 1792 y([29])45 b FJ(H.)27 b(J.)h(Stetter)p FK(,)e Fa(Condition)i (analysis)g(of)g(over)l(determine)l(d)g(algebr)l(aic)h(pr)l(oblems)p FK(,)d(in)f(Computer)f(Algebra)f(in)269 1892 y(Scien)n(ti\014c)28 b(Computing{CASC)f(2000,)f(e.)i(a.)f(V.G.)h(Ganzha,)f(ed.,)h(Springer,) f(2000,)e(pp.)j(345{365.)94 2041 y([30])45 b FJ(J.)27 b(A.)f(Stolan)p FK(,)e Fa(A)n(n)i(impr)l(ove)l(d)1326 2020 y(\024)1318 2041 y(Siljak's)i(algorithm)g(for)f(solving)h(p)l (olynomial)g(e)l(quations)f(c)l(onver)l(ges)f(quadr)l(at-)269 2141 y(ic)l(al)t(ly)32 b(to)d(multiple)i(zer)l(os)p FK(,)d(J.)f (Comput.)h(Appl.)g(Math.,)g(64)f(\(1995\),)f(pp.)i(247{268.)94 2290 y([31])45 b FJ(F.)d(Uhlig)p FK(,)e Fa(Gener)l(al)g(p)l(olynomial)i (r)l(o)l(ots)d(and)h(their)f(multiplicities)i(in)f Fo(O)r FK(\()p Fo(n)p FK(\))g Fa(memory)g(and)g Fo(O)r FK(\()p Fo(n)3523 2260 y Fn(2)3561 2290 y FK(\))f Fa(time)p FK(,)269 2390 y(Linear)27 b(and)g(Multilinear)h(Algebra,)e(46)h(\(1999\),)f(pp.) i(327{359.)94 2539 y([32])45 b FJ(S.)35 b(V)-10 b(an)35 b(Huffel)p FK(,)e Fa(Iter)l(ative)g(algorithms)i(for)f(c)l(omputing)f (the)h(singular)g(subsp)l(ac)l(e)f(of)h(a)g(matrix)f(asso)l(ciate)l(d) 269 2639 y(with)d(its)g(smal)t(lest)g(singular)h(values)p FK(,)d(Linear)f(Alg.)g(Appl.,)i(154-156)24 b(\(1991\),)i(pp.)i (675{709.)94 2788 y([33])45 b FJ(J.)29 b(H.)g(Wilkinson)p FK(,)d Fa(R)l(ounding)i(Err)l(ors)g(in)h(A)n(lgebr)l(aic)g(Pr)l(o)l(c)l (esses)p FK(,)e(Pren)n(tice-Hall,)e(Englew)n(o)r(o)r(d)g(Cli\013s,)h (N.J.,)269 2888 y(1963.)94 3037 y([34])45 b FJ(J.)25 b(R.)g(Winkler)p FK(,)d Fa(Condition)k(numb)l(ers)d(of)i(a)g(ne)l(arly) g(singular)g(simple)g(r)l(o)l(ot)f(of)h(a)g(p)l(olynomial)p FK(,)g(Appl.)d(Numer.)269 3137 y(Math.,)28 b(\(2001\),)e(pp.)i (275{285.)94 3287 y([35])45 b FJ(T.)29 b(J.)f(Ypma)p FK(,)e Fa(Finding)j(a)f(multiple)g(zer)l(o)g(by)g(tr)l(ansformations)h (and)f(Newton-like)g(metho)l(ds)p FK(,)f(SIAM)f(Review,)269 3386 y(25)h(\(1983\),)f(pp.)i(365{378.)94 3536 y([36])45 b FJ(D.)26 b(Y.)g(Y.)f(Yun)p FK(,)f Fa(On)g(squar)l(e-fr)l(e)l(e)i(de)l (c)l(omp)l(osition)h(algorithms)p FK(,)f(in)d(Pro)r(ceedings)e(of)i (1976)e(A)n(CM)i(Symp)r(osium)269 3635 y(of)e(Sym)n(b)r(olic)f(and)h (Algebraic)f(Computation)g(\(ISSA)n(C'76\),)i(R.)f(Janks,)g(ed.,)i(A)n (CM)e(Press,)f(Y)-7 b(orkto)n(wn)20 b(Heigh)n(ts,)269 3735 y(New)28 b(Y)-7 b(ork,)27 b(1976,)f(pp.)i(26{35.)94 3884 y([37])45 b FJ(Z.)29 b(Zeng)p FK(,)e Fa(Multr)l(o)l(ot)i({)g(a)g (matlab)g(p)l(ackage)i(c)l(omputing)e(p)l(olynomial)i(r)l(o)l(ots)d (and)h(multiplicities)p FK(.)37 b(man)n(uscript,)269 3984 y(can)27 b(b)r(e)h(accessed)f(at)g Fp(http://www.neiu.e)o(du)o(/)p Fm(\030)o Fp(zze)o(ng)o(/Pa)o(pe)o(rs)o(/zr)o(oo)o(tpa)o(k.)o(ps)p FK(,)21 b(2003.)94 4133 y([38])p 269 4120 V 236 w(,)g Fa(On)f(il)t(l-c)l(onditione)l(d)k(eigenvalues,)h(multiple)c(r)l(o)l (ots)h(of)g(p)l(olynomials,)k(and)c(their)g(ac)l(cur)l(ate)f(c)l (omputation)p FK(.)269 4233 y(MSRI)28 b(Preprin)n(t)e(No.)i(1998-048,)c (\(1998\))1623 4203 y Fn(2)1661 4233 y FK(.)p 94 5352 1488 4 v 198 5405 a Fw(2)233 5437 y Fz(http://www.neiu.edu/)p Fv(\030)p Fz(zzeng/Papers)q(/mul)q(tipl)q(e.ps)1909 5686 y FN(32)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF SHAR_EOF fi # end of overwriting check if test -f 'readme.txt' then echo shar: will not over-write existing file "'readme.txt'" else cat << "SHAR_EOF" > 'readme.txt' This Matlab package is developed by Zhonggang Zeng (zzeng@neiu.edu) dated Jan. 6, 2004. This package is released through ACM Transaction on Mathematical Software. The author is not responsible for any damage that may be related to using this package. 1. General information on the package There are three directories when the package is unzipped: documentation multroot testsuite There are two papers in "documentation" directory: [1]. "Computing multiple roots of inexact polynomials", Z. Zeng, Math. Comp., to appear. [2]. "MultRoot - A Matlab package for computing polynomial roots and multiplicities", ACM Trans. Math. Software, to appear. The paper [1] presents the overall theory and algorithm, while [2] describes the software package and the test suite in detail. The directory "multroot" contains the Matlab modules of the MultRoot package. The directory "testsuite" contains the Matlab modules that generate test polynomials. 2. Installation Unzip and save all the files, open Matlab, set proper path to the directory containing the Matlab modules, the package is now ready to run. 3. Simple applications To find roots and multiplicities of a polynomial p in Matlab format, simply call >> z = multroot(p) The first column of z lists distinct roots, the 2nd column shows the corresponding multiplicities. At the current version, the code works for polynomials with moderate root multiplicities (such as 30). Example: >> f = poly([ones(1,20),2*ones(1,20),3*ones(1,10),4*ones(1,5)]); >> z = multroot(f) THE STRUCTURE-PRESERVING CONDITION NUMBER: 73.6821 THE BACKWARD ERROR: 8.63e-016 THE ESTIMATED FORWARD ROOT ERROR: 1.27e-013 computed roots multiplicities 4.000000000000002 5 2.999999999999999 10 2.000000000000000 20 1.000000000000000 20 4. Sophisticated applications There are five main programs: 1. MultRoot: calculates roots and corresponding multiplicities, by calling the next two expert programs. 2. GcdRoot: calculates the multiplicity structure and initial estimates of the roots. In many cases, the estimates are accurate enough. The minimum input is the polynomial coefficient vector 3. PejRoot: from given multiplicity structure and initial root estimates, PejRoot calculate the root accurately. The minimum input is the polyn. coef. vector, initial iterate and multiplicity structure. 4. mroot: Same as multroot except that mroot returns a flag and terminates when a nontrivial multiplicity structure is not found. 5. spcond: calculates the structure-preserving condition number. Expert users should read the papers attached in the package. At least the paper [2] above. All modules contain "help" message that can be accessed in Matlab. You may try >> help multroot Zhonggang Zeng Associate Professor SHAR_EOF fi # end of overwriting check cd .. if test ! -d 'Matlab' then mkdir 'Matlab' fi cd 'Matlab' if test ! -d 'Drivers' then mkdir 'Drivers' fi cd 'Drivers' if test -f 'bt01.m' then echo shar: will not over-write existing file "'bt01.m'" else cat << "SHAR_EOF" > 'bt01.m' function [p,z] = bt01 % % Brugnano and Trigiante % p = poly([ones(1,6),-1,-1,i,i,i,-i,-i,-i,2]); z = [1,6; -1, 2; i, 3; -i, 3; 2, 1]; if norm(imag(z(:,1))) == 0 fprintf(' roots multiplicities\n'); fprintf('\n'); fprintf('%25.15f \t \t \t %3g \n', z'); else fprintf(' roots ') fprintf(' \t\t\t\t\t\t multiplicities\n'); fprintf('\n'); fprintf('%22.15f + %22.15f i \t \t %3g \n', ... [real(z(:,1)),imag(z(:,1)),z(:,2)]'); end; SHAR_EOF fi # end of overwriting check if test -f 'bt02.m' then echo shar: will not over-write existing file "'bt02.m'" else cat << "SHAR_EOF" > 'bt02.m' function [p,z] = bt02 % % Brugnano and Trigiante % p = poly([ones(1,10),-1,-1,i,-i,2]); z = [1,10; -1, 2; i, 1; -i, 1]; if norm(imag(z(:,1))) == 0 fprintf(' roots multiplicities\n'); fprintf('\n'); fprintf('%25.15f \t \t \t %3g \n', z'); else fprintf(' roots ') fprintf(' \t\t\t\t\t\t multiplicities\n'); fprintf('\n'); fprintf('%22.15f + %22.15f i \t \t %3g \n', ... [real(z(:,1)),imag(z(:,1)),z(:,2)]'); end; SHAR_EOF fi # end of overwriting check if test -f 'bt03.m' then echo shar: will not over-write existing file "'bt03.m'" else cat << "SHAR_EOF" > 'bt03.m' function [p,z] = bt03 % % Brugnano and Trigiante % p = poly([i*ones(1,5),-i*ones(1,5),0.5i*ones(1,4),... -0.5i*ones(1,4),0.75i,-0.75i]); z = [i,5; -i, 5; 0.5i, 4; -0.5i, 4; .75i, 1; -0.75i, 1]; if norm(imag(z(:,1))) == 0 fprintf(' roots multiplicities\n'); fprintf('\n'); fprintf('%25.15f \t \t \t %3g \n', z'); else fprintf(' roots ') fprintf(' \t\t\t\t\t\t multiplicities\n'); fprintf('\n'); fprintf('%22.15f + %22.15f i \t \t %3g \n', ... [real(z(:,1)),imag(z(:,1)),z(:,2)]'); end; SHAR_EOF fi # end of overwriting check if test -f 'bt04.m' then echo shar: will not over-write existing file "'bt04.m'" else cat << "SHAR_EOF" > 'bt04.m' function [p,z] = bt04 % % Brugnano and Trigiante % p = poly([1,1,1, -1*[1,1,1,1], (.5+i)*[1,1,1], (.5-i)*[1,1,1], ... .5*(1+i)*[1,1], .5*(1-i)*[1,1]]); z = [1,3; -1,4; .5+i,3; .5-i,3; .5*(1+i),2; .5*(1-i),2]; if norm(imag(z(:,1))) == 0 fprintf(' roots multiplicities\n'); fprintf('\n'); fprintf('%25.15f \t \t \t %3g \n', z'); else fprintf(' roots ') fprintf(' \t\t\t\t\t\t multiplicities\n'); fprintf('\n'); fprintf('%22.15f + %22.15f i \t \t %3g \n', ... [real(z(:,1)),imag(z(:,1)),z(:,2)]'); end; SHAR_EOF fi # end of overwriting check if test -f 'farloi01.m' then echo shar: will not over-write existing file "'farloi01.m'" else cat << "SHAR_EOF" > 'farloi01.m' function [p,z] = farloi01 % % Farmer-Loizou % p = conv([1,1,2],[1,1,3]); p = conv(p,p); p = conv(p,p); z = [-0.50000000000000 + 1.65831239517770i -0.50000000000000 - 1.65831239517770i -0.50000000000000 + 1.32287565553230i -0.50000000000000 - 1.32287565553230i]; z = [z,4*ones(4,1)]; if norm(imag(z(:,1))) == 0 fprintf(' roots multiplicities\n'); fprintf('\n'); fprintf('%25.15f \t \t \t %3g \n', z'); else fprintf(' roots ') fprintf(' \t\t\t\t\t\t multiplicities\n'); fprintf('\n'); fprintf('%22.15f + %22.15f i \t \t %3g \n', ... [real(z(:,1)),imag(z(:,1)),z(:,2)]'); end; SHAR_EOF fi # end of overwriting check if test -f 'fib.m' then echo shar: will not over-write existing file "'fib.m'" else cat << "SHAR_EOF" > 'fib.m' function p = fib(n) % % test polynomial suggested by Goedecker % p = [-1,ones(1,n)]; SHAR_EOF fi # end of overwriting check if test -f 'fib05.m' then echo shar: will not over-write existing file "'fib05.m'" else cat << "SHAR_EOF" > 'fib05.m' function [p,z] = fib05 % % test polynomial suggested by Goedecker % n = 5; p = [-1,ones(1,n)]; z = [-.6783507129699967-.4585361872731445*i, 1; ... -.6783507129699967+.4585361872731445*i, 1; ... .19537659464725405-.8488536405462456*i, 1; ... .19537659464725405+.8488536405462456*i, 1; ... 1.9659482366454853, 1]; if norm(imag(z(:,1))) == 0 fprintf(' roots multiplicities\n'); fprintf('\n'); fprintf('%25.15f \t \t \t %3g \n', z'); else fprintf(' roots ') fprintf(' \t\t\t\t\t\t multiplicities\n'); fprintf('\n'); fprintf('%22.15f + %22.15f i \t \t %3g \n', ... [real(z(:,1)),imag(z(:,1)),z(:,2)]'); end; SHAR_EOF fi # end of overwriting check if test -f 'fib10.m' then echo shar: will not over-write existing file "'fib10.m'" else cat << "SHAR_EOF" > 'fib10.m' function [p,z] = fib10 % % test polynomial suggested by Goedecker % n = 10; p = [-1,ones(1,n)]; z = [... -.8990310941957183, 1; ... -.7399333995096503-.5167965365015562*i, 1; ... -.7399333995096503+.5167965365015562*i, 1; ... -.3130259587265545-.8583547482505247*i, 1; ... -.3130259587265545+.8583547482505247*i, 1; ... .2462366020696891-.9013280059509374*i, 1; ... .2462366020696891+.9013280059509374*i, 1; ... .7567289869093243-.6038877405652167*i, 1; ... .7567289869093243+.6038877405652167*i, 1; ... 1.999018632710101, 1; ]; if norm(imag(z(:,1))) == 0 fprintf(' roots multiplicities\n'); fprintf('\n'); fprintf('%25.15f \t \t \t %3g \n', z'); else fprintf(' roots ') fprintf(' \t\t\t\t\t\t multiplicities\n'); fprintf('\n'); fprintf('%22.15f + %22.15f i \t \t %3g \n', ... [real(z(:,1)),imag(z(:,1)),z(:,2)]'); end; SHAR_EOF fi # end of overwriting check if test -f 'fib100.m' then echo shar: will not over-write existing file "'fib100.m'" else cat << "SHAR_EOF" > 'fib100.m' function [p,z] = fib100 % % test polynomial suggested by Goedecker % n = 100; p = [-1,ones(1,n)]; z = [-.9891099736226092, ... -.9871753104718534-.6190245872746059e-1*i, ... -.9871753104718534+.6190245872746059e-1*i, ... -.9813788056446001-.1235640063982463*i, ... -.9813788056446001+.1235640063982463*i, ... -.9717428840956687-.1847446602565987*i, ... -.9717428840956687+.1847446602565987*i, ... -.9583048244621342-.2452062905225305*i, ... -.9583048244621342+.2452062905225305*i, ... -.9411166150274026-.3047135378280153*i, ... -.9411166150274026+.3047135378280153*i, ... -.9202447528605484-.3630347198287647*i, ... -.9202447528605484+.3630347198287647*i, ... -.8957699869173070-.4199427234459649*i, ... -.8957699869173070+.4199427234459649*i, ... -.8677870061092108-.4752158792455410*i, ... -.8677870061092108+.4752158792455410*i, ... -.8364040735640849-.5286388145285358*i, ... -.8364040735640849+.5286388145285358*i, ... -.8017426085137990-.5800032817847029*i, ... -.8017426085137990+.5800032817847029*i, ... -.7639367174532022-.6291089592520593*i, ... -.7639367174532022+.6291089592520593*i, ... -.7231326764169818-.6757642204274601*i, ... -.7231326764169818+.6757642204274601*i, ... -.6794883664183336-.7197868694867672*i, ... -.6794883664183336+.7197868694867672*i, ... -.6331726642844369-.7610048396973263*i, ... -.6331726642844369+.7610048396973263*i, ... -.5843647913085710-.7992568520396451*i, ... -.5843647913085710+.7992568520396451*i, ... -.5332536223172150-.8343930313987525*i, ... -.5332536223172150+.8343930313987525*i, ... -.4800369579227939-.8662754778380540*i, ... -.4800369579227939+.8662754778380540*i, ... -.4249207628992477-.8947787906289414*i, ... -.4249207628992477+.8947787906289414*i, ... -.3681183737790589-.9197905428773492*i, ... -.3681183737790589+.9197905428773492*i, ... -.3098496789278928-.9412117047633791*i, ... -.3098496789278928+.9412117047633791*i, ... -.2503402745082667-.9589570135917125*i, ... -.2503402745082667+.9589570135917125*i, ... -.1898205998989927-.9729552890388070*i, ... -.1898205998989927+.9729552890388070*i, ... -.1285250562957008-.9831496921783526*i, ... -.1285250562957008+.9831496921783526*i, ... -.6669111238377778e-1-.9894979270705116*i, ... -.6669111238377778e-1+.9894979270705116*i, ... -.4558401154059704e-2-.9919723839157337*i, ... -.4558401154059704e-2+.9919723839157337*i, ... .5763218786929409e-1-.9905602230050737*i, ... .5763218786929409e-1+.9905602230050737*i, ... .1196394164912155-.9852633989536015*i, ... .1196394164912155+.9852633989536015*i, ... .1812226010328281-.9760986249939283*i, ... .1812226010328281+.9760986249939283*i, ... .2421425057566853-.9630972774521361*i, ... .2421425057566853+.9630972774521361*i, ... .3021622210555556-.9463052409577104*i, ... .3021622210555556+.9463052409577104*i, ... .3610480204611693-.9257826954965046*i, ... .3610480204611693+.9257826954965046*i, ... .4185701899698170-.9016038471668328*i, ... .4185701899698170+.9016038471668328*i, ... .4745038225339292-.8738566055392182*i, ... .4745038225339292+.8738566055392182*i, ... .5286295698615102-.8426422119868863*i, ... .5286295698615102+.8426422119868863*i, ... .5807343429795305-.8080748254361926*i, ... .5807343429795305+.8080748254361926*i, ... .6306119525307706-.7702810749337576*i, ... .6306119525307706+.7702810749337576*i, ... .6780636798327496-.7293995925445015*i, ... .6780636798327496+.7293995925445015*i, ... .7228987709578736-.6855805456977909*i, ... .7228987709578736+.6855805456977909*i, ... .7649348495515867-.6389851953915993*i, ... .7649348495515867+.6389851953915993*i, ... .8039982514449404-.5897855154613771*i, ... .8039982514449404+.5897855154613771*i, ... .8399242976881979-.5381639173058062*i, ... .8399242976881979+.5381639173058062*i, ... .8725575452576575-.4843131310830981*i, ... .8725575452576575+.4843131310830981*i, ... .9017520886609776-.4284362924194566*i, ... .9017520886609776+.4284362924194566*i, ... .9273720302113955-.3707472629394290*i, ... .9273720302113955+.3707472629394290*i, ... .9492922833419482-.3114711598054638*i, ... .9492922833419482+.3114711598054638*i, ... .9673998997347004-.2508449722447981*i, ... .9673998997347004+.2508449722447981*i, ... .9815960786437149-.1891180053506043*i, ... .9815960786437149+.1891180053506043*i, ... .9917988816037989-.1265517518780266*i, ... .9917988816037989+.1265517518780266*i, ... .9979464229745987-.6341873580445710e-1*i, ... .9979464229745987+.6341873580445710e-1*i, ... 2.000000000000000]; z = [z',ones(n,1)]; if norm(imag(z(:,1))) == 0 fprintf(' roots multiplicities\n'); fprintf('\n'); fprintf('%25.15f \t \t \t %3g \n', z'); else fprintf(' roots ') fprintf(' \t\t\t\t\t\t multiplicities\n'); fprintf('\n'); fprintf('%22.15f + %22.15f i \t \t %3g \n', ... [real(z(:,1)),imag(z(:,1)),z(:,2)]'); end; SHAR_EOF fi # end of overwriting check if test -f 'fib15.m' then echo shar: will not over-write existing file "'fib15.m'" else cat << "SHAR_EOF" > 'fib15.m' function [p,z] = fib15 % % test polynomial suggested by Goedecker % n = 15; p = [-1,ones(1,n)]; z = [-.9115849521765970-.1896359497733640*i, ... -.9115849521765970+.1896359497733640*i, ... -.7620785395636261-.5388639564255547*i, ... -.7620785395636261+.5388639564255547*i, ... -.4858038058618722-.8023902123527991*i, ... -.4858038058618722+.8023902123527991*i, ... -.1248130033927766-.9371029958793902*i, ... -.1248130033927766+.9371029958793902*i, ... .2657642744875577-.9183694775911604*i, ... .2657642744875577+.9183694775911604*i, ... .6253568882929779-.7424114494678755*i, ... .6253568882929779+.7424114494678755*i, ... .8931744004970845-.4240640341420445*i, ... .8931744004970845+.4240640341420445*i, ... 1.999969475434503]; z = [z',ones(n,1)]; if norm(imag(z(:,1))) == 0 fprintf(' roots multiplicities\n'); fprintf('\n'); fprintf('%25.15f \t \t \t %3g \n', z'); else fprintf(' roots ') fprintf(' \t\t\t\t\t\t multiplicities\n'); fprintf('\n'); fprintf('%22.15f + %22.15f i \t \t %3g \n', ... [real(z(:,1)),imag(z(:,1)),z(:,2)]'); end; SHAR_EOF fi # end of overwriting check if test -f 'fib150.m' then echo shar: will not over-write existing file "'fib150.m'" else cat << "SHAR_EOF" > 'fib150.m' function [p,z] = fib150 % % test polynomial suggested by Goedecker % n = 150; p = [-1,ones(1,n)]; z = [-.9927187559253492, ... -.9918530882118113-.4147913261323188e-1*i, ... -.9918530882118113+.4147913261323188e-1*i, ... -.9892575836821723-.8288617318453414e-1*i, ... -.9892575836821723+.8288617318453414e-1*i, ... -.9849367355771876-.1241491541448754*i, ... -.9849367355771876+.1241491541448754*i, ... -.9788980239954073-.1651963566541725*i, ... -.9788980239954073+.1651963566541725*i, ... -.9711519029549269-.2059564344240104*i, ... -.9711519029549269+.2059564344240104*i, ... -.9617117823114389-.2463585368912788*i, ... -.9617117823114389+.2463585368912788*i, ... -.9505940045641273-.2863324315280673*i, ... -.9505940045641273+.2863324315280673*i, ... -.9378178165898683-.3258086250746164*i, ... -.9378178165898683+.3258086250746164*i, ... -.9234053363550660-.3647184834839322*i, ... -.9234053363550660+.3647184834839322*i, ... -.9073815146632301-.4029943503688435*i, ... -.9073815146632301+.4029943503688435*i, ... -.8897740920050947-.4405696637447960*i, ... -.8897740920050947+.4405696637447960*i, ... -.8706135505866639-.4773790708645437*i, ... -.8706135505866639+.4773790708645437*i, ... -.8499330616190360-.5133585409440989*i, ... -.8499330616190360+.5133585409440989*i, ... -.8277684279621956-.5484454755828406*i, ... -.8277684279621956+.5484454755828406*i, ... -.8041580222231623-.5825788166845422*i, ... -.8041580222231623+.5825788166845422*i, ... -.7791427204169229-.6156991516902623*i, ... -.7791427204169229+.6156991516902623*i, ... -.7527658313064592-.6477488159385299*i, ... -.7527658313064592+.6477488159385299*i, ... -.7250730215458899-.6786719919730538*i, ... -.7250730215458899+.6786719919730538*i, ... -.6961122367582722-.7084148056232615*i, ... -.6961122367582722+.7084148056232615*i, ... -.6659336186869521-.7369254186883471*i, ... -.6659336186869521+.7369254186883471*i, ... -.6345894185664953-.7641541180611353*i, ... -.6345894185664953+.7641541180611353*i, ... -.6021339068661845-.7900534011339715*i, ... -.6021339068661845+.7900534011339715*i, ... -.5686232795658151-.8145780573349891*i, ... -.5686232795658151+.8145780573349891*i, ... -.5341155611300741-.8376852456494883*i, ... -.5341155611300741+.8376852456494883*i, ... -.4986705043541354-.8593345679877696*i, ... -.4986705043541354+.8593345679877696*i, ... -.4623494872592649-.8794881382675888*i, ... -.4623494872592649+.8794881382675888*i, ... -.4252154072232028-.8981106470864277*i, ... -.4252154072232028+.8981106470864277*i, ... -.3873325725358873-.9151694218659986*i, ... -.3873325725358873+.9151694218659986*i, ... -.3487665915767304-.9306344823588107*i, ... -.3487665915767304+.9306344823588107*i, ... -.3095842598151608-.9444785914142191*i, ... -.3095842598151608+.9444785914142191*i, ... -.2698534448415497-.9566773009091463*i, ... -.2698534448415497+.9566773009091463*i, ... -.2296429696409571-.9672089927566195*i, ... -.2296429696409571+.9672089927566195*i, ... -.1890224943274280-.9760549149134032*i, ... -.1890224943274280+.9760549149134032*i, ... -.1480623965618780-.9831992123163535*i, ... -.1480623965618780+.9831992123163535*i, ... -.1068336508819937-.9886289526856903*i, ... -.1068336508819937+.9886289526856903*i, ... -.6540770717811572e-1-.9923341471422300*i, ... -.6540770717811572e-1+.9923341471422300*i, ... -.2385636855484892e-1-.9943077655947946*i, ... -.2385636855484892e-1+.9943077655947946*i, ... .1774833117573815e-1-.9945457468636005*i, ... .1774833117573815e-1+.9945457468636005*i, ... .5933425011689494e-1-.9930470035155510*i, ... .5933425011689494e-1+.9930470035155510*i, ... .1008292607663695-.9898134213981652*i, ... .1008292607663695+.9898134213981652*i, ... .1421613720845280-.9848498538705906*i, ... .1421613720845280+.9848498538705906*i, ... .1832588508751938-.9781641107430547*i, ... .1832588508751938+.9781641107430547*i, ... .2240503421931883-.9697669419505880*i, ... .2240503421931883+.9697669419505880*i, ... .2644649884815679-.9596720160034037*i, ... .2644649884815679+.9596720160034037*i, ... .3044325471292468-.9478958932755882*i, ... .3044325471292468+.9478958932755882*i, ... .3438835061260140-.9344579942165983*i, ... .3438835061260140+.9344579942165983*i, ... .3827491974769262-.9193805625975399*i, ... .3827491974769262+.9193805625975399*i, ... .4209619080219071-.9026886239377113*i, ... .4209619080219071+.9026886239377113*i, ... .4584549872896030-.8844099392981356*i, ... .4584549872896030+.8844099392981356*i, ... .4951629519980681-.8645749546799259*i, ... .4951629519980681+.8645749546799259*i, ... .5310215868002686-.8432167463288874*i, ... .5310215868002686+.8432167463288874*i, ... .5659680408622439-.8203709623267845*i, ... .5659680408622439+.8203709623267845*i, ... .5999409198599586-.7960757609476053*i, ... .5999409198599586+.7960757609476053*i, ... .6328803729932399-.7703717463775087*i, ... .6328803729932399+.7703717463775087*i, ... .6647281746501417-.7433019025433064*i, ... .6647281746501417+.7433019025433064*i, ... .6954278004244162-.7149115259686257*i, ... .6954278004244162+.7149115259686257*i, ... .7249244973086152-.6852481587793065*i, ... .7249244973086152+.6852481587793065*i, ... .7531653480767721-.6543615232056414*i, ... .7531653480767721+.6543615232056414*i, ... .7800993301600754-.6223034591666788*i, ... .7800993301600754+.6223034591666788*i, ... .8056773697373049-.5891278667468168*i, ... .8056773697373049+.5891278667468168*i, ... .8298523923415625-.5548906555452699*i, ... .8298523923415625+.5548906555452699*i, ... .8525793720535097-.5196497029280600*i, ... .8525793720535097+.5196497029280600*i, ... .8738153823196024-.4834648230425889*i, ... .8738153823196024+.4834648230425889*i, ... .8935196525762829-.4463977479358605*i, ... .8935196525762829+.4463977479358605*i, ... .9116536360890624-.4085121210914591*i, ... .9116536360890624+.4085121210914591*i, ... .9281810955442824-.3698735020064003*i, ... .9281810955442824+.3698735020064003*i, ... .9430682136510916-.3305493779521499*i, ... .9430682136510916+.3305493779521499*i, ... .9562837358786358-.2906091758178016*i, ... .9562837358786358+.2906091758178016*i, ... .9677991509344817-.2501242631678211*i, ... .9677991509344817+.2501242631678211*i, ... .9775889111839516-.2091679239550551*i, ... .9775889111839516+.2091679239550551*i, ... .9856306896705420-.1678152916898196*i, ... .9856306896705420+.1678152916898196*i, ... .9919056630215224-.1261432225318404*i, ... .9919056630215224+.1261432225318404*i, ... .9963988013915535-.8423009394365190e-1*i, ... .9963988013915535+.8423009394365190e-1*i, ... .9990991395939178-.4215552183469491e-1*i, ... .9990991395939178+.4215552183469491e-1*i, ... 2.000000000000000]; z = [z',ones(n,1)]; if norm(imag(z(:,1))) == 0 fprintf(' roots multiplicities\n'); fprintf('\n'); fprintf('%25.15f \t \t \t %3g \n', z'); else fprintf(' roots ') fprintf(' \t\t\t\t\t\t multiplicities\n'); fprintf('\n'); fprintf('%22.15f + %22.15f i \t \t %3g \n', ... [real(z(:,1)),imag(z(:,1)),z(:,2)]'); end; SHAR_EOF fi # end of overwriting check if test -f 'fib20.m' then echo shar: will not over-write existing file "'fib20.m'" else cat << "SHAR_EOF" > 'fib20.m' function [p,z] = fib20 % % test polynomial suggested by Goedecker % n = 20; p = [-1,ones(1,n)]; z = [-.9473885474109854, ... -.9029230998844866-.2884421994216345*i, ... -.9029230998844866+.2884421994216345*i, ... -.7734751441751908-.5504669225440185*i, ... -.7734751441751908+.5504669225440185*i, ... -.5705411148090332-.7619498890625472*i, ... -.5705411148090332+.7619498890625472*i, ... -.3121525651549263-.9031354642584495*i, ... -.3121525651549263+.9031354642584495*i, ... -.2129876087300105e-1-.9602891496489760*i, ... -.2129876087300105e-1+.9602891496489760*i, ... .2760438985065760-.9267417870492216*i, ... .2760438985065760+.9267417870492216*i, ... .5529956567245064-.8031600819414702*i, ... .5529956567245064+.8031600819414702*i, ... .7834063973613306-.5969423289081712*i, ... .7834063973613306+.5969423289081712*i, ... .9416394828514235-.3213776783384489*i, ... .9416394828514235+.3213776783384489*i, ... 1.999999046316589]; z = [z',ones(n,1)]; if norm(imag(z(:,1))) == 0 fprintf(' roots multiplicities\n'); fprintf('\n'); fprintf('%25.15f \t \t \t %3g \n', z'); else fprintf(' roots ') fprintf(' \t\t\t\t\t\t multiplicities\n'); fprintf('\n'); fprintf('%22.15f + %22.15f i \t \t %3g \n', ... [real(z(:,1)),imag(z(:,1)),z(:,2)]'); end; SHAR_EOF fi # end of overwriting check if test -f 'fib30.m' then echo shar: will not over-write existing file "'fib30.m'" else cat << "SHAR_EOF" > 'fib30.m' function [p,z] = fib30 % % test polynomial suggested by Goedecker % n = 30; p = [-1,ones(1,n)]; z = [-.9644254204388659, ... -.9439446810207413-.1984294062681023*i, ... -.9439446810207413+.1984294062681023*i, ... -.8833406981383851-.3885717512211873*i, ... -.8833406981383851+.3885717512211873*i, ... -.7850939960122262-.5624744046156477*i, ... -.7850939960122262+.5624744046156477*i, ... -.6532262984053521-.7128393846266873*i, ... -.6532262984053521+.7128393846266873*i, ... -.4931368697016981-.8333156264808382*i, ... -.4931368697016981+.8333156264808382*i, ... -.3113834310599106-.9187504784857605*i, ... -.3113834310599106+.9187504784857605*i, ... -.1154175582352257-.9653888511274815*i, ... -.1154175582352257+.9653888511274815*i, ... .8671260421202705e-1-.9710099019085114*i, ... .8671260421202705e-1+.9710099019085114*i, ... .2866757308842761-.9349927196966469*i, ... .2866757308842761+.9349927196966469*i, ... .4761639390711853-.8583045796885356*i, ... .4761639390711853+.8583045796885356*i, ... .6471568601471165-.7434108410885516*i, ... .6471568601471165+.7434108410885516*i, ... .7920805342279742-.5941279591518842*i, ... .7920805342279742+.5941279591518842*i, ... .9037637390940364-.4155315939426088*i, ... .9037637390940364+.4155315939426088*i, ... .9752028356220179-.2142999451221434*i, ... .9752028356220179+.2142999451221434*i, ... 1.999999999068677]; z = [z',ones(n,1)]; if norm(imag(z(:,1))) == 0 fprintf(' roots multiplicities\n'); fprintf('\n'); fprintf('%25.15f \t \t \t %3g \n', z'); else fprintf(' roots ') fprintf(' \t\t\t\t\t\t multiplicities\n'); fprintf('\n'); fprintf('%22.15f + %22.15f i \t \t %3g \n', ... [real(z(:,1)),imag(z(:,1)),z(:,2)]'); end; SHAR_EOF fi # end of overwriting check if test -f 'fib50.m' then echo shar: will not over-write existing file "'fib50.m'" else cat << "SHAR_EOF" > 'fib50.m' function [p,z] = fib50 % % test polynomial suggested by Goedecker % n = 50; p = [-1,ones(1,n)]; z = [-.9784087187483260, ... -.9708270453807137-.1218355210630181*i, ... -.9708270453807137+.1218355210630181*i, ... -.9481969430968524-.2418020920979234*i, ... -.9481969430968524+.2418020920979234*i, ... -.9108614275411458-.3580588620158148*i, ... -.9108614275411458+.3580588620158148*i, ... -.8593864256832723-.4688207429141271*i, ... -.8593864256832723+.4688207429141271*i, ... -.7945522211328693-.5723852109383657*i, ... -.7945522211328693+.5723852109383657*i, ... -.7173416644328307-.6671578269195402*i, ... -.7173416644328307+.6671578269195402*i, ... -.6289253327402240-.7516760773713197*i, ... -.6289253327402240+.7516760773713197*i, ... -.5306438727753174-.8246311589916703*i, ... -.5306438727753174+.8246311589916703*i, ... -.4239878085206529-.8848873569443235*i, ... -.4239878085206529+.8848873569443235*i, ... -.3105751412168900-.9314986982161571*i, ... -.3105751412168900+.9314986982161571*i, ... -.1921271147696794-.9637225955421171*i, ... -.1921271147696794+.9637225955421171*i, ... -.7044256696033078e-1-.9810302342100871*i, ... -.7044256696033078e-1+.9810302342100871*i, ... .5262866000569139e-1-.9831134936221445*i, ... .5262866000569139e-1+.9831134936221445*i, ... .1752127097133740-.9698882397909325*i, ... .1752127097133740+.9698882397909325*i, ... .2954384705313263-.9414938807153827*i, ... .2954384705313263+.9414938807153827*i, ... .4114633335560060-.8982891618708299*i, ... .4114633335560060+.8982891618708299*i, ... .5214969674899803-.8408443381786473*i, ... .5214969674899803+.8408443381786473*i, ... .6238218132212410-.7699301944269942*i, ... .6238218132212410+.7699301944269942*i, ... .7168085997048416-.6865051209865677*i, ... .7168085997048416+.6865051209865677*i, ... .7989249239308063-.5917030159705396*i, ... .7989249239308063+.5917030159705396*i, ... .8687357356619744-.4868277941747678*i, ... .8687357356619744+.4868277941747678*i, ... .9248989976696934-.3733647458567231*i, ... .9248989976696934+.3733647458567231*i, ... .9661725191057798-.2530209270696782*i, ... .9661725191057798+.2530209270696782*i, ... .9914691930342274-.1277916638512058*i, ... .9914691930342274+.1277916638512058*i, ... 1.999999999999999]; z = [z',ones(n,1)]; if norm(imag(z(:,1))) == 0 fprintf(' roots multiplicities\n'); fprintf('\n'); fprintf('%25.15f \t \t \t %3g \n', z'); else fprintf(' roots ') fprintf(' \t\t\t\t\t\t multiplicities\n'); fprintf('\n'); fprintf('%22.15f + %22.15f i \t \t %3g \n', ... [real(z(:,1)),imag(z(:,1)),z(:,2)]'); end; SHAR_EOF fi # end of overwriting check if test -f 'fibsq04.m' then echo shar: will not over-write existing file "'fibsq04.m'" else cat << "SHAR_EOF" > 'fibsq04.m' function [p,z] = fibsq04 % % test polynomial suggested by Goedecker % n = 4; p = [-1,ones(1,n)]; p = conv(p,p); z = [-.7748041132154339, ... -.7637893113374573e-1-.8147036471703865*i, ... -.7637893113374573e-1+.8147036471703865*i, ... 1.927561975482925]; z = [z',2*ones(n,1)]; if norm(imag(z(:,1))) == 0 fprintf(' roots multiplicities\n'); fprintf('\n'); fprintf('%25.15f \t \t \t %3g \n', z'); else fprintf(' roots ') fprintf(' \t\t\t\t\t\t multiplicities\n'); fprintf('\n'); fprintf('%22.15f + %22.15f i \t \t %3g \n', ... [real(z(:,1)),imag(z(:,1)),z(:,2)]'); end; SHAR_EOF fi # end of overwriting check if test -f 'fibsq08.m' then echo shar: will not over-write existing file "'fibsq08.m'" else cat << "SHAR_EOF" > 'fibsq08.m' function [p,z] = fibsq08 % % test polynomial suggested by Goedecker % n = 8; p = [-1,ones(1,n)]; p = conv(p,p); z = [-.8762862300182460, ... -.6416053894725764-.6063952206057722*i, ... -.6416053894725764+.6063952206057722*i, ... -.4694032461065018e-1-.9030234661229178*i, ... -.4694032461065018e-1+.9030234661229178*i, ... .6286732392246423-.7084725692273577*i, ... .6286732392246423+.7084725692273577*i, ... 1.996031179735415]; z = [z',2*ones(n,1)]; if norm(imag(z(:,1))) == 0 fprintf(' roots multiplicities\n'); fprintf('\n'); fprintf('%25.15f \t \t \t %3g \n', z'); else fprintf(' roots ') fprintf(' \t\t\t\t\t\t multiplicities\n'); fprintf('\n'); fprintf('%22.15f + %22.15f i \t \t %3g \n', ... [real(z(:,1)),imag(z(:,1)),z(:,2)]'); end; SHAR_EOF fi # end of overwriting check if test -f 'fibsq16.m' then echo shar: will not over-write existing file "'fibsq16.m'" else cat << "SHAR_EOF" > 'fibsq16.m' function [p,z] = fibsq16 % % test polynomial suggested by Goedecker % square of Fibocacci polynomial % n = 16; p = [-1,ones(1,n)]; p = conv(p,p); z = [-.9349216524104182, ... -.8673161854070643-.3515310211756676*i, ... -.8673161854070643+.3515310211756676*i, ... -.6736211580446324-.6537256870201036*i, ... -.6736211580446324+.6537256870201036*i, ... -.3799784643132753-.8637046142557905*i, ... -.3799784643132753+.8637046142557905*i, ... -.2606735312578153e-1-.9505412093420680*i, ... -.2606735312578153e-1+.9505412093420680*i, ... .3400699024566086-.8988910323302336*i, ... .3400699024566086+.8988910323302336*i, ... .6677317831101087-.7098987824372997*i, ... .6677317831101087+.7098987824372997*i, ... .9066499318552733-.3989926258132711*i, ... .9066499318552733+.3989926258132711*i, ... 1.999984739347944]; z = [z',2*ones(n,1)]; if norm(imag(z(:,1))) == 0 fprintf(' roots multiplicities\n'); fprintf('\n'); fprintf('%25.15f \t \t \t %3g \n', z'); else fprintf(' roots ') fprintf(' \t\t\t\t\t\t multiplicities\n'); fprintf('\n'); fprintf('%22.15f + %22.15f i \t \t %3g \n', ... [real(z(:,1)),imag(z(:,1)),z(:,2)]'); end; SHAR_EOF fi # end of overwriting check if test -f 'fibsq24.m' then echo shar: will not over-write existing file "'fibsq24.m'" else cat << "SHAR_EOF" > 'fibsq24.m' function [p,z] = fibsq24 % % test polynomial suggested by Goedecker % square of Fibocacci polynomial % n = 24; p = [-1,ones(1,n)]; p = conv(p,p); z = [-.9558467190743565, ... -.9244083656460181-.2442575811931259*i, ... -.9244083656460181+.2442575811931259*i, ... -.8320677914606713-.4727784060949767*i, ... -.8320677914606713+.4727784060949767*i, ... -.6846250033032301-.6707962641470166*i, ... -.6846250033032301+.6707962641470166*i, ... -.4913433576989534-.8254218906657225*i, ... -.4913433576989534+.8254218906657225*i, ... -.2643730603203030-.9264240880183555*i, ... -.2643730603203030+.9264240880183555*i, ... -.1800084671534713e-1-.9668298465915332*i, ... -.1800084671534713e-1+.9668298465915332*i, ... .2322158170552186-.9432943625623267*i, ... .2322158170552186+.9432943625623267*i, ... .4703449187206489-.8561989307649443*i, ... .4703449187206489+.8561989307649443*i, ... .6808619003199168-.7094507040001145*i, ... .6808619003199168+.7094507040001145*i, ... .8490194865023282-.5100588510854889*i, ... .8490194865023282+.5100588510854889*i, ... .9602996918859324-.2681678908797330*i, ... .9602996918859324+.2681678908797330*i, ... 1.999999940395313]; z = [z',2*ones(n,1)]; if norm(imag(z(:,1))) == 0 fprintf(' roots multiplicities\n'); fprintf('\n'); fprintf('%25.15f \t \t \t %3g \n', z'); else fprintf(' roots ') fprintf(' \t\t\t\t\t\t multiplicities\n'); fprintf('\n'); fprintf('%22.15f + %22.15f i \t \t %3g \n', ... [real(z(:,1)),imag(z(:,1)),z(:,2)]'); end; SHAR_EOF fi # end of overwriting check if test -f 'fibsq32.m' then echo shar: will not over-write existing file "'fibsq32.m'" else cat << "SHAR_EOF" > 'fibsq32.m' function [p,z] = fibsq32 % % test polynomial suggested by Goedecker % square of Fibocacci polynomial % n = 32; p = [-1,ones(1,n)]; p = conv(p,p); z = [-.9665892387165332, ... -.9485092065867169-.1867222182129376*i, ... -.9485092065867169+.1867222182129376*i, ... -.8949223987510440-.3665688458356083*i, ... -.8949223987510440+.3665688458356083*i, ... -.8077651569138829-.5329094943253877*i, ... -.8077651569138829+.5329094943253877*i, ... -.6901871862102008-.6795950061744879*i, ... -.6901871862102008+.6795950061744879*i, ... -.5464383280299697-.8011754158225671*i, ... -.5464383280299697+.8011754158225671*i, ... -.3817161237254617-.8930914814237271*i, ... -.3817161237254617+.8930914814237271*i, ... -.2019801534661303-.9518321713734013*i, ... -.2019801534661303+.9518321713734013*i, ... -.1374081948052525e-1-.9750513807247308*i, ... -.1374081948052525e-1+.9750513807247308*i, ... .1761678077924798-.9616381291177512*i, ... .1761678077924798+.9616381291177512*i, ... .3608186423377573-.9117356125589683*i, ... .3608186423377573+.9117356125589683*i, ... .5334081767292508-.8267064135873833*i, ... .5334081767292508+.8267064135873833*i, ... .6874310990889808-.7090471025782057*i, ... .6874310990889808+.7090471025782057*i, ... .8167660514846921-.5622778345619681*i, ... .8167660514846921+.5622778345619681*i, ... .9156129648696007-.3909102172601782*i, ... .9156129648696007+.3909102172601782*i, ... .9783492503358520-.2007808902007197*i, ... .9783492503358520+.2007808902007197*i, ... 1.999999999767169]; z = [z',2*ones(n,1)]; if norm(imag(z(:,1))) == 0 fprintf(' roots multiplicities\n'); fprintf('\n'); fprintf('%25.15f \t \t \t %3g \n', z'); else fprintf(' roots ') fprintf(' \t\t\t\t\t\t multiplicities\n'); fprintf('\n'); fprintf('%22.15f + %22.15f i \t \t %3g \n', ... [real(z(:,1)),imag(z(:,1)),z(:,2)]'); end; SHAR_EOF fi # end of overwriting check if test -f 'fibsq48.m' then echo shar: will not over-write existing file "'fibsq48.m'" else cat << "SHAR_EOF" > 'fibsq48.m' function [p,z] = fibsq48 % % test polynomial suggested by Goedecker % square of Fibocacci polynomial % n = 48; p = [-1,ones(1,n)]; p = conv(p,p); z = [-.9775253106342052, ... -.9693128478789493-.1267362311985826*i, ... -.9693128478789493+.1267362311985826*i, ... -.9448102919885515-.2513655659338360*i, ... -.9448102919885515+.2513655659338360*i, ... -.9044199317254038-.3718154066011398*i, ... -.9044199317254038+.3718154066011398*i, ... -.8488049242636358-.4860811780173362*i, ... -.8488049242636358+.4860811780173362*i, ... -.7788784384948876-.5922589089475263*i, ... -.7788784384948876+.5922589089475263*i, ... -.6957887129626554-.6885761217415814*i, ... -.6957887129626554+.6885761217415814*i, ... -.6009002821775153-.7734205049875106*i, ... -.6009002821775153+.7734205049875106*i, ... -.4957716929578976-.8453658760449938*i, ... -.4957716929578976+.8453658760449938*i, ... -.3821300978164688-.9031949785779086*i, ... -.3821300978164688+.9031949785779086*i, ... -.2618431760355581-.9459187037026260*i, ... -.2618431760355581+.9459187037026260*i, ... -.1368888970077769-.9727913710515751*i, ... -.1368888970077769+.9727913710515751*i, ... -.9323708922186879e-2-.9833217572983870*i, ... -.9323708922186879e-2+.9833217572983870*i, ... .1187501830426266-.9772796154416013*i, ... .1187501830426266+.9772796154416013*i, ... .2452176769219828-.9546974936920340*i, ... .2452176769219828+.9546974936920340*i, ... .3679828414167529-.9158677545985760*i, ... .3679828414167529+.9158677545985760*i, ... .4849990112751703-.8613348568446792*i, ... .4849990112751703+.8613348568446792*i, ... .5942953455083113-.7918833021200295*i, ... .5942953455083113+.7918833021200295*i, ... .6939977395882224-.7085224187149837*i, ... .6939977395882224+.7085224187149837*i, ... .7823414572835585-.6124708814099550*i, ... .7823414572835585+.6124708814099550*i, ... .8576732672059729-.5051474483453174*i, ... .8576732672059729+.5051474483453174*i, ... .9184452635324498-.3881803987843682*i, ... .9184452635324498+.3881803987843682*i, ... .9632174723875938-.2634528222232032*i, ... .9632174723875938+.2634528222232032*i, ... .9907153993859500-.1331861055266858*i, ... .9907153993859500+.1331861055266858*i, ... 1.999999999999996]; z = [z',2*ones(n,1)]; if norm(imag(z(:,1))) == 0 fprintf(' roots multiplicities\n'); fprintf('\n'); fprintf('%25.15f \t \t \t %3g \n', z'); else fprintf(' roots ') fprintf(' \t\t\t\t\t\t multiplicities\n'); fprintf('\n'); fprintf('%22.15f + %22.15f i \t \t %3g \n', ... [real(z(:,1)),imag(z(:,1)),z(:,2)]'); end; SHAR_EOF fi # end of overwriting check if test -f 'fl.m' then echo shar: will not over-write existing file "'fl.m'" else cat << "SHAR_EOF" > 'fl.m' function p = fl(k) % % generalization of Farmer-Loizou example: % % (x-1)^4k * (x-2)^3k * (x-3)^2K * (X-4)^k % p = poly([ones(1,4*k),2*ones(1,3*k),3*ones(1,2*k),4*ones(1,k)]); SHAR_EOF fi # end of overwriting check if test -f 'fl01.m' then echo shar: will not over-write existing file "'fl01.m'" else cat << "SHAR_EOF" > 'fl01.m' function [p,z] = fl01() % % generalization of Farmer-Loizou example: % % (x-1)^4k * (x-2)^3k * (x-3)^2K * (X-4)^k % k = 1; p = poly([ones(1,4*k),2*ones(1,3*k),3*ones(1,2*k),4*ones(1,k)]); z = [4,k;3,2*k;2,3*k;1,4*k]; if norm(imag(z(:,1))) == 0 fprintf(' roots multiplicities\n'); fprintf('\n'); fprintf('%25.15f \t \t \t %3g \n', z'); else fprintf(' roots ') fprintf(' \t\t\t\t\t\t multiplicities\n'); fprintf('\n'); fprintf('%22.15f + %22.15f i \t \t %3g \n', ... [real(z(:,1)),imag(z(:,1)),z(:,2)]'); end; SHAR_EOF fi # end of overwriting check if test -f 'fl02.m' then echo shar: will not over-write existing file "'fl02.m'" else cat << "SHAR_EOF" > 'fl02.m' function [p,z] = fl02() % % generalization of Farmer-Loizou example: % % (x-1)^4k * (x-2)^3k * (x-3)^2K * (X-4)^k % k = 2; p = poly([ones(1,4*k),2*ones(1,3*k),3*ones(1,2*k),4*ones(1,k)]); z = [4,k;3,2*k;2,3*k;1,4*k]; if norm(imag(z(:,1))) == 0 fprintf(' roots multiplicities\n'); fprintf('\n'); fprintf('%25.15f \t \t \t %3g \n', z'); else fprintf(' roots ') fprintf(' \t\t\t\t\t\t multiplicities\n'); fprintf('\n'); fprintf('%22.15f + %22.15f i \t \t %3g \n', ... [real(z(:,1)),imag(z(:,1)),z(:,2)]'); end; SHAR_EOF fi # end of overwriting check if test -f 'fl03.m' then echo shar: will not over-write existing file "'fl03.m'" else cat << "SHAR_EOF" > 'fl03.m' function [p,z] = fl03() % % generalization of Farmer-Loizou example: % % (x-1)^4k * (x-2)^3k * (x-3)^2K * (X-4)^k % k = 3; p = poly([ones(1,4*k),2*ones(1,3*k),3*ones(1,2*k),4*ones(1,k)]); z = [4,k;3,2*k;2,3*k;1,4*k]; if norm(imag(z(:,1))) == 0 fprintf(' roots multiplicities\n'); fprintf('\n'); fprintf('%25.15f \t \t \t %3g \n', z'); else fprintf(' roots ') fprintf(' \t\t\t\t\t\t multiplicities\n'); fprintf('\n'); fprintf('%22.15f + %22.15f i \t \t %3g \n', ... [real(z(:,1)),imag(z(:,1)),z(:,2)]'); end; SHAR_EOF fi # end of overwriting check if test -f 'fl04.m' then echo shar: will not over-write existing file "'fl04.m'" else cat << "SHAR_EOF" > 'fl04.m' function [p,z] = fl04() % % generalization of Farmer-Loizou example: % % (x-1)^4k * (x-2)^3k * (x-3)^2K * (X-4)^k % k = 4; p = poly([ones(1,4*k),2*ones(1,3*k),3*ones(1,2*k),4*ones(1,k)]); z = [4,k;3,2*k;2,3*k;1,4*k]; if norm(imag(z(:,1))) == 0 fprintf(' roots multiplicities\n'); fprintf('\n'); fprintf('%25.15f \t \t \t %3g \n', z'); else fprintf(' roots ') fprintf(' \t\t\t\t\t\t multiplicities\n'); fprintf('\n'); fprintf('%22.15f + %22.15f i \t \t %3g \n', ... [real(z(:,1)),imag(z(:,1)),z(:,2)]'); end; SHAR_EOF fi # end of overwriting check if test -f 'fl05.m' then echo shar: will not over-write existing file "'fl05.m'" else cat << "SHAR_EOF" > 'fl05.m' function [p,z] = fl05() % % generalization of Farmer-Loizou example: % % (x-1)^4k * (x-2)^3k * (x-3)^2K * (X-4)^k % k = 5; p = poly([ones(1,4*k),2*ones(1,3*k),3*ones(1,2*k),4*ones(1,k)]); z = [4,k;3,2*k;2,3*k;1,4*k]; if norm(imag(z(:,1))) == 0 fprintf(' roots multiplicities\n'); fprintf('\n'); fprintf('%25.15f \t \t \t %3g \n', z'); else fprintf(' roots ') fprintf(' \t\t\t\t\t\t multiplicities\n'); fprintf('\n'); fprintf('%22.15f + %22.15f i \t \t %3g \n', ... [real(z(:,1)),imag(z(:,1)),z(:,2)]'); end; SHAR_EOF fi # end of overwriting check if test -f 'fl06.m' then echo shar: will not over-write existing file "'fl06.m'" else cat << "SHAR_EOF" > 'fl06.m' function [p,z] = fl06() % % generalization of Farmer-Loizou example: % % (x-1)^4k * (x-2)^3k * (x-3)^2K * (X-4)^k % k = 6; p = poly([ones(1,4*k),2*ones(1,3*k),3*ones(1,2*k),4*ones(1,k)]); z = [4,k;3,2*k;2,3*k;1,4*k]; if norm(imag(z(:,1))) == 0 fprintf(' roots multiplicities\n'); fprintf('\n'); fprintf('%25.15f \t \t \t %3g \n', z'); else fprintf(' roots ') fprintf(' \t\t\t\t\t\t multiplicities\n'); fprintf('\n'); fprintf('%22.15f + %22.15f i \t \t %3g \n', ... [real(z(:,1)),imag(z(:,1)),z(:,2)]'); end; SHAR_EOF fi # end of overwriting check if test -f 'fl07.m' then echo shar: will not over-write existing file "'fl07.m'" else cat << "SHAR_EOF" > 'fl07.m' function [p,z] = fl07() % % generalization of Farmer-Loizou example: % % (x-1)^4k * (x-2)^3k * (x-3)^2K * (X-4)^k % k = 7; p = poly([ones(1,4*k),2*ones(1,3*k),3*ones(1,2*k),4*ones(1,k)]); z = [4,k;3,2*k;2,3*k;1,4*k]; if norm(imag(z(:,1))) == 0 fprintf(' roots multiplicities\n'); fprintf('\n'); fprintf('%25.15f \t \t \t %3g \n', z'); else fprintf(' roots ') fprintf(' \t\t\t\t\t\t multiplicities\n'); fprintf('\n'); fprintf('%22.15f + %22.15f i \t \t %3g \n', ... [real(z(:,1)),imag(z(:,1)),z(:,2)]'); end; SHAR_EOF fi # end of overwriting check if test -f 'henrici.m' then echo shar: will not over-write existing file "'henrici.m'" else cat << "SHAR_EOF" > 'henrici.m' function [p,z] = henrici % % M. Petkovic testing polynomials, page 123 % p = poly([-4.1,-3.8,-2.05,-1.85,1.95,2.15,3.9,4.05]); z = [[-4.1,-3.8,-2.05,-1.85,1.95,2.15,3.9,4.05]',ones(8,1)]; if norm(imag(z(:,1))) == 0 fprintf(' roots multiplicities\n'); fprintf('\n'); fprintf('%25.15f \t \t \t %3g \n', z'); else fprintf(' roots ') fprintf(' \t\t\t\t\t\t multiplicities\n'); fprintf('\n'); fprintf('%22.15f + %22.15f i \t \t %3g \n', ... [real(z(:,1)),imag(z(:,1)),z(:,2)]'); end; SHAR_EOF fi # end of overwriting check if test -f 'igyp00.m' then echo shar: will not over-write existing file "'igyp00.m'" else cat << "SHAR_EOF" > 'igyp00.m' function [p,z] = igyp00 % % generalization of Igarash and Ypma % p = poly([2.35,2.37,2.39]); z = [2.35,1;2.37,1;2.39,1]; if norm(imag(z(:,1))) == 0 fprintf(' roots multiplicities\n'); fprintf('\n'); fprintf('%25.15f \t \t \t %3g \n', z'); else fprintf(' roots ') fprintf(' \t\t\t\t\t\t multiplicities\n'); fprintf('\n'); fprintf('%22.15f + %22.15f i \t \t %3g \n', ... [real(z(:,1)),imag(z(:,1)),z(:,2)]'); end; SHAR_EOF fi # end of overwriting check if test -f 'igyp01.m' then echo shar: will not over-write existing file "'igyp01.m'" else cat << "SHAR_EOF" > 'igyp01.m' function [p,z] = igyp01 % % generalization of Igarash and Ypma % p = poly([2.35*[1,1,1],2.56]); z = [2.35,3;2.56,1]; if norm(imag(z(:,1))) == 0 fprintf(' roots multiplicities\n'); fprintf('\n'); fprintf('%25.15f \t \t \t %3g \n', z'); else fprintf(' roots ') fprintf(' \t\t\t\t\t\t multiplicities\n'); fprintf('\n'); fprintf('%22.15f + %22.15f i \t \t %3g \n', ... [real(z(:,1)),imag(z(:,1)),z(:,2)]'); end; SHAR_EOF fi # end of overwriting check if test -f 'igyp02a.m' then echo shar: will not over-write existing file "'igyp02a.m'" else cat << "SHAR_EOF" > 'igyp02a.m' function [p,z] = igyp02a % % generalization of Igarash and Ypma % m = 8; p = poly([10*(1+i)*ones(1,m),-1*ones(1,10-m)]); z = [10*(1+i),m; -1, 10-m]; if norm(imag(z(:,1))) == 0 fprintf(' roots multiplicities\n'); fprintf('\n'); fprintf('%25.15f \t \t \t %3g \n', z'); else fprintf(' roots ') fprintf(' \t\t\t\t\t\t multiplicities\n'); fprintf('\n'); fprintf('%22.15f + %22.15f i \t \t %3g \n', ... [real(z(:,1)),imag(z(:,1)),z(:,2)]'); end; SHAR_EOF fi # end of overwriting check if test -f 'igyp02b.m' then echo shar: will not over-write existing file "'igyp02b.m'" else cat << "SHAR_EOF" > 'igyp02b.m' function [p,z] = igyp02b % % generalization of Igarash and Ypma % m = 7; p = poly([10*(1+i)*ones(1,m),-1*ones(1,10-m)]); z = [10*(1+i),m; -1, 10-m]; if norm(imag(z(:,1))) == 0 fprintf(' roots multiplicities\n'); fprintf('\n'); fprintf('%25.15f \t \t \t %3g \n', z'); else fprintf(' roots ') fprintf(' \t\t\t\t\t\t multiplicities\n'); fprintf('\n'); fprintf('%22.15f + %22.15f i \t \t %3g \n', ... [real(z(:,1)),imag(z(:,1)),z(:,2)]'); end; SHAR_EOF fi # end of overwriting check if test -f 'igyp02c.m' then echo shar: will not over-write existing file "'igyp02c.m'" else cat << "SHAR_EOF" > 'igyp02c.m' function [p,z] = igyp02c % % generalization of Igarash and Ypma % m = 3; p = poly([10*(1+i)*ones(1,m),-1*ones(1,10-m)]); z = [10*(1+i),m; -1, 10-m]; if norm(imag(z(:,1))) == 0 fprintf(' roots multiplicities\n'); fprintf('\n'); fprintf('%25.15f \t \t \t %3g \n', z'); else fprintf(' roots ') fprintf(' \t\t\t\t\t\t multiplicities\n'); fprintf('\n'); fprintf('%22.15f + %22.15f i \t \t %3g \n', ... [real(z(:,1)),imag(z(:,1)),z(:,2)]'); end; SHAR_EOF fi # end of overwriting check if test -f 'igyp03.m' then echo shar: will not over-write existing file "'igyp03.m'" else cat << "SHAR_EOF" > 'igyp03.m' function [p,z] = igyp03 % % generalization of Igarash and Ypma % p = poly([10*(1+i)*[1,1,1],1,i,2,2i,3,4i,5]); z = [10*(1+i),3; 1,1; i,1; 2,1; 2i,1; 3,1; 4i,1; 5,1]; if norm(imag(z(:,1))) == 0 fprintf(' roots multiplicities\n'); fprintf('\n'); fprintf('%25.15f \t \t \t %3g \n', z'); else fprintf(' roots ') fprintf(' \t\t\t\t\t\t multiplicities\n'); fprintf('\n'); fprintf('%22.15f + %22.15f i \t \t %3g \n', ... [real(z(:,1)),imag(z(:,1)),z(:,2)]'); end; SHAR_EOF fi # end of overwriting check if test -f 'iliev00.m' then echo shar: will not over-write existing file "'iliev00.m'" else cat << "SHAR_EOF" > 'iliev00.m' function [p,z] = iliev00 % % generalization of Iliev example: % % (x-1) (x-2)^2 (x-3)^3 % k = 0; p = poly([1,2,2,3,3,3]); z = [1,1; 2,2; 3,3]; if norm(imag(z(:,1))) == 0 fprintf(' roots multiplicities\n'); fprintf('\n'); fprintf('%25.15f \t \t \t %3g \n', z'); else fprintf(' roots ') fprintf(' \t\t\t\t\t\t multiplicities\n'); fprintf('\n'); fprintf('%22.15f + %22.15f i \t \t %3g \n', ... [real(z(:,1)),imag(z(:,1)),z(:,2)]'); end; SHAR_EOF fi # end of overwriting check if test -f 'iliev01.m' then echo shar: will not over-write existing file "'iliev01.m'" else cat << "SHAR_EOF" > 'iliev01.m' function [p,z] = iliev01 % % generalization of Iliev example: % % (x-1) (x-2)^2 (x-3)^3 % k = 2; l = 1*k; m = 2*k; n = 4*k; p = poly([ones(1,l),2*ones(1,m),3*ones(1,n)]); z = [1,l; 2,m; 3,n]; if norm(imag(z(:,1))) == 0 fprintf(' roots multiplicities\n'); fprintf('\n'); fprintf('%25.15f \t \t \t %3g \n', z'); else fprintf(' roots ') fprintf(' \t\t\t\t\t\t multiplicities\n'); fprintf('\n'); fprintf('%22.15f + %22.15f i \t \t %3g \n', ... [real(z(:,1)),imag(z(:,1)),z(:,2)]'); end; SHAR_EOF fi # end of overwriting check if test -f 'iliev02.m' then echo shar: will not over-write existing file "'iliev02.m'" else cat << "SHAR_EOF" > 'iliev02.m' function [p,z] = iliev02 % % generalization of Iliev example: % % (x-1) (x-2)^2 (x-3)^3 % k = 4; l = 1*k; m = 2*k; n = 4*k; p = poly([ones(1,l),2*ones(1,m),3*ones(1,n)]); z = [1,l; 2,m; 3,n]; if norm(imag(z(:,1))) == 0 fprintf(' roots multiplicities\n'); fprintf('\n'); fprintf('%25.15f \t \t \t %3g \n', z'); else fprintf(' roots ') fprintf(' \t\t\t\t\t\t multiplicities\n'); fprintf('\n'); fprintf('%22.15f + %22.15f i \t \t %3g \n', ... [real(z(:,1)),imag(z(:,1)),z(:,2)]'); end; SHAR_EOF fi # end of overwriting check if test -f 'iliev03.m' then echo shar: will not over-write existing file "'iliev03.m'" else cat << "SHAR_EOF" > 'iliev03.m' function [p,z] = iliev03 % % generalization of Iliev example: % % (x-1) (x-2)^2 (x-3)^3 % k = 8; l = 1*k; m = 2*k; n = 3*k; p = poly([ones(1,l),2*ones(1,m),3*ones(1,n)]); z = [1,l; 2,m; 3,n]; if norm(imag(z(:,1))) == 0 fprintf(' roots multiplicities\n'); fprintf('\n'); fprintf('%25.15f \t \t \t %3g \n', z'); else fprintf(' roots ') fprintf(' \t\t\t\t\t\t multiplicities\n'); fprintf('\n'); fprintf('%22.15f + %22.15f i \t \t %3g \n', ... [real(z(:,1)),imag(z(:,1)),z(:,2)]'); end; SHAR_EOF fi # end of overwriting check if test -f 'inex01.m' then echo shar: will not over-write existing file "'inex01.m'" else cat << "SHAR_EOF" > 'inex01.m' function [p,z] = inex01 % % test polynomial suggested by Goedecker % p = poly([(10/11)*[1,1,1,1,1],(20/11)*[1,1,1],... (30/11)*[1,1]]); p = round(10^7*p)/10^7; z = [10/11, 5; 20/11, 3; 30/11, 2]; z = z(3:-1:1,:); fprintf('\n'); fprintf(' Coefficients are rounded up at the 7-th digits after\n'); fprintf(' decimal point. Originally, \n'); if norm(imag(z(:,1))) == 0 fprintf(' roots multiplicities\n'); fprintf('\n'); fprintf('%25.15f \t \t \t %3g \n', z'); else fprintf(' roots ') fprintf(' \t\t\t\t\t\t multiplicities\n'); fprintf('\n'); fprintf('%22.15f + %22.15f i \t \t %3g \n', ... [real(z(:,1)),imag(z(:,1)),z(:,2)]'); end; SHAR_EOF fi # end of overwriting check if test -f 'inex02.m' then echo shar: will not over-write existing file "'inex02.m'" else cat << "SHAR_EOF" > 'inex02.m' function [p,z] = inex02 % % test polynomial suggested by Goedecker % p = poly([(10/11)*[1,1,1,1,1],(20/11)*[1,1,1],... (30/11)*[1,1]]); p = round(10^6*p)/10^6; z = [10/11, 5; 20/11, 3; 30/11, 2]; z = z(3:-1:1,:); fprintf('\n'); fprintf(' Coefficients are rounded up at the 6-th digits after\n'); fprintf(' decimal point. Originally, \n'); if norm(imag(z(:,1))) == 0 fprintf(' roots multiplicities\n'); fprintf('\n'); fprintf('%25.15f \t \t \t %3g \n', z'); else fprintf(' roots ') fprintf(' \t\t\t\t\t\t multiplicities\n'); fprintf('\n'); fprintf('%22.15f + %22.15f i \t \t %3g \n', ... [real(z(:,1)),imag(z(:,1)),z(:,2)]'); end; SHAR_EOF fi # end of overwriting check if test -f 'inex03.m' then echo shar: will not over-write existing file "'inex03.m'" else cat << "SHAR_EOF" > 'inex03.m' function [p,z] = inex03 % % test polynomial suggested by Goedecker % p = poly([(10/11)*[1,1,1,1,1],(20/11)*[1,1,1],... (30/11)*[1,1]]); p = round(10^5*p)/10^5; z = [10/11, 5; 20/11, 3; 30/11, 2]; z = z(3:-1:1,:); fprintf('\n'); fprintf(' Coefficients are rounded up at the 5-th digits after\n'); fprintf(' decimal point. Originally, \n'); if norm(imag(z(:,1))) == 0 fprintf(' roots multiplicities\n'); fprintf('\n'); fprintf('%25.15f \t \t \t %3g \n', z'); else fprintf(' roots ') fprintf(' \t\t\t\t\t\t multiplicities\n'); fprintf('\n'); fprintf('%22.15f + %22.15f i \t \t %3g \n', ... [real(z(:,1)),imag(z(:,1)),z(:,2)]'); end; SHAR_EOF fi # end of overwriting check if test -f 'inex04.m' then echo shar: will not over-write existing file "'inex04.m'" else cat << "SHAR_EOF" > 'inex04.m' function [p,z] = inex04 % % test polynomial suggested by Goedecker % p = poly([(10/11)*[1,1,1,1,1],(20/11)*[1,1,1],... (30/11)*[1,1]]); p = round(10^4*p)/10^4; z = [10/11, 5; 20/11, 3; 30/11, 2]; z = z(3:-1:1,:); fprintf('\n'); fprintf(' Coefficients are rounded up at the 4-th digits after\n'); fprintf(' decimal point. Originally, \n'); if norm(imag(z(:,1))) == 0 fprintf(' roots multiplicities\n'); fprintf('\n'); fprintf('%25.15f \t \t \t %3g \n', z'); else fprintf(' roots ') fprintf(' \t\t\t\t\t\t multiplicities\n'); fprintf('\n'); fprintf('%22.15f + %22.15f i \t \t %3g \n', ... [real(z(:,1)),imag(z(:,1)),z(:,2)]'); end; SHAR_EOF fi # end of overwriting check if test -f 'jt01a.m' then echo shar: will not over-write existing file "'jt01a.m'" else cat << "SHAR_EOF" > 'jt01a.m' function [p,z] = jt01a % % test polynomial suggested by Jenkins and Traub % a = 10^(10); p = [1,-1,-a^2,a^2]; z = [a,1;-a,1;1,1]; if norm(imag(z(:,1))) == 0 fprintf(' roots multiplicities\n'); fprintf('\n'); fprintf('%25.5f \t \t \t %3g \n', z'); else fprintf(' roots ') fprintf(' \t\t\t\t\t\t multiplicities\n'); fprintf('\n'); fprintf('%22.15f + %22.15f i \t \t %3g \n', ... [real(z(:,1)),imag(z(:,1)),z(:,2)]'); end; SHAR_EOF fi # end of overwriting check if test -f 'jt01b.m' then echo shar: will not over-write existing file "'jt01b.m'" else cat << "SHAR_EOF" > 'jt01b.m' function [p,z] = jt01b % % test polynomial suggested by Jenkins and Traub % a = 10^(-10) p = [1,-1,-a^2,a^2]; z = [a,1;-a,1;1,1]; if norm(imag(z(:,1))) == 0 fprintf(' roots multiplicities\n'); fprintf('\n'); fprintf('%25.15f \t \t \t %3g \n', z'); else fprintf(' roots ') fprintf(' \t\t\t\t\t\t multiplicities\n'); fprintf('\n'); fprintf('%22.15f + %22.15f i \t \t %3g \n', ... [real(z(:,1)),imag(z(:,1)),z(:,2)]'); end; SHAR_EOF fi # end of overwriting check if test -f 'jt02.m' then echo shar: will not over-write existing file "'jt02.m'" else cat << "SHAR_EOF" > 'jt02.m' function [p,z] = jt02 % % test polynomial suggested by Jenkins and Traub % p = poly([1:17]); z = [1:17]; z = [z',ones(17,1)]; fprintf('\n'); fprintf(' Ill-conditioned polynomial \n'); if norm(imag(z(:,1))) == 0 fprintf(' roots multiplicities\n'); fprintf('\n'); fprintf('%25.15f \t \t \t %3g \n', z'); else fprintf(' roots ') fprintf(' \t\t\t\t\t\t multiplicities\n'); fprintf('\n'); fprintf('%22.15f + %22.15f i \t \t %3g \n', ... [real(z(:,1)),imag(z(:,1)),z(:,2)]'); end; SHAR_EOF fi # end of overwriting check if test -f 'jt03.m' then echo shar: will not over-write existing file "'jt03.m'" else cat << "SHAR_EOF" > 'jt03.m' function [p,z] = jt03 % % test polynomial suggested by Jenkins and Traub % z = 1./10.^[1:8]; p = poly(z); z = [z',ones(8,1)]; fprintf('\n'); if norm(imag(z(:,1))) == 0 fprintf(' roots multiplicities\n'); fprintf('\n'); fprintf('%25.15f \t \t \t %3g \n', z'); else fprintf(' roots ') fprintf(' \t\t\t\t\t\t multiplicities\n'); fprintf('\n'); fprintf('%22.15f + %22.15f i \t \t %3g \n', ... [real(z(:,1)),imag(z(:,1)),z(:,2)]'); end; SHAR_EOF fi # end of overwriting check if test -f 'jt04.m' then echo shar: will not over-write existing file "'jt04.m'" else cat << "SHAR_EOF" > 'jt04.m' function [p,z] = jt04 % % test polynomial suggested by Jenkins and Traub % p = poly([0.1,0.1,0.1,0.5,0.6,0.7]); z = [0.5, 1; 0.6, 1; 0.7, 1; 0.1, 3]; fprintf('\n'); if norm(imag(z(:,1))) == 0 fprintf(' roots multiplicities\n'); fprintf('\n'); fprintf('%25.15f \t \t \t %3g \n', z'); else fprintf(' roots ') fprintf(' \t\t\t\t\t\t multiplicities\n'); fprintf('\n'); fprintf('%22.15f + %22.15f i \t \t %3g \n', ... [real(z(:,1)),imag(z(:,1)),z(:,2)]'); end; SHAR_EOF fi # end of overwriting check if test -f 'jt05.m' then echo shar: will not over-write existing file "'jt05.m'" else cat << "SHAR_EOF" > 'jt05.m' function [p,z] = jt05 % % test polynomial suggested by Jenkins and Traub % p = poly([0.1*ones(1,4),0.2*ones(1,3),0.3,0.3,0.4]); z = [0.4, 1; 0.3, 2; 0.2, 3; 0.1, 4]; fprintf('\n'); if norm(imag(z(:,1))) == 0 fprintf(' roots multiplicities\n'); fprintf('\n'); fprintf('%25.15f \t \t \t %3g \n', z'); else fprintf(' roots ') fprintf(' \t\t\t\t\t\t multiplicities\n'); fprintf('\n'); fprintf('%22.15f + %22.15f i \t \t %3g \n', ... [real(z(:,1)),imag(z(:,1)),z(:,2)]'); end; SHAR_EOF fi # end of overwriting check if test -f 'jt06.m' then echo shar: will not over-write existing file "'jt06.m'" else cat << "SHAR_EOF" > 'jt06.m' function [p,z] = jt06 % % test polynomial suggested by Jenkins and Traub % p = poly([.1,1.001,.998,1.00002,.99999]); z = [.1,1.001,.998,1.00002,.99999]; z = [z',ones(5,1)]; fprintf('\n'); fprintf(' Ill-conditioned polynomial \n'); if norm(imag(z(:,1))) == 0 fprintf(' roots multiplicities\n'); fprintf('\n'); fprintf('%25.15f \t \t \t %3g \n', z'); else fprintf(' roots ') fprintf(' \t\t\t\t\t\t multiplicities\n'); fprintf('\n'); fprintf('%22.15f + %22.15f i \t \t %3g \n', ... [real(z(:,1)),imag(z(:,1)),z(:,2)]'); end; SHAR_EOF fi # end of overwriting check if test -f 'jt07a.m' then echo shar: will not over-write existing file "'jt07a.m'" else cat << "SHAR_EOF" > 'jt07a.m' function [p,z] = jt07a % % test polynomial suggested by Jenkins and Traub % a = 10^(-10); p = poly([.001, .01, .1, .1+a*i, .1-a*i, 1, -10]); z = [.001, .01, .1, .1+a*i, .1-a*i, 1, -10]; z = [z',ones(7,1)]; y = [-10, 1; 1, 1; 0.01, 1; 0.001, 1; 0.1, 3]; fprintf('\n'); fprintf(' Ill-conditioned polynomial \n'); fprintf(' It is constructed using\n') fprintf(' roots ') fprintf(' \t\t\t\t\t\t multiplicities\n'); fprintf('\n'); fprintf('%22.15f + %22.15f i \t \t %3g \n', ... [real(z(:,1)),imag(z(:,1)),z(:,2)]'); fprintf('\n'); fprintf(' However, the polynomial is closer to having \n'); fprintf(' roots multiplicities\n'); fprintf('\n'); fprintf('%25.15f \t \t \t %3g \n', y'); SHAR_EOF fi # end of overwriting check if test -f 'jt07b.m' then echo shar: will not over-write existing file "'jt07b.m'" else cat << "SHAR_EOF" > 'jt07b.m' function [p,z] = jt07b % % test polynomial suggested by Jenkins and Traub % a = 10^(-9); p = poly([.001, .01, .1, .1+a*i, .1-a*i, 1, -10]); z = [.001, .01, .1, .1+a*i, .1-a*i, 1, -10]; z = [z',ones(7,1)]; y = [-10, 1; 1, 1; 0.01, 1; 0.001, 1; 0.1, 3]; fprintf('\n'); fprintf(' Ill-conditioned polynomial \n'); fprintf(' It is constructed using\n') fprintf(' roots ') fprintf(' \t\t\t\t\t\t multiplicities\n'); fprintf('\n'); fprintf('%22.15f + %22.15f i \t \t %3g \n', ... [real(z(:,1)),imag(z(:,1)),z(:,2)]'); fprintf('\n'); fprintf(' However, the polynomial is closer to having \n'); fprintf(' roots multiplicities\n'); fprintf('\n'); fprintf('%25.15f \t \t \t %3g \n', y'); SHAR_EOF fi # end of overwriting check if test -f 'jt07c.m' then echo shar: will not over-write existing file "'jt07c.m'" else cat << "SHAR_EOF" > 'jt07c.m' function [p,z] = jt07c % % test polynomial suggested by Jenkins and Traub % a = 10^(-8); p = poly([.001, .01, .1, .1+a*i, .1-a*i, 1, -10]); z = [.001, .01, .1, .1+a*i, .1-a*i, 1, -10]; z = [z',ones(7,1)]; y = [-10, 1; 1, 1; 0.01, 1; 0.001, 1; 0.1, 3]; fprintf('\n'); fprintf(' Ill-conditioned polynomial \n'); fprintf(' It is constructed using\n') fprintf(' roots ') fprintf(' \t\t\t\t\t\t multiplicities\n'); fprintf('\n'); fprintf('%22.15f + %22.15f i \t \t %3g \n', ... [real(z(:,1)),imag(z(:,1)),z(:,2)]'); fprintf('\n'); fprintf(' However, the polynomial is closer to having \n'); fprintf(' roots multiplicities\n'); fprintf('\n'); fprintf('%25.15f \t \t \t %3g \n', y'); SHAR_EOF fi # end of overwriting check if test -f 'jt07d.m' then echo shar: will not over-write existing file "'jt07d.m'" else cat << "SHAR_EOF" > 'jt07d.m' function [p,z] = jt07d % % test polynomial suggested by Jenkins and Traub % a = 10^(-7); p = poly([.001, .01, .1, .1+a*i, .1-a*i, 1, -10]); z = [.001, .01, .1, .1+a*i, .1-a*i, 1, -10]; z = [z',ones(7,1)]; y = [-10, 1; 1, 1; 0.01, 1; 0.001, 1; 0.1, 3]; fprintf('\n'); fprintf(' Ill-conditioned polynomial \n'); fprintf(' It is constructed using\n') fprintf(' roots ') fprintf(' \t\t\t\t\t\t multiplicities\n'); fprintf('\n'); fprintf('%22.15f + %22.15f i \t \t %3g \n', ... [real(z(:,1)),imag(z(:,1)),z(:,2)]'); fprintf('\n'); fprintf(' However, the polynomial is closer to having \n'); fprintf(' roots multiplicities\n'); fprintf('\n'); fprintf('%25.15f \t \t \t %3g \n', y'); SHAR_EOF fi # end of overwriting check if test -f 'jt08.m' then echo shar: will not over-write existing file "'jt08.m'" else cat << "SHAR_EOF" > 'jt08.m' function [p,z] = jt08 % % test polynomial suggested by Jenkins and Traub % p = poly([-1,-1,-1,-1,-1]); z = [-1, 5]; fprintf(' roots multiplicities\n'); fprintf('\n'); fprintf('%25.15f \t \t \t %3g \n', z'); SHAR_EOF fi # end of overwriting check if test -f 'jt09.m' then echo shar: will not over-write existing file "'jt09.m'" else cat << "SHAR_EOF" > 'jt09.m' function [p,z] = jt09 % % test polynomial suggested by Jenkins and Traub % p1 = [1,zeros(1,9),-10^(-20)]; p2 = [1,zeros(1,9),10^20]; p = conv(p1,p2); z1 = [-0.01000000000000, ... -0.00809016994375 + 0.00587785252292i, ... -0.00809016994375 - 0.00587785252292i, ... -0.00309016994375 + 0.00951056516295i, ... -0.00309016994375 - 0.00951056516295i, ... 0.00309016994375 + 0.00951056516295i, ... 0.00309016994375 - 0.00951056516295i, ... 0.01000000000000, ... 0.00809016994375 + 0.00587785252292i, ... 0.00809016994375 - 0.00587785252292i]; ... z2 = [... -0.95105651629515 + 0.30901699437495i, ... -0.95105651629515 - 0.30901699437495i, ... -0.58778525229247 + 0.80901699437495i, ... -0.58778525229247 - 0.80901699437495i, ... -0.00000000000000 + 1.00000000000000i, ... -0.00000000000000 - 1.00000000000000i, ... 0.95105651629515 + 0.30901699437495i, ... 0.95105651629515 - 0.30901699437495i, ... 0.58778525229247 + 0.80901699437495i, ... 0.58778525229247 - 0.80901699437495i]*100; ... z = [[z1';z2'],ones(20,1)]; fprintf('\n'); if norm(imag(z(:,1))) == 0 fprintf(' roots multiplicities\n'); fprintf('\n'); fprintf('%25.15f \t \t \t %3g \n', z'); else fprintf(' roots ') fprintf(' \t\t\t\t\t\t multiplicities\n'); fprintf('\n'); fprintf('%22.15f + %22.15f i \t \t %3g \n', ... [real(z(:,1)),imag(z(:,1)),z(:,2)]'); end; SHAR_EOF fi # end of overwriting check if test -f 'jt10a.m' then echo shar: will not over-write existing file "'jt10a.m'" else cat << "SHAR_EOF" > 'jt10a.m' function [p,z] = jt10a % % test polynomial suggested by Jenkins and Traub % a = 10^3; p = poly([a,1,1/a]); z = [[a,1,1/a]',ones(3,1)]; fprintf('\n'); if norm(imag(z(:,1))) == 0 fprintf(' roots multiplicities\n'); fprintf('\n'); fprintf('%25.15f \t \t \t %3g \n', z'); else fprintf(' roots ') fprintf(' \t\t\t\t\t\t multiplicities\n'); fprintf('\n'); fprintf('%22.15f + %22.15f i \t \t %3g \n', ... [real(z(:,1)),imag(z(:,1)),z(:,2)]'); end; SHAR_EOF fi # end of overwriting check if test -f 'jt10b.m' then echo shar: will not over-write existing file "'jt10b.m'" else cat << "SHAR_EOF" > 'jt10b.m' function [p,z] = jt10b % % test polynomial suggested by Jenkins and Traub % a = 10^6; p = poly([a,1,1/a]); z = [[a,1,1/a]',ones(3,1)]; fprintf('\n'); if norm(imag(z(:,1))) == 0 fprintf(' roots multiplicities\n'); fprintf('\n'); fprintf('%25.8f \t \t \t %3g \n', z'); else fprintf(' roots ') fprintf(' \t\t\t\t\t\t multiplicities\n'); fprintf('\n'); fprintf('%22.15f + %22.15f i \t \t %3g \n', ... [real(z(:,1)),imag(z(:,1)),z(:,2)]'); end; SHAR_EOF fi # end of overwriting check if test -f 'jt10c.m' then echo shar: will not over-write existing file "'jt10c.m'" else cat << "SHAR_EOF" > 'jt10c.m' function [p,z] = jt10c % % test polynomial suggested by Jenkins and Traub % a = 10^9; p = poly([a,1,1/a]); z = [[a,1,1/a]',ones(3,1)]; fprintf('\n'); if norm(imag(z(:,1))) == 0 fprintf(' roots multiplicities\n'); fprintf('\n'); fprintf('%25.10f \t \t \t %3g \n', z'); else fprintf(' roots ') fprintf(' \t\t\t\t\t\t multiplicities\n'); fprintf('\n'); fprintf('%22.15f + %22.15f i \t \t %3g \n', ... [real(z(:,1)),imag(z(:,1)),z(:,2)]'); end; SHAR_EOF fi # end of overwriting check if test -f 'jt11a.m' then echo shar: will not over-write existing file "'jt11a.m'" else cat << "SHAR_EOF" > 'jt11a.m' function [p,z] = jt11a % % test polynomial suggested by Jenkins and Traub % m = 15; y = exp(i*pi*[(1-m):(m-1)]/(2*m)); z = 0.9*exp(i*pi*[m:3*m]/(2*m)); p1 = poly(y); p2 = poly(z); p = conv(p1,p2); z = [[y';z'],ones(4*m,1)]; fprintf('\n'); %fprintf(' Roots are all simple and sensitive, see plot '); %plot(real(z(:,1)),imag(z(:,1)),'+'); SHAR_EOF fi # end of overwriting check if test -f 'jt11b.m' then echo shar: will not over-write existing file "'jt11b.m'" else cat << "SHAR_EOF" > 'jt11b.m' function [p,z] = jt11b % % test polynomial suggested by Jenkins and Traub % m = 20; y = exp(i*pi*[(1-m):(m-1)]/(2*m)); z = 0.9*exp(i*pi*[m:3*m]/(2*m)); u = [y,z]; p = poly(u); z = [[y';z'],ones(4*m,1)]; fprintf('\n'); %fprintf(' Roots are all simple and sensitive, see plot '); %plot(real(z(:,1)),imag(z(:,1)),'+'); SHAR_EOF fi # end of overwriting check if test -f 'jt11c.m' then echo shar: will not over-write existing file "'jt11c.m'" else cat << "SHAR_EOF" > 'jt11c.m' function [p,z] = jt11c % % test polynomial suggested by Jenkins and Traub % m = 25; y = exp(i*pi*[(1-m):(m-1)]/(2*m)); z = 0.9*exp(i*pi*[m:3*m]/(2*m)); u = [y,z]; p = poly(u); z = [[y';z'],ones(4*m,1)]; fprintf('\n'); %fprintf(' Roots are all simple and sensitive, see plot '); %plot(real(z(:,1)),imag(z(:,1)),'+'); SHAR_EOF fi # end of overwriting check if test -f 'large01.m' then echo shar: will not over-write existing file "'large01.m'" else cat << "SHAR_EOF" > 'large01.m' function [p,z] = large01 % % test polynomial suggested by Jenkins and Traub % p = [ 1.0 ... 0 ... -0.28000000000000 ... -0.59200000000000 ... -0.96890000000000 ... -0.62020000000000 ... 0.54382200000000 ... 0.08842680000000 ... 0.43204191000000 ... -0.15503004000000 ... 0.44119337520000 ... -0.65443195750000 ... -0.35708097260000 ... -0.55597791620000 ... 0.63916879370000 ... 0.14637576020000 ... -0.06709193010000 ... -0.03193754010000 ... 0.50857405010000 ... -0.14968258190000 ... 0.63273024920000 ]; z = [-1.0 + 0.3i, ... -1.0 - 0.30i, ... -0.9 + 0.4i, ... -0.9 - 0.4i, ... -0.7 + 0.7i, ... -0.7 - 0.7i, ... -0.4 + 0.9i, ... -0.4 - 0.9i, ... -0.0 + 1.10000000000i, ... -0.000000000000 - 1.10000000000i, ... 1.2, ... 1.00000000000000, ... 0.9 + 0.4i, ... 0.9 - 0.4i, ... 0.6 + 0.600000000000i, ... 0.6 - 0.600000000000i, ... 0.4 + 0.900000000000i, ... 0.4 - 0.900000000000i, ... 0.00000000000 + 0.8i, ... 0.00000000000 - 0.8i]; z = [z',ones(20,1)]; fprintf('\n'); fprintf(' Coefficients are rounded up at the 10-th digit, \n'); fprintf(' It is originally constructed using\n') fprintf(' roots ') fprintf(' \t\t multiplicities\n'); fprintf('\n'); fprintf('%12.1f + %6.1f i \t \t %3g \n', ... [real(z(:,1)),imag(z(:,1)),z(:,2)]'); fprintf('\n'); SHAR_EOF fi # end of overwriting check if test -f 'large02.m' then echo shar: will not over-write existing file "'large02.m'" else cat << "SHAR_EOF" > 'large02.m' function [p,z] = large02 % % test polynomial suggested by Z Zeng % p = [ 1.0 ... 0 ... -0.28000000000000 ... -0.59200000000000 ... -0.96890000000000 ... -0.62020000000000 ... 0.54382200000000 ... 0.08842680000000 ... 0.43204191000000 ... -0.15503004000000 ... 0.44119337520000 ... -0.65443195750000 ... -0.35708097260000 ... -0.55597791620000 ... 0.63916879370000 ... 0.14637576020000 ... -0.06709193010000 ... -0.03193754010000 ... 0.50857405010000 ... -0.14968258190000 ... 0.63273024920000 ]; p = conv(p,p); z = [-1.0 + 0.3i, ... -1.0 - 0.30i, ... -0.9 + 0.4i, ... -0.9 - 0.4i, ... -0.7 + 0.7i, ... -0.7 - 0.7i, ... -0.4 + 0.9i, ... -0.4 - 0.9i, ... -0.0 + 1.10000000000i, ... -0.000000000000 - 1.10000000000i, ... 1.2, ... 1.00000000000000, ... 0.9 + 0.4i, ... 0.9 - 0.4i, ... 0.6 + 0.600000000000i, ... 0.6 - 0.600000000000i, ... 0.4 + 0.900000000000i, ... 0.4 - 0.900000000000i, ... 0.00000000000 + 0.8i, ... 0.00000000000 - 0.8i]; z = [z',2*ones(20,1)]; fprintf('\n'); fprintf(' Coefficients are rounded up at the 10-th digit, \n'); fprintf(' It is originally constructed using\n') fprintf(' roots ') fprintf(' \t\t multiplicities\n'); fprintf('\n'); fprintf('%12.1f + %6.1f i \t \t %3g \n', ... [real(z(:,1)),imag(z(:,1)),z(:,2)]'); fprintf('\n'); SHAR_EOF fi # end of overwriting check if test -f 'large03.m' then echo shar: will not over-write existing file "'large03.m'" else cat << "SHAR_EOF" > 'large03.m' function [p,z] = large03 % % test polynomial suggested by Z Zeng % p = [ 1.0 ... 0 ... -0.28000000000000 ... -0.59200000000000 ... -0.96890000000000 ... -0.62020000000000 ... 0.54382200000000 ... 0.08842680000000 ... 0.43204191000000 ... -0.15503004000000 ... 0.44119337520000 ... -0.65443195750000 ... -0.35708097260000 ... -0.55597791620000 ... 0.63916879370000 ... 0.14637576020000 ... -0.06709193010000 ... -0.03193754010000 ... 0.50857405010000 ... -0.14968258190000 ... 0.63273024920000 ]; p = conv(p,p); p = conv(p,p); z = [-1.0 + 0.3i, ... -1.0 - 0.30i, ... -0.9 + 0.4i, ... -0.9 - 0.4i, ... -0.7 + 0.7i, ... -0.7 - 0.7i, ... -0.4 + 0.9i, ... -0.4 - 0.9i, ... -0.0 + 1.10000000000i, ... -0.000000000000 - 1.10000000000i, ... 1.2, ... 1.00000000000000, ... 0.9 + 0.4i, ... 0.9 - 0.4i, ... 0.6 + 0.600000000000i, ... 0.6 - 0.600000000000i, ... 0.4 + 0.900000000000i, ... 0.4 - 0.900000000000i, ... 0.00000000000 + 0.8i, ... 0.00000000000 - 0.8i]; z = [z',4*ones(20,1)]; fprintf('\n'); fprintf(' Coefficients are rounded up at the 10-th digit, \n'); fprintf(' It is originally constructed using\n') fprintf(' roots ') fprintf(' \t\t multiplicities\n'); fprintf('\n'); fprintf('%12.1f + %6.1f i \t \t %3g \n', ... [real(z(:,1)),imag(z(:,1)),z(:,2)]'); fprintf('\n'); SHAR_EOF fi # end of overwriting check if test -f 'large04.m' then echo shar: will not over-write existing file "'large04.m'" else cat << "SHAR_EOF" > 'large04.m' function [p,z] = large04 % % test polynomial suggested by Jenkins and Traub % p = [ 1.0 ... 0 ... -0.28000000000000 ... -0.59200000000000 ... -0.96890000000000 ... -0.62020000000000 ... 0.54382200000000 ... 0.08842680000000 ... 0.43204191000000 ... -0.15503004000000 ... 0.44119337520000 ... -0.65443195750000 ... -0.35708097260000 ... -0.55597791620000 ... 0.63916879370000 ... 0.14637576020000 ... -0.06709193010000 ... -0.03193754010000 ... 0.50857405010000 ... -0.14968258190000 ... 0.63273024920000 ]; p = conv(p,p); p = conv(p,p); p = conv(p,p); z = [-1.0 + 0.3i, ... -1.0 - 0.30i, ... -0.9 + 0.4i, ... -0.9 - 0.4i, ... -0.7 + 0.7i, ... -0.7 - 0.7i, ... -0.4 + 0.9i, ... -0.4 - 0.9i, ... -0.0 + 1.10000000000i, ... -0.000000000000 - 1.10000000000i, ... 1.2, ... 1.00000000000000, ... 0.9 + 0.4i, ... 0.9 - 0.4i, ... 0.6 + 0.600000000000i, ... 0.6 - 0.600000000000i, ... 0.4 + 0.900000000000i, ... 0.4 - 0.900000000000i, ... 0.00000000000 + 0.8i, ... 0.00000000000 - 0.8i]; z = [z',8*ones(20,1)]; fprintf('\n'); fprintf(' Coefficients are rounded up at the 10-th digit, \n'); fprintf(' It is originally constructed using\n') fprintf(' roots ') fprintf(' \t\t multiplicities\n'); fprintf('\n'); fprintf('%12.1f + %6.1f i \t \t %3g \n', ... [real(z(:,1)),imag(z(:,1)),z(:,2)]'); fprintf('\n'); SHAR_EOF fi # end of overwriting check if test -f 'large05.m' then echo shar: will not over-write existing file "'large05.m'" else cat << "SHAR_EOF" > 'large05.m' function [p,z] = large05 % % test polynomial suggested by Jenkins and Traub % p = [ 1.0 ... 0 ... -0.28000000000000 ... -0.59200000000000 ... -0.96890000000000 ... -0.62020000000000 ... 0.54382200000000 ... 0.08842680000000 ... 0.43204191000000 ... -0.15503004000000 ... 0.44119337520000 ... -0.65443195750000 ... -0.35708097260000 ... -0.55597791620000 ... 0.63916879370000 ... 0.14637576020000 ... -0.06709193010000 ... -0.03193754010000 ... 0.50857405010000 ... -0.14968258190000 ... 0.63273024920000 ]; p = conv(p,p); p = conv(p,p); p = conv(p,p); p = conv(p,p); z = [-1.0 + 0.3i, ... -1.0 - 0.30i, ... -0.9 + 0.4i, ... -0.9 - 0.4i, ... -0.7 + 0.7i, ... -0.7 - 0.7i, ... -0.4 + 0.9i, ... -0.4 - 0.9i, ... -0.0 + 1.10000000000i, ... -0.000000000000 - 1.10000000000i, ... 1.2, ... 1.00000000000000, ... 0.9 + 0.4i, ... 0.9 - 0.4i, ... 0.6 + 0.600000000000i, ... 0.6 - 0.600000000000i, ... 0.4 + 0.900000000000i, ... 0.4 - 0.900000000000i, ... 0.00000000000 + 0.8i, ... 0.00000000000 - 0.8i]; z = [z',16*ones(20,1)]; fprintf('\n'); fprintf(' Coefficients are rounded up at the 10-th digit, \n'); fprintf(' It is originally constructed using\n') fprintf(' roots ') fprintf(' \t\t multiplicities\n'); fprintf('\n'); fprintf('%12.1f + %6.1f i \t \t %3g \n', ... [real(z(:,1)),imag(z(:,1)),z(:,2)]'); fprintf('\n'); SHAR_EOF fi # end of overwriting check if test -f 'lgd.m' then echo shar: will not over-write existing file "'lgd.m'" else cat << "SHAR_EOF" > 'lgd.m' function p = lgd(n) % % test polynomial suggested by Goedecker % Legendre polynomial of degree n % p0 = [1]; p1 = [1,0]; if n == 0 p = p0; elseif n == 1 p = p1;; elseif n > 1 for k = 1:n-1 q1 = (2*k+1)*[p1,0]; m = length(q1)-length(p0); q0 = k*[zeros(1,m),p0]; p = (q1-q0)/(k+1); % p0 = p1; p1 = p; end; end; SHAR_EOF fi # end of overwriting check if test -f 'lgd05.m' then echo shar: will not over-write existing file "'lgd05.m'" else cat << "SHAR_EOF" > 'lgd05.m' function [p,z] = lgd05 % % test polynomial suggested by Goedecker % Legendre polynomial of degree 5 % p = lgd(5); z = [ 0, ... 0.90617984593866, ... 0.53846931010568, ... -0.90617984593866, ... -0.53846931010568]; z = [z',ones(5,1)]; fprintf('\n'); if norm(imag(z(:,1))) == 0 fprintf(' roots multiplicities\n'); fprintf('\n'); fprintf('%25.14f \t \t \t %3g \n', z'); else fprintf(' roots ') fprintf(' \t\t\t\t\t\t multiplicities\n'); fprintf('\n'); fprintf('%22.15f + %22.15f i \t \t %3g \n', ... [real(z(:,1)),imag(z(:,1)),z(:,2)]'); end; SHAR_EOF fi # end of overwriting check if test -f 'lgd10.m' then echo shar: will not over-write existing file "'lgd10.m'" else cat << "SHAR_EOF" > 'lgd10.m' function [p,z] = lgd10 % % test polynomial suggested by Goedecker % Legendre polynomial of degree 10 % p = lgd(10); z = [0.97390652851717 0.86506336668898 0.67940956829903 -0.97390652851717 -0.86506336668899 -0.67940956829903 0.43339539412925 -0.43339539412925 0.14887433898163 -0.14887433898163]; z = [z,ones(10,1)]; fprintf('\n'); if norm(imag(z(:,1))) == 0 fprintf(' roots multiplicities\n'); fprintf('\n'); fprintf('%25.14f \t \t \t %3g \n', z'); else fprintf(' roots ') fprintf(' \t\t\t\t\t\t multiplicities\n'); fprintf('\n'); fprintf('%22.15f + %22.15f i \t \t %3g \n', ... [real(z(:,1)),imag(z(:,1)),z(:,2)]'); end; SHAR_EOF fi # end of overwriting check if test -f 'lgd100.m' then echo shar: will not over-write existing file "'lgd100.m'" else cat << "SHAR_EOF" > 'lgd100.m' function [p,z] = lgd100 % % test polynomial suggested by Goedecker % Legendre polynomial of degree 100 % p = lgd(100); z = [-.9997137267734412, ... -.9984919506395958, ... -.9962951347331251, ... -.9931249370374435, ... -.9889843952429917, ... -.9838775407060570, ... -.9778093584869183, ... -.9707857757637063, ... -.9628136542558155, ... -.9539007829254917, ... -.9440558701362560, ... -.9332885350430795, ... -.9216092981453340, ... -.9090295709825297, ... -.8955616449707270, ... -.8812186793850184, ... -.8660146884971646, ... -.8499645278795913, ... -.8330838798884008, ... -.8153892383391763, ... -.7968978923903145, ... -.7776279096494955, ... -.7575981185197072, ... -.7368280898020207, ... -.7153381175730564, ... -.6931491993558020, ... -.6702830156031410, ... -.6467619085141293, ... -.6226088602037078, ... -.5978474702471787, ... -.5725019326213812, ... -.5465970120650942, ... -.5201580198817631, ... -.4932107892081909, ... -.4657816497733580, ... -.4378974021720315, ... -.4095852916783015, ... -.3808729816246300, ... -.3517885263724217, ... -.3223603439005292, ... -.2926171880384720, ... -.2625881203715035, ... -.2323024818449740, ... -.2017898640957360, ... -.1710800805386033, ... -.1402031372361140, ... -.1091892035800611, ... -.7806858281343664e-1, ... -.4687168242159163e-1, ... -.1562898442154308e-1, ... .1562898442154308e-1, ... .4687168242159163e-1, ... .7806858281343664e-1, ... .1091892035800611, ... .1402031372361140, ... .1710800805386033, ... .2017898640957360, ... .2323024818449740, ... .2625881203715035, ... .2926171880384720, ... .3223603439005292, ... .3517885263724217, ... .3808729816246300, ... .4095852916783015, ... .4378974021720315, ... .4657816497733580, ... .4932107892081909, ... .5201580198817631, ... .5465970120650942, ... .5725019326213812, ... .5978474702471787, ... .6226088602037078, ... .6467619085141293, ... .6702830156031410, ... .6931491993558020, ... .7153381175730564, ... .7368280898020207, ... .7575981185197072, ... .7776279096494955, ... .7968978923903145, ... .8153892383391763, ... .8330838798884008, ... .8499645278795913, ... .8660146884971646, ... .8812186793850184, ... .8955616449707270, ... .9090295709825297, ... .9216092981453340, ... .9332885350430795, ... .9440558701362560, ... .9539007829254917, ... .9628136542558155, ... .9707857757637063, ... .9778093584869183, ... .9838775407060570, ... .9889843952429917, ... .9931249370374435, ... .9962951347331251, ... .9984919506395958, ... .9997137267734412]; %plot(real(z),imag(z),'.'); z = [z',ones(100,1)]; %fprintf('Legendre polynomial, roots are sensitive, see plot. \n'); SHAR_EOF fi # end of overwriting check if test -f 'lgd15.m' then echo shar: will not over-write existing file "'lgd15.m'" else cat << "SHAR_EOF" > 'lgd15.m' function [p,z] = lgd15 % % test polynomial suggested by Goedecker % Legendre polynomial of degree 15 % p = lgd(15); z = [ 0 0.98799251802063 0.93727339240040 0.84820658341064 0.72441773136009 0.57097217260857 0.39415134707756 -0.98799251802067 -0.93727339240030 -0.84820658341075 -0.72441773136004 0.20119409399743 -0.57097217260858 -0.39415134707756 -0.20119409399743]; z = [z,ones(15,1)]; fprintf('\n'); if norm(imag(z(:,1))) == 0 fprintf(' roots multiplicities\n'); fprintf('\n'); fprintf('%25.14f \t \t \t %3g \n', z'); else fprintf(' roots ') fprintf(' \t\t\t\t\t\t multiplicities\n'); fprintf('\n'); fprintf('%22.14f + %22.14f i \t \t %3g \n', ... [real(z(:,1)),imag(z(:,1)),z(:,2)]'); end; SHAR_EOF fi # end of overwriting check if test -f 'lgd20.m' then echo shar: will not over-write existing file "'lgd20.m'" else cat << "SHAR_EOF" > 'lgd20.m' function [p,z] = lgd20 % % test polynomial suggested by Goedecker % Legendre polynomial of degree 20 % p = lgd(20); z = [-.9931285991850949, ... -.9639719272779138, ... -.9122344282513259, ... -.8391169718222188, ... -.7463319064601508, ... -.6360536807265150, ... -.5108670019508271, ... -.3737060887154196, ... -.2277858511416451, ... -.7652652113349733e-1, ... .7652652113349733e-1, ... .2277858511416451, ... .3737060887154196, ... .5108670019508271, ... .6360536807265150, ... .7463319064601508, ... .8391169718222188, ... .9122344282513259, ... .9639719272779138, ... .9931285991850949]; z = [z',ones(20,1)]; fprintf('Legendre polynomial, roots are sensitive \n'); fprintf(' roots multiplicities\n'); fprintf('\n'); fprintf('%25.14f \t \t \t %3g \n', z'); SHAR_EOF fi # end of overwriting check if test -f 'lgd24.m' then echo shar: will not over-write existing file "'lgd24.m'" else cat << "SHAR_EOF" > 'lgd24.m' function [p,z] = lgd24 % % test polynomial suggested by Goedecker % Legendre polynomial of degree 24 % p = lgd(24); z = [-.9951872199970214, ... -.9747285559713095, ... -.9382745520027328, ... -.8864155270044010, ... -.8200019859739029, ... -.7401241915785544, ... -.6480936519369756, ... -.5454214713888395, ... -.4337935076260451, ... -.3150426796961634, ... -.1911188674736163, ... -.6405689286260563e-1, ... .6405689286260563e-1, ... .1911188674736163, ... .3150426796961634, ... .4337935076260451, ... .5454214713888395, ... .6480936519369756, ... .7401241915785544, ... .8200019859739029, ... .8864155270044010, ... .9382745520027328, ... .9747285559713095, ... .9951872199970214]; z = [z',ones(24,1)]; fprintf('Legendre polynomial, roots are sensitive \n'); fprintf(' roots multiplicities\n'); fprintf('\n'); fprintf('%25.14f \t \t \t %3g \n', z'); SHAR_EOF fi # end of overwriting check if test -f 'lgd50.m' then echo shar: will not over-write existing file "'lgd50.m'" else cat << "SHAR_EOF" > 'lgd50.m' function [p,z] = lgd50 % % test polynomial suggested by Goedecker % Legendre polynomial of degree 50 % p = lgd(50); z = [-.9988664044200711, -.9940319694320907, -.9853540840480059, ... -.9728643851066921, -.9566109552428079, -.9366566189448779, ... -.9130785566557919, -.8859679795236130, -.8554297694299461, ... -.8215820708593359, -.7845558329003993, -.7444943022260685, ... -.7015524687068223, -.6558964656854394, -.6077029271849502, ... -.5571583045146501, -.5044581449074642, -.4498063349740388, ... -.3934143118975651, -.3355002454194374, -.2762881937795320, ... -.2160072368760418, -.1548905899981459, -.9317470156008614e-1, ... -.3109833832718888e-1, .3109833832718888e-1, ... .9317470156008614e-1, .1548905899981459, .2160072368760418, ... .2762881937795320, .3355002454194374, .3934143118975651, ... .4498063349740388, .5044581449074642, .5571583045146501, ... .6077029271849502, .6558964656854394, .7015524687068223, ... .7444943022260685, .7845558329003993, .8215820708593359, ... .8554297694299461, .8859679795236130, .9130785566557919, ... .9366566189448779, .9566109552428079, .9728643851066921, ... .9853540840480059, .9940319694320907, .9988664044200711]; z = [z',ones(50,1)]; fprintf('Legendre polynomial, roots are sensitive \n'); fprintf(' roots multiplicities\n'); fprintf('\n'); fprintf('%25.14f \t \t \t %3g \n', z'); SHAR_EOF fi # end of overwriting check if test -f 'miyak00.m' then echo shar: will not over-write existing file "'miyak00.m'" else cat << "SHAR_EOF" > 'miyak00.m' function [p,z] = miyak00 % % test polynomial suggested by Goedecker % square of Fibocacci polynomial % p = poly([(1.1+1.1*i)*ones(1,4),(3.2+2.3*i)*[1,1],2.1+1.5*i]); z = [[(1.1+1.1*i),(3.2+2.3*i),2.1+1.5*i].',[4,2,1]']; if norm(imag(z(:,1))) == 0 fprintf(' roots multiplicities\n'); fprintf('\n'); fprintf('%25.15f \t \t \t %3g \n', z'); else fprintf(' roots ') fprintf(' \t\t\t\t\t\t multiplicities\n'); fprintf('\n'); fprintf('%22.15f + %22.15f i \t \t %3g \n', ... [real(z(:,1)),imag(z(:,1)),z(:,2)]'); end; SHAR_EOF fi # end of overwriting check if test -f 'miyak02.m' then echo shar: will not over-write existing file "'miyak02.m'" else cat << "SHAR_EOF" > 'miyak02.m' function [p,z] = miyak02 % % test polynomial suggested by Goedecker % square of Fibocacci polynomial % p = poly([(1.1+1.1*i)*ones(1,4),(3.2+2.3*i)*[1,1],2.1+1.5*i]); p = conv(p,p); z = [[(1.1+1.1*i),(3.2+2.3*i),2.1+1.5*i].',2*[4,2,1]']; if norm(imag(z(:,1))) == 0 fprintf(' roots multiplicities\n'); fprintf('\n'); fprintf('%25.15f \t \t \t %3g \n', z'); else fprintf(' roots ') fprintf(' \t\t\t\t\t\t multiplicities\n'); fprintf('\n'); fprintf('%22.15f + %22.15f i \t \t %3g \n', ... [real(z(:,1)),imag(z(:,1)),z(:,2)]'); end; SHAR_EOF fi # end of overwriting check if test -f 'miyak04.m' then echo shar: will not over-write existing file "'miyak04.m'" else cat << "SHAR_EOF" > 'miyak04.m' function [p,z] = miyak04 % % test polynomial suggested by Goedecker % square of Fibocacci polynomial % p = poly([(1.1+1.1*i)*ones(1,4),(3.2+2.3*i)*[1,1],2.1+1.5*i]); p = conv(p,p); p = conv(p,p); z = [[(1.1+1.1*i),(3.2+2.3*i),2.1+1.5*i].',4*[4,2,1]']; if norm(imag(z(:,1))) == 0 fprintf(' roots multiplicities\n'); fprintf('\n'); fprintf('%25.15f \t \t \t %3g \n', z'); else fprintf(' roots ') fprintf(' \t\t\t\t\t\t multiplicities\n'); fprintf('\n'); fprintf('%22.15f + %22.15f i \t \t %3g \n', ... [real(z(:,1)),imag(z(:,1)),z(:,2)]'); end; SHAR_EOF fi # end of overwriting check if test -f 'miyak08.m' then echo shar: will not over-write existing file "'miyak08.m'" else cat << "SHAR_EOF" > 'miyak08.m' function [p,z] = miyak08 % % test polynomial suggested by Goedecker % square of Fibocacci polynomial % p = poly([(1.1+1.1*i)*ones(1,4),(3.2+2.3*i)*[1,1],2.1+1.5*i]); p = poly([(1.1+1.1*i)*ones(1,32),(3.2+2.3*i)*ones(1,16), ... (2.1+1.5*i)*ones(1,8)]); z = [[(1.1+1.1*i),(3.2+2.3*i),2.1+1.5*i].',8*[4,2,1]']; if norm(imag(z(:,1))) == 0 fprintf(' roots multiplicities\n'); fprintf('\n'); fprintf('%25.15f \t \t \t %3g \n', z'); else fprintf(' roots ') fprintf(' \t\t\t\t\t\t multiplicities\n'); fprintf('\n'); fprintf('%22.15f + %22.15f i \t \t %3g \n', ... [real(z(:,1)),imag(z(:,1)),z(:,2)]'); end; SHAR_EOF fi # end of overwriting check if test -f 'near01.m' then echo shar: will not over-write existing file "'near01.m'" else cat << "SHAR_EOF" > 'near01.m' function [p,z] = near01 % % test polynomial suggested by Z. Zeng % e = 0.1; p = poly([(1-e)*ones(1,20),ones(1,20),-0.5*[1,1,1,1,1]]); z = [1-e, 20; 1, 20; -0.5, 5]; if norm(imag(z(:,1))) == 0 fprintf(' roots multiplicities\n'); fprintf('\n'); fprintf('%25.15f \t \t \t %3g \n', z'); else fprintf(' roots ') fprintf(' \t\t\t\t\t\t multiplicities\n'); fprintf('\n'); fprintf('%22.15f + %22.15f i \t \t %3g \n', ... [real(z(:,1)),imag(z(:,1)),z(:,2)]'); end; SHAR_EOF fi # end of overwriting check if test -f 'near02.m' then echo shar: will not over-write existing file "'near02.m'" else cat << "SHAR_EOF" > 'near02.m' function [p,z] = near02 % % test polynomial suggested by Z. Zeng % e = 0.01; p = poly([(1-e)*ones(1,20),ones(1,20),-0.5*[1,1,1,1,1]]); z = [1-e, 20; 1, 20; -0.5, 5]; if norm(imag(z(:,1))) == 0 fprintf(' roots multiplicities\n'); fprintf('\n'); fprintf('%25.15f \t \t \t %3g \n', z'); else fprintf(' roots ') fprintf(' \t\t\t\t\t\t multiplicities\n'); fprintf('\n'); fprintf('%22.15f + %22.15f i \t \t %3g \n', ... [real(z(:,1)),imag(z(:,1)),z(:,2)]'); end; SHAR_EOF fi # end of overwriting check if test -f 'near03.m' then echo shar: will not over-write existing file "'near03.m'" else cat << "SHAR_EOF" > 'near03.m' function [p,z] = near03 % % test polynomial suggested by Z. Zeng % It is suggested that MultRoot input gamma = 10. % e = 0.001; p = poly([(1-e)*ones(1,20),ones(1,20),-0.5*[1,1,1,1,1]]); z = [1-e, 20; 1, 20; -0.5, 5]; if norm(imag(z(:,1))) == 0 fprintf(' roots multiplicities\n'); fprintf('\n'); fprintf('%25.15f \t \t \t %3g \n', z'); else fprintf(' roots ') fprintf(' \t\t\t\t\t\t multiplicities\n'); fprintf('\n'); fprintf('%22.15f + %22.15f i \t \t %3g \n', ... [real(z(:,1)),imag(z(:,1)),z(:,2)]'); end; fprintf('To use MultRoot, set input gamma = 5\n') SHAR_EOF fi # end of overwriting check if test -f 'petk01.m' then echo shar: will not over-write existing file "'petk01.m'" else cat << "SHAR_EOF" > 'petk01.m' function [p,z] = petk01 % % M. Petkovic testing polynomials (p. 109) % p1 = poly([-1, -1, 3, 3, 3, -i*[1,1,1,1]]); p2 = [1,-2,5]; p2 = conv(p2,p2); p = conv(p1,p2); z = [-1,2; 3, 3; -i, 4; 1+2*i, 2; 1-2*i, 2]; if norm(imag(z(:,1))) == 0 fprintf(' roots multiplicities\n'); fprintf('\n'); fprintf('%25.15f \t \t \t %3g \n', z'); else fprintf(' roots ') fprintf(' \t\t\t\t\t\t multiplicities\n'); fprintf('\n'); fprintf('%22.15f + %22.15f i \t \t %3g \n', ... [real(z(:,1)),imag(z(:,1)),z(:,2)]'); end; SHAR_EOF fi # end of overwriting check if test -f 'petk02.m' then echo shar: will not over-write existing file "'petk02.m'" else cat << "SHAR_EOF" > 'petk02.m' function [p,z] = petk02 % % M. Petkovic testing polynomials, p118 % p = poly([1,1,-i,-i,-i,5*i,5*i,-5*i,-5*i]); z = [1,2; -i,3; 5*i,2; -5*i,2]; if norm(imag(z(:,1))) == 0 fprintf(' roots multiplicities\n'); fprintf('\n'); fprintf('%25.15f \t \t \t %3g \n', z'); else fprintf(' roots ') fprintf(' \t\t\t\t\t\t multiplicities\n'); fprintf('\n'); fprintf('%22.15f + %22.15f i \t \t %3g \n', ... [real(z(:,1)),imag(z(:,1)),z(:,2)]'); end; SHAR_EOF fi # end of overwriting check if test -f 'petk03.m' then echo shar: will not over-write existing file "'petk03.m'" else cat << "SHAR_EOF" > 'petk03.m' function [p,z] = petk03 % % M. Petkovic testing polynomials, p134 % p1 = [1,-1-99*i/70]; p2 = [1,2,3]; p2 = conv(p2,p2); p = conv(p1,p2); p = conv(p,[1,1]); z = [-1.00000000000000 + 1.41421356237309i, 2;... 1.00000000000000 + 1.41421356237309i, 2; ... -1-99*i/70, 1; -1, 1]; if norm(imag(z(:,1))) == 0 fprintf(' roots multiplicities\n'); fprintf('\n'); fprintf('%25.15f \t \t \t %3g \n', z'); else fprintf(' roots ') fprintf(' \t\t\t\t\t\t multiplicities\n'); fprintf('\n'); fprintf('%22.15f + %22.15f i \t \t %3g \n', ... [real(z(:,1)),imag(z(:,1)),z(:,2)]'); end; SHAR_EOF fi # end of overwriting check if test -f 'petk04.m' then echo shar: will not over-write existing file "'petk04.m'" else cat << "SHAR_EOF" > 'petk04.m' function [p,z] = petk04 % % M. Petkovic testing polynomials, p139 % z = [3, -1*[1,1,1], 2*i*[1,1,1], ... (-2+i)*[1,1], (-2-i)*[1,1], ... (2+i)*[1,1], (2-i)*[1,1]]; p = poly(z); z = [3, 1; -1, 3; 2*i, 3; -2+i, 2; -2-i, 2; ... 2+i, 2; 2-i, 2]; if norm(imag(z(:,1))) == 0 fprintf(' roots multiplicities\n'); fprintf('\n'); fprintf('%25.15f \t \t \t %3g \n', z'); else fprintf(' roots ') fprintf(' \t\t\t\t\t\t multiplicities\n'); fprintf('\n'); fprintf('%22.15f + %22.15f i \t \t %3g \n', ... [real(z(:,1)),imag(z(:,1)),z(:,2)]'); end; SHAR_EOF fi # end of overwriting check if test -f 'petk05.m' then echo shar: will not over-write existing file "'petk05.m'" else cat << "SHAR_EOF" > 'petk05.m' function [p,z] = petk05 % % M. Petkovic testing polynomials, p142 % p = [1,-7,20,-28,-18,110,-92,-44,345,225]; z = [3.00000000000000 + 0.00000000000000i 1.00000000000000 + 2.00000000000000i 1.00000000000000 - 2.00000000000000i -1.00000000000000 - 0.00000000000000i]; z = [z, [2,2,2,3]']; if norm(imag(z(:,1))) == 0 fprintf(' roots multiplicities\n'); fprintf('\n'); fprintf('%25.15f \t \t \t %3g \n', z'); else fprintf(' roots ') fprintf(' \t\t\t\t\t\t multiplicities\n'); fprintf('\n'); fprintf('%22.15f + %22.15f i \t \t %3g \n', ... [real(z(:,1)),imag(z(:,1)),z(:,2)]'); end; SHAR_EOF fi # end of overwriting check if test -f 'petk06.m' then echo shar: will not over-write existing file "'petk06.m'" else cat << "SHAR_EOF" > 'petk06.m' function [p,z] = petk06 % % M. Petkovic testing polynomials, page 146 % y = [-1*[1,1,1,1],3*[1,1,1],-i,-i]; p1 = poly(y); p2 = [1,-2,5]; p2 = conv(p2,p2); p = conv(p1,p2); z = [-1,4; 3,3; -i,2; 1+2*i,2; 1-2*i,2]; if norm(imag(z(:,1))) == 0 fprintf(' roots multiplicities\n'); fprintf('\n'); fprintf('%25.15f \t \t \t %3g \n', z'); else fprintf(' roots ') fprintf(' \t\t\t\t\t\t multiplicities\n'); fprintf('\n'); fprintf('%22.15f + %22.15f i \t \t %3g \n', ... [real(z(:,1)),imag(z(:,1)),z(:,2)]'); end; SHAR_EOF fi # end of overwriting check if test -f 'petk07.m' then echo shar: will not over-write existing file "'petk07.m'" else cat << "SHAR_EOF" > 'petk07.m' function [p,z] = petk07 % % M. Petkovic testing polynomials, page 147 % y = [1*[1,1,1],-2+i, -2-i, 5*i,5*i, -5*i, -5*i]; p = poly(y); z = [1, 3; -2+i,1; -2-i,1; 5*i,2; , -5*i, 2]; if norm(imag(z(:,1))) == 0 fprintf(' roots multiplicities\n'); fprintf('\n'); fprintf('%25.15f \t \t \t %3g \n', z'); else fprintf(' roots ') fprintf(' \t\t\t\t\t\t multiplicities\n'); fprintf('\n'); fprintf('%22.15f + %22.15f i \t \t %3g \n', ... [real(z(:,1)),imag(z(:,1)),z(:,2)]'); end; SHAR_EOF fi # end of overwriting check if test -f 'toh01.m' then echo shar: will not over-write existing file "'toh01.m'" else cat << "SHAR_EOF" > 'toh01.m' function [p,z] = toh01 % % generalization of K.C. Toh and L. N. Trefethen % k = [-10:9]'; z = 2*(k+0.5)/19 + i*sin(2*pi*(k+0.5)/19); p = poly(z); z = [z,ones(20,1)]; if norm(imag(z(:,1))) == 0 fprintf(' roots multiplicities\n'); fprintf('\n'); fprintf('%25.15f \t \t \t %3g \n', z'); else fprintf(' roots ') fprintf(' \t\t\t\t\t\t multiplicities\n'); fprintf('\n'); fprintf('%22.15f + %22.15f i \t \t %3g \n', ... [real(z(:,1)),imag(z(:,1)),z(:,2)]'); end; SHAR_EOF fi # end of overwriting check if test -f 'toh02.m' then echo shar: will not over-write existing file "'toh02.m'" else cat << "SHAR_EOF" > 'toh02.m' function [p,z] = toh02 % % generalization of K.C. Toh and L. N. Trefethen % k = [20:-1:0]'; p1 = 10.^k; p2 = ones(21,1); for j = 20:-1:0, p2(21-j)=factorial(j); end; p = p1./p2; z = 2*(k+0.5)/19 + i*sin(2*pi*(k+0.5)/19); z = [ 1.08045842459084 + 0.92291979046775i 1.08045842459084 - 0.92291979046775i 0.57624370699834 + 0.97554880155012i 0.57624370699834 - 0.97554880155012i 0.23672740822527 + 0.94133582606706i 0.23672740822527 - 0.94133582606706i -0.01684064074376 + 0.86388496107331i -0.01684064074376 - 0.86388496107331i -0.21255469312515 + 0.76040546396802i -0.21255469312515 - 0.76040546396802i -0.36406222857634 + 0.63986739119691i -0.36406222857634 - 0.63986739119691i -0.47903273599273 + 0.50773158889098i -0.47903273599273 - 0.50773158889098i -0.56209888019876 + 0.36770724207462i -0.56209888019876 - 0.36770724207462i -0.64273026408505 + 0.07449713941656i -0.64273026408505 - 0.07449713941656i -0.61611009709266 + 0.22255230874526i -0.61611009709266 - 0.22255230874526i]; z = [z,ones(20,1)]; if norm(imag(z(:,1))) == 0 fprintf(' roots multiplicities\n'); fprintf('\n'); fprintf('%25.15f \t \t \t %3g \n', z'); else fprintf(' roots ') fprintf(' \t\t\t\t\t\t multiplicities\n'); fprintf('\n'); fprintf('%22.15f + %22.15f i \t \t %3g \n', ... [real(z(:,1)),imag(z(:,1)),z(:,2)]'); end; SHAR_EOF fi # end of overwriting check if test -f 'toh03.m' then echo shar: will not over-write existing file "'toh03.m'" else cat << "SHAR_EOF" > 'toh03.m' function [p,z] = toh03 % % generalization of K.C. Toh and L. N. Trefethen % k = [1:20]'; z = 10/11 - 2.^(-k); p = poly(z); z = [z,ones(20,1)]; if norm(imag(z(:,1))) == 0 fprintf(' Illconditioned polynomial, constructed with\n'); fprintf(' roots multiplicities\n'); fprintf('\n'); fprintf('%25.15f \t \t \t %3g \n', z'); else fprintf(' roots ') fprintf(' \t\t\t\t\t\t multiplicities\n'); fprintf('\n'); fprintf('%22.15f + %22.15f i \t \t %3g \n', ... [real(z(:,1)),imag(z(:,1)),z(:,2)]'); end; SHAR_EOF fi # end of overwriting check if test -f 'toh04.m' then echo shar: will not over-write existing file "'toh04.m'" else cat << "SHAR_EOF" > 'toh04.m' function [p,z] = toh04 % % generalization of K.C. Toh and L. N. Trefethen % p = poly((10/11)*ones(1,20)); z = [10/11,20]; if norm(imag(z(:,1))) == 0 fprintf(' roots multiplicities\n'); fprintf('\n'); fprintf('%25.15f \t \t \t %3g \n', z'); else fprintf(' roots ') fprintf(' \t\t\t\t\t\t multiplicities\n'); fprintf('\n'); fprintf('%22.15f + %22.15f i \t \t %3g \n', ... [real(z(:,1)),imag(z(:,1)),z(:,2)]'); end; SHAR_EOF fi # end of overwriting check if test -f 'toh05.m' then echo shar: will not over-write existing file "'toh05.m'" else cat << "SHAR_EOF" > 'toh05.m' function [p,z] = toh05 % % generalization of K.C. Toh and L. N. Trefethen % k = [-19:0]'; z = 2.^k; p = poly(z); z = [z,ones(20,1)]; if norm(imag(z(:,1))) == 0 fprintf(' Illconditioned polynomial, constructed with\n'); fprintf(' roots multiplicities\n'); fprintf('\n'); fprintf('%25.15f \t \t \t %3g \n', z'); else fprintf(' roots ') fprintf(' \t\t\t\t\t\t multiplicities\n'); fprintf('\n'); fprintf('%22.15f + %22.15f i \t \t %3g \n', ... [real(z(:,1)),imag(z(:,1)),z(:,2)]'); end; SHAR_EOF fi # end of overwriting check if test -f 'toh06a.m' then echo shar: will not over-write existing file "'toh06a.m'" else cat << "SHAR_EOF" > 'toh06a.m' function [p,z] = toh06a % % generalization of K.C. Toh and L. N. Trefethen % p = ones(1,21); z = [ 0.95557280578614 + 0.29475517441090i 0.95557280578614 - 0.29475517441090i 0.82623877431600 + 0.56332005806362i 0.82623877431600 - 0.56332005806362i 0.62348980185873 + 0.78183148246803i 0.62348980185873 - 0.78183148246803i 0.36534102436640 + 0.93087374864420i 0.36534102436640 - 0.93087374864420i 0.07473009358642 + 0.99720379718118i 0.07473009358642 - 0.99720379718118i -0.98883082622513 + 0.14904226617617i -0.98883082622513 - 0.14904226617617i -0.90096886790242 + 0.43388373911756i -0.90096886790242 - 0.43388373911756i -0.73305187182983 + 0.68017273777092i -0.73305187182983 - 0.68017273777092i -0.50000000000000 + 0.86602540378444i -0.50000000000000 - 0.86602540378444i -0.22252093395631 + 0.97492791218182i -0.22252093395631 - 0.97492791218182i]; z = [z,ones(20,1)]; if norm(imag(z(:,1))) == 0 fprintf(' roots multiplicities\n'); fprintf('\n'); fprintf('%25.15f \t \t \t %3g \n', z'); else fprintf(' roots ') fprintf(' \t\t\t\t\t\t multiplicities\n'); fprintf('\n'); fprintf('%22.15f + %22.15f i \t \t %3g \n', ... [real(z(:,1)),imag(z(:,1)),z(:,2)]'); end; SHAR_EOF fi # end of overwriting check if test -f 'toh06b.m' then echo shar: will not over-write existing file "'toh06b.m'" else cat << "SHAR_EOF" > 'toh06b.m' function [p,z] = toh06a % % generalization of K.C. Toh and L. N. Trefethen % p = ones(1,11); p = conv(p,p); z = [ 0.84125353283118 + 0.54064081745560i 0.84125353283118 - 0.54064081745560i 0.41541501300189 + 0.90963199535452i 0.41541501300189 - 0.90963199535452i -0.95949297361449 + 0.28173255684143i -0.95949297361449 - 0.28173255684143i -0.65486073394529 + 0.75574957435426i -0.65486073394529 - 0.75574957435426i -0.14231483827329 + 0.98982144188093i -0.14231483827329 - 0.98982144188093i]; z = [z,2*ones(10,1)]; if norm(imag(z(:,1))) == 0 fprintf(' roots multiplicities\n'); fprintf('\n'); fprintf('%25.15f \t \t \t %3g \n', z'); else fprintf(' roots ') fprintf(' \t\t\t\t\t\t multiplicities\n'); fprintf('\n'); fprintf('%22.15f + %22.15f i \t \t %3g \n', ... [real(z(:,1)),imag(z(:,1)),z(:,2)]'); end; SHAR_EOF fi # end of overwriting check if test -f 'toh06c.m' then echo shar: will not over-write existing file "'toh06c.m'" else cat << "SHAR_EOF" > 'toh06c.m' function [p,z] = toh06c % % generalization of K.C. Toh and L. N. Trefethen % p = ones(1,6); p = conv(p,p); p = conv(p,p); z = [ 0.50000000000000 + 0.86602540378444i 0.50000000000000 - 0.86602540378444i -1.00000000000000 -0.50000000000000 + 0.86602540378444i -0.50000000000000 - 0.86602540378444i]; z = [z,4*ones(5,1)]; if norm(imag(z(:,1))) == 0 fprintf(' roots multiplicities\n'); fprintf('\n'); fprintf('%25.15f \t \t \t %3g \n', z'); else fprintf(' roots ') fprintf(' \t\t\t\t\t\t multiplicities\n'); fprintf('\n'); fprintf('%22.15f + %22.15f i \t \t %3g \n', ... [real(z(:,1)),imag(z(:,1)),z(:,2)]'); end; SHAR_EOF fi # end of overwriting check if test -f 'triple.m' then echo shar: will not over-write existing file "'triple.m'" else cat << "SHAR_EOF" > 'triple.m' function p = triple(k1,k2,k3) % % test polynomial suggested by Goedecker % square of Fibocacci polynomial % p = poly([0.9*ones(1,k1),ones(1,k2),1.1*ones(1,k3)]); SHAR_EOF fi # end of overwriting check if test -f 'triple01.m' then echo shar: will not over-write existing file "'triple01.m'" else cat << "SHAR_EOF" > 'triple01.m' function [p,z] = triple01 % % test polynomial suggested by Goedecker % p = triple(5,5,5); z = [[0.9,1,1.1]',[5,5,5]']; fprintf(' roots multiplicities\n'); fprintf('\n'); fprintf('%25.15f \t \t \t %3g \n', z'); SHAR_EOF fi # end of overwriting check if test -f 'triple02.m' then echo shar: will not over-write existing file "'triple02.m'" else cat << "SHAR_EOF" > 'triple02.m' function [p,z] = triple02 % % test polynomial suggested by Goedecker % p = triple(10,10,10); z = [[0.9,1,1.1]',[10,10,10]']; fprintf(' roots multiplicities\n'); fprintf('\n'); fprintf('%25.15f \t \t \t %3g \n', z'); SHAR_EOF fi # end of overwriting check if test -f 'triple03.m' then echo shar: will not over-write existing file "'triple03.m'" else cat << "SHAR_EOF" > 'triple03.m' function [p,z] = triple03 % % test polynomial suggested by Goedecker % p = triple(18,10,16); z = [[0.9,1,1.1]',[18,10,16]']; fprintf(' roots multiplicities\n'); fprintf('\n'); fprintf('%25.15f \t \t \t %3g \n', z'); fprintf(' multroot works when tol = 1.0d-9'); SHAR_EOF fi # end of overwriting check if test -f 'triple04.m' then echo shar: will not over-write existing file "'triple04.m'" else cat << "SHAR_EOF" > 'triple04.m' function [p,z] = triple04 % % test polynomial suggested by Goedecker % p = triple(20,15,10); z = [[0.9,1,1.1]',[20,15,10]']; fprintf(' roots multiplicities\n'); fprintf('\n'); fprintf('%25.15f \t \t \t %3g \n', z'); fprintf(' multroot should work when tol = 1.0d-12'); SHAR_EOF fi # end of overwriting check if test -f 'twin01.m' then echo shar: will not over-write existing file "'twin01.m'" else cat << "SHAR_EOF" > 'twin01.m' function [p,z] = twin01() k = 4; p = poly([-0.2*ones(1,k),0.39*ones(1,k),0.40*ones(1,k)]); z = [-0.2, k; 0.39, k; 0.4, k]; fprintf(' roots multiplicities\n'); fprintf('\n'); fprintf('%25.15f \t \t \t %3g \n', z'); SHAR_EOF fi # end of overwriting check if test -f 'twin02.m' then echo shar: will not over-write existing file "'twin02.m'" else cat << "SHAR_EOF" > 'twin02.m' function [p,z] = twin02() k = 8; p = poly([-0.2*ones(1,k),0.39*ones(1,k),0.40*ones(1,k)]); z = [-0.2, k; 0.39, k; 0.4, k]; fprintf(' roots multiplicities\n'); fprintf('\n'); fprintf('%25.15f \t \t \t %3g \n', z'); SHAR_EOF fi # end of overwriting check if test -f 'twin03.m' then echo shar: will not over-write existing file "'twin03.m'" else cat << "SHAR_EOF" > 'twin03.m' function [p,z] = twin03() k = 12; %p = poly([-0.2*ones(1,k),0.39*ones(1,k),0.40*ones(1,k)]); p = [1.0 -7.080 22.95060000000000 -44.65322000000000 57.09185295000000 ... -48.65042284080000 25.44460281708400 -4.460667417035280 ... -4.351465724097838 3.804141115315636 -1.063009474987107 ... -0.2419285008678686 0.2814258796961311 -0.06584793423745971 ... -0.01673633807069614 0.01282876495641771 -0.001513177304657989 ... -0.0008994022741137765 0.0003209100515379872 ... 0.00001075739874607803 -0.00002462942659459002 ... 0.3041172258562284e-5 0.1002335654574647e-5 ... -0.2782381055469692e-6 -0.1610441794212416e-7 ... 0.1297429644396243e-7 -0.5130969955638040e-9 ... -0.3793232099259401e-9 0.3906127656218143e-10 ... 0.7116422374880351e-11 -0.1160791690488553e-11 ... -0.8086286771997257-13 0.1980701247647230e-13 ... 0.4633473519647368e-15 -0.1928476904073257e-15 ... -0.6545032025137130e-18 0.8508541632678268e-18]; p = poly([-0.2*ones(1,k),0.39*ones(1,k),0.40*ones(1,k)]); z = [-0.2, k; 0.39, k; 0.4, k]; fprintf(' roots multiplicities\n'); fprintf('\n'); fprintf('%25.15f \t \t \t %3g \n', z'); SHAR_EOF fi # end of overwriting check if test -f 'twin04.m' then echo shar: will not over-write existing file "'twin04.m'" else cat << "SHAR_EOF" > 'twin04.m' function [p,z] = twin04() k = 16; p = poly([-0.2*ones(1,k),0.39*ones(1,k),0.40*ones(1,k)]); z = [-0.2, k; 0.39, k; 0.4, k]; fprintf(' roots multiplicities\n'); fprintf('\n'); fprintf('%25.15f \t \t \t %3g \n', z'); SHAR_EOF fi # end of overwriting check if test -f 'uhlig01.m' then echo shar: will not over-write existing file "'uhlig01.m'" else cat << "SHAR_EOF" > 'uhlig01.m' function [p,z] = uhlig01 % % test polynomial used by F. Uhlig % a = 0.01; p1 = [1,0,0,0,-a^4]; p2 = poly([a,a,a,a]); p = conv(p1,p2); z = [-0.01000000000000 - 0.00000000000000i -0.00000000000000 + 0.01000000000000i -0.00000000000000 - 0.01000000000000i 0.01000000000000 - 0.00000000000000i]; z = [z, [1,1,1,5]']; if norm(imag(z(:,1))) == 0 fprintf(' roots multiplicities\n'); fprintf('\n'); fprintf('%25.15f \t \t \t %3g \n', z'); else fprintf(' roots ') fprintf(' \t\t\t\t\t\t multiplicities\n'); fprintf('\n'); fprintf('%22.15f + %22.15f i \t \t %3g \n', ... [real(z(:,1)),imag(z(:,1)),z(:,2)]'); end; SHAR_EOF fi # end of overwriting check if test -f 'uhlig02.m' then echo shar: will not over-write existing file "'uhlig02.m'" else cat << "SHAR_EOF" > 'uhlig02.m' function [p,z] = uhlig02 % % test polynomial used by F. Uhlig % a = 0.001; p1 = [1,0,0,0,-a^4]; p2 = poly([a,a,a,a]); p = conv(p1,p2); z = [ -0.00000000000000 + 0.00100000000000i -0.00000000000000 - 0.00100000000000i -0.00100000000000 + 0.00000000000000i 0.00100000000000 + 0.00000000000000i]; z = [z, [1,1,1,5]']; if norm(imag(z(:,1))) == 0 fprintf(' roots multiplicities\n'); fprintf('\n'); fprintf('%25.15f \t \t \t %3g \n', z'); else fprintf(' roots ') fprintf(' \t\t\t\t\t\t multiplicities\n'); fprintf('\n'); fprintf('%22.15f + %22.15f i \t \t %3g \n', ... [real(z(:,1)),imag(z(:,1)),z(:,2)]'); end; SHAR_EOF fi # end of overwriting check if test -f 'uhlig03.m' then echo shar: will not over-write existing file "'uhlig03.m'" else cat << "SHAR_EOF" > 'uhlig03.m' function [p,z] = uhlig03 % % test polynomial used by F. Uhlig % r = [(3/11)*ones(1,12),11/3,11/3,(2*i/7)*ones(1,4),... (2.5+i/4)*ones(1,2),1/4]; p = poly(r); z = [3/11,12; 11/3, 2; 2*i/7, 4; 2.5+i/4, 2; 1/4, 1]; if norm(imag(z(:,1))) == 0 fprintf(' roots multiplicities\n'); fprintf('\n'); fprintf('%25.15f \t \t \t %3g \n', z'); else fprintf(' roots ') fprintf(' \t\t\t\t\t\t multiplicities\n'); fprintf('\n'); fprintf('%22.15f + %22.15f i \t \t %3g \n', ... [real(z(:,1)),imag(z(:,1)),z(:,2)]'); end; SHAR_EOF fi # end of overwriting check if test -f 'uhlig04.m' then echo shar: will not over-write existing file "'uhlig04.m'" else cat << "SHAR_EOF" > 'uhlig04.m' function [p,z] = uhlig04 % % test polynomial used by F. Uhlig % r = [(3/11)*ones(1,12),11/3,11/3,(2*i/7)*ones(1,4),... (2.5+i/4)*ones(1,2),1/8]; p = poly(r); z = [3/11,12; 11/3, 2; 2*i/7, 4; 2.5+i/4, 2; 1/8, 1]; if norm(imag(z(:,1))) == 0 fprintf(' roots multiplicities\n'); fprintf('\n'); fprintf('%25.15f \t \t \t %3g \n', z'); else fprintf(' roots ') fprintf(' \t\t\t\t\t\t multiplicities\n'); fprintf('\n'); fprintf('%22.15f + %22.15f i \t \t %3g \n', ... [real(z(:,1)),imag(z(:,1)),z(:,2)]'); end; SHAR_EOF fi # end of overwriting check if test -f 'uhlig05.m' then echo shar: will not over-write existing file "'uhlig05.m'" else cat << "SHAR_EOF" > 'uhlig05.m' function [p,z] = uhlig05 % % test polynomial used by F. Uhlig % p = poly([-1*ones(1,6),2,2]); z = [-1, 6; 2, 2]; if norm(imag(z(:,1))) == 0 fprintf(' roots multiplicities\n'); fprintf('\n'); fprintf('%25.15f \t \t \t %3g \n', z'); else fprintf(' roots ') fprintf(' \t\t\t\t\t\t multiplicities\n'); fprintf('\n'); fprintf('%22.15f + %22.15f i \t \t %3g \n', ... [real(z(:,1)),imag(z(:,1)),z(:,2)]'); end; SHAR_EOF fi # end of overwriting check if test -f 'uhlig06.m' then echo shar: will not over-write existing file "'uhlig06.m'" else cat << "SHAR_EOF" > 'uhlig06.m' function [p,z] = uhlig06 % % test polynomial used by F. Uhlig % p = [1,0,0,0,0,0,0,0,-1]; z = [-1.00000000000000 -0.70710678118655 + 0.70710678118655i -0.70710678118655 - 0.70710678118655i 0 + 1.00000000000000i 0 - 1.00000000000000i 1.00000000000000 0.70710678118655 + 0.70710678118655i 0.70710678118655 - 0.70710678118655i]; z = [z,ones(8,1)]; if norm(imag(z(:,1))) == 0 fprintf(' roots multiplicities\n'); fprintf('\n'); fprintf('%25.15f \t \t \t %3g \n', z'); else fprintf(' roots ') fprintf(' \t\t\t\t\t\t multiplicities\n'); fprintf('\n'); fprintf('%22.15f + %22.15f i \t \t %3g \n', ... [real(z(:,1)),imag(z(:,1)),z(:,2)]'); end; SHAR_EOF fi # end of overwriting check cd .. if test ! -d 'Src' then mkdir 'Src' fi cd 'Src' if test -f 'backsub.m' then echo shar: will not over-write existing file "'backsub.m'" else cat << "SHAR_EOF" > 'backsub.m' function x = backsub(A,b) % % assume A is an upper triangular matrix % backsub performs backward substitution to solve Ax=b % syntax: >>x = backsub(A,b) % n = size(A,2); x = zeros(n,1); B = diagproc(A); x(n) = b(n)/B(n,n); for k = (n-1):-1:1 x(k) = (b(k) - B(k,k+1:n)*x(k+1:n))/B(k,k); end; SHAR_EOF fi # end of overwriting check if test -f 'cauchymt.m' then echo shar: will not over-write existing file "'cauchymt.m'" else cat << "SHAR_EOF" > 'cauchymt.m' function A = cauchymt(f,k) % % program to generate a k-column Cauchy matrix of polynomial f % n = length(f); m = n + k - 1; A = zeros(m,k); for j = 1:k A(j:j+n-1,j) = f.'; end; SHAR_EOF fi # end of overwriting check if test -f 'diagproc.m' then echo shar: will not over-write existing file "'diagproc.m'" else cat << "SHAR_EOF" > 'diagproc.m' function B = diagproc(A) % making zero diagonal entries of A nonzero n = size(A,2); d = diag(A); I = find(d); B = A; for j = 1:n if d(j) == 0 [l,k] = min(abs(I-j)); B(j,j) = eps*d(I(k)); end end SHAR_EOF fi # end of overwriting check if test -f 'fixend' then echo shar: will not over-write existing file "'fixend'" else cat << "SHAR_EOF" > 'fixend' #!/usr/local/bin/perl use strict; use warnings; open(OLD,"<$ARGV[0]"); my @text = ; close OLD; while($text[$#text] =~ /^\s*$/){ $#text--; } chomp $text[$#text]; $text[$#text] .= "\n"; open (NEW, ">New/$ARGV[0]"); print NEW @text; close NEW SHAR_EOF fi # end of overwriting check if test -f 'forsub.m' then echo shar: will not over-write existing file "'forsub.m'" else cat << "SHAR_EOF" > 'forsub.m' function x = forsub(A,b) % % assume A is an lower triangular matrix % forsub performs forward substitution to solve Ax=b % n = size(A,1); x = zeros(n,1); B = diagproc(A); x(1) = b(1)/B(1,1); for k = 2:n x(k) = (b(k) - B(k,1:k-1)*x(1:k-1))/B(k,k); end; SHAR_EOF fi # end of overwriting check if test -f 'gcdroot.m' then echo shar: will not over-write existing file "'gcdroot.m'" else cat << "SHAR_EOF" > 'gcdroot.m' function [z, bkerr] = gcdroot( p, tol, thresh, growf) % GCDROOT calculates the multiplicity structure and initial root % approximation of a polynomial p given as an (n+1)-row vector % p = (a_1, ..., a_n+1) % % GcdRoot is developed by Zhonggang Zeng (email: zzeng@neiu.edu) % This code is freely released for research exchange only. The author % is not responsible for any demage from using this code. % % INPUT : p -- polynomial coefficients. % tol -- initial residual tolerance (optional); % thresh -- zero singular value shreshold (optional); % gamma -- residual growth factor (optional) % % OUTPUT : z -- roots of polynomial p and multiplicities: % z(:,1) = all distinct roots % z(:,2) = corresponding multiplicities % bkerr -- backward error of z % % CALL : The simplest way to call gcdroot is: % >> z = gcdroot(p) % One can also set the 1, 2, or 3 parameters such as % >> [z,bkerr] = gcdroot(p,1e-12,1e-8,10); % >> z = gcdroot(p,1e-15); % >> [z,bke] = gcdroot(p,1e-10,1e-7); % etc. % % set default parameters if not provided if nargin == 1 tol = 1.e-10; thresh = tol*100; gamma = 100; elseif nargin == 2 thresh = tol*100; gamma = 100; elseif nargin == 3 gamma = 100; else gamma = growf; end; drop = 5.0d-5; % make the polynomial monic if p(1) == 0, p = p(min(find(p)):n+1); n = length(p)-1; end; if p(1) ~= 1, f = p/(p(1)); end; n = length(p)-1; % get polynomial degree q = polyder(p)/n; % the derevative ff = p; gg = q; % back up the polynomial nf = norm(ff,inf); % the largest coefficient mx = n; wtol = tol; s0 = 0; s = 0; wthrh = thresh; k = n; % the degree of working polynomial while k >= 1 if k == 1 % the polynomial is linear, GCD = 1 h = 1; u = f; v = 1; m = 1; else U = []; R = []; % initialize input of sylup for m = 1:k % when m = k, then gcd = 1 if m == k, h = 1; u = ff; break; end; % Update the Sylvester discriminant matrix with QR decomp. [U,R] = sylup(ff,gg,m,U,R); s0 = s; % Compute the smallest singular value [s,x] = uminsv(R,tol); if s < wthrh*nf | m == mx | s < drop*s0 % a gcd is suspected, get the triplet u0 = x(1:2:2*m+1).'/x(1); v0 = x(2:2:2*m).'/x(2); g0 = lsqdiv(ff,u0); g0 = g0.'/g0(1); % refine and confirm the gcd [h,u,v,rec] = zgcdgn0(ff, gg, g0, u0, v0); if rec < wtol | m == mx % gcd is confirmed wtol = max([wtol,rec*gamma]); wthrh = max([wthrh,(s/nf)*100]); break; end; end; end; end; if k == n % the first gcd, all distinct roots z = roots(u); l = ones(m,1); id = [1:m]; lid = m; if m == 1, l = [n]; break; end; else t = roots(u); % match roots (avoid matching the same root twice) id0 = []; lid0 = 0; for j = 1:m [s,jj] = min(abs(z(id) - t(j))); l(id(jj)) = l(id(jj)) + 1; id0 = [id0,id(jj)]; lid0 = lid0+1; id(jj) = id(lid); lid = lid-1; id = id(1:lid); end; id = id0; lid = lid0; % if there is only one root left, done if m == 1, l(id(1)) = l(id(1)) + k - 1; end; end; if m > 1, k = k - m; else, k = 0; break; end; if k > 0 f = h; g = polyder(f)/k; ff = f; gg = g; nf = norm(ff); mx = m; end; end; ff = [1]; ll = l; % form the base polynomial mm = max(ll); [yy,jj] = sort(abs(z)); y = z(jj); ll = l(jj); for kk = 1:mm id = find(ll>0); ff = conv(ff,poly(y(id))); ll = ll - 1; end; z = [z, l]; % solution w = ones(1,length(p)); for j = 1:length(p), if abs(p(j))>1, w(j)=1/abs(p(j));end,end; bkerr = norm( (ff-p).*w , inf ); SHAR_EOF fi # end of overwriting check if test -f 'hessqr.m' then echo shar: will not over-write existing file "'hessqr.m'" else cat << "SHAR_EOF" > 'hessqr.m' function [B,t] = hessqr(A) % % assume A is a Hessenberg matrix % output B -- upper triangular % c -- rotation used % [m,n] = size(A); if m < n, return; end; B = A; for j = 1:n if j < m d = sqrt(B(j,j)^2+B(j+1,j)^2); c = B(j,j)/d; s = B(j+1,j)/d; T = [c,s;-s,c]; B(j:j+1,j:n) = T*B(j:j+1,j:n); t(1,j) = c; t(2,j) = s; end; end; SHAR_EOF fi # end of overwriting check if test -f 'hqrt.m' then echo shar: will not over-write existing file "'hqrt.m'" else cat << "SHAR_EOF" > 'hqrt.m' function z = hqrt(t,b) % % transformation using the t from hessqr % [m,n] = size(b); z = b; k = size(t,2); for j = 1:k T = [t(1,j),t(2,j); -t(2,j), t(1,j)]; z(j:j+1,:)=T*z(j:j+1,:); end; SHAR_EOF fi # end of overwriting check if test -f 'lsqdiv.m' then echo shar: will not over-write existing file "'lsqdiv.m'" else cat << "SHAR_EOF" > 'lsqdiv.m' function u = lsqdiv( f, v, w) % % polynomial division u = f/v using least squares method % generate a k-column Cauchy matrix of polynomial f % m = length(f); k = length(v); n = m - k + 1; A = zeros(m,n); if nargin == 2 % construct weights, if not provided w = ones(m,1); for j = 1:m, if abs(f(j)) > 1, w(j) = 1/abs(f(j)); end; end end; % % construct the augmented Cauchy matrix % kk = k-1; for j = 1:n, A(j:j+kk, j) = v.'; end; A = [A,f.']; for j = 1:m, A(j,:) = A(j,:)*w(j); end; % scale % % QR decomposition of the Cauchy matrix % U = []; for j = 1:n x = A(j:j+kk, j); s = norm(x); if x(1) ~= 0, s = s*x(1)/abs(x(1)); end; u = x; u(1) = u(1) + s; u = u/norm(u); A(j,j) = -s; A(j+1:j+kk, j) = zeros(kk,1); ll = [j+1:min(j+kk,n), n+1]; for l = ll s = 2*u'*A(j:j+kk, l); A(j:j+kk, l) = A(j:j+kk, l) - s*u; end; U = [U,u]; end; % % banded backward substitution % u = zeros(n,1); b = A(1:n,n+1); u(n) = b(n)/A(n,n); % % triangular part % for j = (n-1):-1:max(1,n-kk) u(j) = (b(j) - A(j,j+1:n)*u(j+1:n))/A(j,j); end; % % banded part % if n-k >=1 for j = n-k:-1:1 u(j) = (b(j)-A(j,j+1:j+k)*u(j+1:j+k))/A(j,j); end; end; SHAR_EOF fi # end of overwriting check if test -f 'mroot.m' then echo shar: will not over-write existing file "'mroot.m'" else cat << "SHAR_EOF" > 'mroot.m' function [z,ferr,bkerr,pjcnd,job] = mroot( p, tol, thresh, gamma ) % MRoot calculates roots and multiplicities of a polynomial p, % p(x) = a_1 x^n + a_2 x^n-1 + ... + a_n x + a_n+1, % given as an (n+1)-row vector p = (a_1, ..., a_n+1) % % MRoot is developed by Zhonggang Zeng (email: zzeng@neiu.edu) % This code is freely released for research exchange only. The author % is not responsible for any demage from using this code. % % INPUT : p -- polynomial coefficients. % tol -- initial residual tolerance (optional); % thresh -- zero singular value shreshold (optional); % gamma -- residual growth factor (optional) % % OUTPUT : z -- roots of polynomial p and multiplicities: % z(:,1) = distinct roots % z(:,2) = corresponding multiplicities % pjcnd -- the structure-preserving condition number % bkerr -- backward error of z % ferr -- the estimated forward error % job -- status flag. job = 1 when a % nontrivial multiplicity structure is found, % job = 0 otherwise. If job = 0, the other % output items should be discarded. % % CALL: >> [z,ferr,berr,cond,job] = multroot(p, tol,thresh,gamma) % % set default parameters if not provided if nargin == 1 tol = 1.e-10; thresh = tol*100; gamma = 100; elseif nargin == 2 thresh = tol*100; gamma = 100; elseif nargin == 3 gamma = 100; end; % % transpose if p is a column vector % [m,n] = size(p); if m > n, p = p'; end; % % clear leading/trailing zeros % n = length(p); q = p; if q(1) == 0 | q(n) == 0 jj = find(p); j1 = min(jj); j2 = max(jj); q = p(j1:j2); q = q/q(1); else j1 = 1; j2 = n; end; % % scaling % m = length(q)-1; k = ceil(log(abs(q(m+1)))/(m*log(2))); c = 2^(-k); q = q.*(c.^[0:m]); % [y,bke] = gcdroot(q, tol, thresh, gamma); if bke < 1.0d-2 [z,bkerr,pjcnd,job] = pejroot(q,y(:,1).',y(:,2)'); if job == 1 if j2 < n z = [z;0,n-j2]; end; % % show off results % z(:,1) = z(:,1)/c; fprintf('\n'); fprintf(' !!!THE COMPUTATION IS SUCCESSFUL!!!\n'); fprintf('\n'); fprintf('THE PEJORATIVE CONDITION NUMBER: \t\t\t\t %g \n',pjcnd); fprintf('THE BACKWARD ERROR: %6.2e \n',bkerr); fprintf('THE ESTIMATED FORWARD ROOT ERROR: %6.2e \n',... 2*bkerr*pjcnd); fprintf('\n'); if norm(imag(z(:,1))) == 0 fprintf(' computed roots multiplicities\n'); fprintf('\n'); fprintf('%25.15f \t \t \t %3g \n', z'); else fprintf(' computed roots ') fprintf(' \t\t\t\t\t\t multiplicities\n'); fprintf('\n'); fprintf('%22.15f + %22.15f i \t \t %3g \n', ... [real(z(:,1)),imag(z(:,1)),z(:,2)]'); end; else z = y; bkerr = bke; pjcnd = 1; ferr = 0; job = 0; end; else z = y; bkerr = bke; pjcnd = 1; ferr = 0; job = 0; end; SHAR_EOF fi # end of overwriting check if test -f 'multroot.m' then echo shar: will not over-write existing file "'multroot.m'" else cat << "SHAR_EOF" > 'multroot.m' function [z,f_err, b_err, cond] = ... multroot( p, tol, thresh, growf ) % MultRoot calculates roots and multiplicities of a polynomial p, % p(x) = a_1 x^n + a_2 x^n-1 + ... + a_n x + a_n+1, % given as an (n+1)-row vector p = (a_1, ..., a_n+1) % % MultRoot is developed by Zhonggang Zeng (email: zzeng@neiu.edu) % This code is freely released for research exchange only. The author % is not responsible for any demage from using this code. % % INPUT : p -- polynomial coefficients. % tol -- initial residual tolerance (optional); % thresh -- zero singular value shreshold (optional); % growf -- residual growth factor (optional) % % OUTPUT : z -- roots of polynomial p and multiplicities: % z(:,1) = distinct roots % z(:,2) = corresponding multiplicities % f_err -- the estimated forward error % b_err -- backward error of z % cond -- the structure-preserving condition number % % CALL : The simplest way to call: % >> z = multroot(p); % For expert users, 1, 2 or 3 parameters can be entered % For example: % >> [z,ferr,berr,cond] = multroot(p, 1e-12,12-9,10); % >> [z,cond] = multroot(p,1e-9); % etc. % % set default parameters if not provided if nargin == 1 tol = 1.e-10; thresh = tol*100; gamma = 100; elseif nargin == 2 thresh = tol*100; gamma = 100; elseif nargin == 3 gamma = 100; else gamma = growf; end; % % transpose if p is a column vector % [m,n] = size(p); if m > n, p = p.'; end; % % clear leading/trailing zeros % n = length(p); q = p; if q(1) == 0 | q(n) == 0 jj = find(p); j1 = min(jj); j2 = max(jj); q = p(j1:j2); q = q/q(1); else j1 = 1; j2 = n; end; % % scaling % m = length(q)-1; %k = ceil(log(abs(q(m+1)))/(m*log(2))); k = log(abs(q(m+1)))/(m*log(2)); c = 2^(-k); %c = 1; q = q.*(c.^[0:m]); % [y,bke] = gcdroot(q, tol, thresh, gamma); if bke < 1.0d-2 & max(y(:,2)) > 1 [z,bkerr,pjcnd,job] = pejroot(q,y(:,1).',y(:,2)'); if job == 1 if j2 < n z = [z;0,n-j2]; end; % % show off results % z(:,1) = z(:,1)/c; pjcnd = spcond(z); else [z,bkerr,pjcnd] = rtsinfo(p); end; else [z,bkerr,pjcnd] = rtsinfo(p); end; ferr = 2*bkerr*pjcnd; fprintf('\n'); fprintf('THE CONDITION NUMBER: %g \n',pjcnd); fprintf('THE BACKWARD ERROR: %6.2e \n',bkerr); fprintf('THE ESTIMATED FORWARD ROOT ERROR: %6.2e \n',... 2*bkerr*pjcnd); fprintf('\n'); if norm(imag(z(:,1))) == 0 fprintf(' computed roots multiplicities\n'); fprintf('\n'); fprintf('%25.15f %3g \n', z'); else fprintf(' computed roots ') fprintf(' multiplicities\n'); fprintf('\n'); fprintf('%22.15f + %22.15f i %3g \n', ... [real(z(:,1)),imag(z(:,1)),z(:,2)]'); end; f_err = ferr; b_err = bkerr; cond = pjcnd; SHAR_EOF fi # end of overwriting check if test -f 'pejroot.m' then echo shar: will not over-write existing file "'pejroot.m'" else cat << "SHAR_EOF" > 'pejroot.m' function [y, bkerr, pjcnd, job ] = ... pejroot( f, y0, l, noi, tol) % PEJROOT calculates (multiple) roots of polynomial % % This code is freely released for research exchange only. The author % is not responsible for any demage caused by using this code. % % Calling syntax: The simplest way to call is % % © z = pejroot(f,y,m) % % where the input items are % f --- (row vector) the polynomial % y --- (row vector) the initial iterate (of the roots) % m --- (row vector) the multiplicity structure % % For more advanced usage: % % © [z, e, c] = pejroot(f, y, l, noi, tol) % % The output % z --- (matrix) distinct roots (1st column) and corresponding % multiplicities (2nd column) % e --- (scalar) the backward error of the roots % c --- (scalar) the pejorative condition number % % % Additional input parameters % % noi --- (integer) the number of iteration allowed % (default: 10) % tol --- (numeric) the error tolerance % (default: 1.0d-8) if nargin == 3, noi = 10; tol = 1.e-6; end; m = length(l); % number of variables n = sum(l); % number of equations if abs(f(1)) ~= 0 % make the polynomial monic f = f / f(1); else, jj = find(f); j = min(jj); f = f(j:length(f)); end; if length(f) ~= n+1 % exit if the input is wrong fprintf('In put error'); job = 1; return; end; % sort initial values to enhance accuracy % it is interesting, sorting really improves accuracy [yy,jj] = sort(1./abs(y0)); y0 = y0(jj); l = l(jj); job = 0; % initialze job y = conj(y0'); % pass the initial iterate h = conj(f(2:n+1)'); % make the RHS delta = zeros(1,noi); % space for sizes of the correction bcker = zeros(1,noi); % space for backward errors w = ones(n,1); % set weight for j = 1:n if abs(f(j+1)) >= 1.0 w(j) = abs(1.0/f(j+1)); end end; for k = 1:noi % evaluate the coefficient operator and its Jacobian Df = zeros(n,m); % open the space for A ff = [1]; ll = l - ones(1,m); % form the base polynomial mm = max(ll); lll = ll; for kk = 1:mm id = find(lll>0); ff = conv(ff,poly(y(id))); lll = lll - 1; end; for j = 1:m % remaining polynomial multiplication if m > 1 id = [1:(j-1),(j+1):m]; g = poly(y(id)); else g = [1]; end; g = conv(-l(j)*g,ff); Df(:,j) = conj(g'); end g = conv(-g/l(m), [1,-y(m)]); % evalulate the coef. operator c = conj(g(2:n+1)'); b = c - h; % RHS of the polyn. system d = scalsq(Df,b,w); delta(k) = norm(d,2); bcker(k) = norm(w.*b,inf); if delta(k) < tol, job = 1; end % convergence criterion 1 if k > 1 if delta(k) > delta(k-1) & bcker(k) > bcker(k-1) if job == 1, bkerr = bcker(k-1); break; end; elseif delta(k) < delta(k-1) % criterion 2 if delta(k)^2/(delta(k-1)-delta(k)) < tol, job = 1; end end end y = y - d; % correct the roots bkerr = bcker(k); A = Df; end if job == 1 s = svd(A); pjcnd = 1/s(m); % get pej. cond. number [ll,jj] = sort(l); y = [y(jj),l(jj)']; % sort by multiplicities else pjcnd = 0; end; SHAR_EOF fi # end of overwriting check if test -f 'rtsinfo.m' then echo shar: will not over-write existing file "'rtsinfo.m'" else cat << "SHAR_EOF" > 'rtsinfo.m' function [z,bkerr,cond] = rtsinfo(p) % roots, backward error and condition number using "roots" n = length(p)-1; z = [roots(p),ones(n,1)]; p = p/p(1); cond = spcond(z); [yy,jj]=sort(abs(z)); y = z(jj); ff = [1]; w = ones(1,n+1); for k = 1:n ff = conv(ff,poly(y(k))); if abs(p(k+1)) > 1 w(k+1) = 1/abs(p(k+1)); end end bkerr = norm( (ff-p).*w, inf); SHAR_EOF fi # end of overwriting check if test -f 'scalsq.m' then echo shar: will not over-write existing file "'scalsq.m'" else cat << "SHAR_EOF" > 'scalsq.m' function x = scalsq(A,b,w) % % Solving least squares problem with iterative refinement % [m,n] = size(A); if nargin == 2 w = ones(m,1); for j = 1:m if abs(b(j)) > 1, w(j) = 1/abs(b(j)); end; end end; for j = 1:m A(j,:) = A(j,:)*w(j); end; b = b.*w; [Q,R] = qr(A); d = Q'*b; S = R(1:n,1:n); x = backsub(S,d(1:n)); % one step refinement bb = [b;zeros(n,1)]; B = [eye(m),A;A',zeros(n,n)]; r = b - A*x; %for j = 1:3 rr = bb - B*[r;x]; %disp(norm(rr)); s = Q'*rr(1:m); c = forsub(S',rr(m+1:m+n)); c2 = backsub(S,s(1:n)-c); c1 = Q*[c;s(n+1:m)]; %r = r + c1; x = x + c2; %end; SHAR_EOF fi # end of overwriting check if test -f 'spcond.m' then echo shar: will not over-write existing file "'spcond.m'" else cat << "SHAR_EOF" > 'spcond.m' function cond = ... spcond(z) % SPCond calculates the struture preserving condition number % of roots z(:,1) with multiplicity structure z(:,2) % % Input: z --- mx2 matrix, z(:,1) is the root vector % z(:,2) is the multiplicity structure % % Output: cond -- the structure preserving condition number % % Syntax: >> cond = spcond(z) % m = size(z,1); % number of distinct roots n = sum(z(:,2)); % degree of the polynomial % sort [yy,jj] = sort(1./abs(z(:,1))); y = z(jj,1); l = z(jj,2)'; ff = [1]; ll = l - ones(1,m); % form the base polynomial mm = max(ll); lll = ll; for kk = 1:mm id = find(lll>0); ff = conv(ff,poly(y(id))); lll = lll - 1; end; for j = 1:m % remaining polynomial multiplication if m > 1 id = [1:(j-1),(j+1):m]; g = poly(y(id)); else g = [1]; end; g = conv(-l(j)*g,ff); J(:,j) = conj(g'); end g = conv(-g/l(m), [1,-y(m)]); % evalulate the coef. operator c = conj(g(2:n+1)'); for j = 1:n % scale J if abs(c(j)) >= 1.0 s = abs(1.0/c(j)); J(j,:) = J(j,:)*s; end end; [s,x] = zminsv(J,1.0e-16); % finding smallest singular value cond = 1/s; SHAR_EOF fi # end of overwriting check if test -f 'sylmat.m' then echo shar: will not over-write existing file "'sylmat.m'" else cat << "SHAR_EOF" > 'sylmat.m' function A = sylmat(g,u,v) % % Subroutine of gcdgn % mg = length(g)-1; mu = length(u)-1; mv = length(v)-1; np = mg + mu; nq = mg + mv; m = mg + mu + mv; n = np + nq; A = zeros(n, m); for j = 1:mg A(j:j+mu,j) = u.'; A(np+j:np+j+mv,j) = v.'; end; for j = 1:mu A(j:j+mg,j+mg) = g.'; end; for j = 1:mv A(np+j:np+j+mg, j+mg+mu) = g.'; end SHAR_EOF fi # end of overwriting check if test -f 'sylup.m' then echo shar: will not over-write existing file "'sylup.m'" else cat << "SHAR_EOF" > 'sylup.m' function [U,R] = sylup(f,g,k,U,R) % % updating the Sylvester discriminant matrix and QR decomposition % n = length(f); if k == 1 % % construct S_1 % S = zeros(n,3); S(1:n-1,1) = g.'; S(:,2) = f.'; S(2:n,3) = g.'; % % triangularize it % U = zeros(n,3); R = S; for j = 1:3 if R(j,j) ~= 0 s = norm(R(j:n,j))*R(j,j)/abs(R(j,j)); else s = norm(R(j:n,j)); end; u = R(j:n,j); u(1) = u(1) + s; u = u/norm(u); U(j:n,j) = u; R(j:n,j:3) = R(j:n,j:3) - 2*u*(u'*R(j:n,j:3)); end; else % % extend S_j % R = [R; zeros(1,2*k-1)]; U = [U; zeros(1,2*k-1)]; R = [R, [zeros(k-1,1);f.'],[zeros(k,1);g.']]; U = [U, zeros(n+k-1,2)]; % % update previous transformation % jj = [2*k, 2*k+1]; for j = 1:2*k-1 m = n+max(0,floor((j-2)/2)); u = U(j:m,j); R(j:m,jj) = R(j:m,jj) - 2*u*(u'*R(j:m,jj)); end; % % additional transformations % m = n+k-1; for j = 2*k:2*k+1 if R(j,j) ~= 0 s = norm(R(j:m,j))*R(j,j)/abs(R(j,j)); else s = norm(R(j:m,j)); end; u = R(j:m,j); u(1) = u(1) + s; t = norm(u); if t ~= 0, u = u/t; end; U(j:m,j) = u; R(j:m,j:2*k+1) = R(j:m,j:2*k+1) - 2*u*(u'*R(j:m,j:2*k+1)); end; end; SHAR_EOF fi # end of overwriting check if test -f 'sylves.m' then echo shar: will not over-write existing file "'sylves.m'" else cat << "SHAR_EOF" > 'sylves.m' function A = sylves(f,g,k) % % The k-th sylvester matrix of f(x), f'(x) = g(x) % l = length(f); m = l+k-1; n = 2*k+1; A = zeros(m,n); for j = 1:k+1; A(j:j+l-2,j) = g'; end; for j = 1:k A(j:j+l-1,j+k+1) = f'; end; SHAR_EOF fi # end of overwriting check if test -f 'uminsv.m' then echo shar: will not over-write existing file "'uminsv.m'" else cat << "SHAR_EOF" > 'uminsv.m' function [s,x] = uminsv(R,tol,nit) % % Find mininum singular value of a triangular matrix R % if nargin == 2, nit = 3; end; [m,n] = size(R); % get the dimensions of R scale = norm(R,inf); % get the magnitude of rows as scaler a = 2*rand(1,n)-1; % random initial vector a = scale*a/norm(a); % set the first row b = [scale;zeros(m,1)]; [T,trans] = hessqr([a;R]); % Hessenberg QR decomp. of stacked matrix z = hqrt(trans,b); % same Q on b x = backsub(T(1:n,1:n),z(1:n)); x = x/norm(x); % geting the new vector r = [scale*x';R]*x-b; ss = [norm(r(2:m+1))]; cr = []; for k = 1:nit [T,trans] = hessqr([2*scale*x';R]); z = hqrt(trans,r); u = backsub(T(1:n,1:n),z(1:n)); y = x - u; y = y / norm(y); r = [scale*y';R]*y-b; s = norm(r(2:m+1)); ss = [ss,s]; cr = [cr,norm(x-y)]; %disp(cr(k)); if k == 1 if cr(1) < tol, break; end; else if cr(k) < cr(k-1) if cr(k)^2/(cr(k-1)-cr(k)) 'zgcdgn0.m' function [g,u,v,res] = zgcdgn0(p, q, g0, u0, v0) % % Finds extended GCD of polynomial p and q by Gauss-Newton % iteration, such that % % conv(g,u) = p, conv(g,v) = q % % Calling syntax: % [g,u,v,res] = gcdgn(p, q, g0, u0, v0) % % INPUT: p, q -- polynomial coefficients % g0, u0, v0 -- initial iterates % OUTPUT: g, u, v as described above % res % m = length(g0)-1; m1 = m+1; % degree of g (i.e. gcd) n = length(u0)-1; n1 = n+1; % degree of u k = length(v0)-1; k1 = k+1; % degree of v lp = length(p); % length of p lq = length(q); % length of q = p' % making all polynomials monic if p(1) ~= 1, p = p/(p(1)); end; if q(1) ~= 1, q = q/(q(1)); end; if g0(1) ~= 1, g0 = g0/(g0(1)); end; if u0(1) ~= 1, u0 = u0/(u0(1)); end; if v0(1) ~= 1, v0 = v0/(v0(1)); end; s = conv(g0,u0)-p; t = conv(g0,v0)-q; b = [s(2:lp).'; t(2:lq).']; x = [g0(2:m1).'; u0(2:n1).'; v0(2:k1).']; w = ones(length(b),1); for j = 2:lp if abs(p(j)) > 1 w(j-1) = 1/abs(p(j)); end end; for j = 2:lq if abs(q(j)) > 1 w(lp-2+j) = 1/abs(q(j)); end end; %bke = [max(abs(b))]; bke = [max(abs(b.*w))]; %fprintf(' g-n %g \n',bke(1)); j = 1; while j > 0 A = sylmat(g0,u0,v0); d = scalsq(A,b,w); y = x - d; g = [1,y(1:m).']; u = [1,y(m+1:m+n).']; v = [1,y(m+n+1:m+n+k).']; s = conv(g,u)-p; t = conv(g,v)-q; b = [s(2:lp).'; t(2:lq).']; bke = [bke,max(abs(b).*w)]; j = j + 1; %fprintf(' g-n %g,%g \n',bke(j), norm(d)); if bke(j) >= bke(j-1) g = g0; u = u0; v = v0; res = bke(j-1); break; end; g0 = g; u0 = u; v0 = v; x = y; end SHAR_EOF fi # end of overwriting check if test -f 'zminsv.m' then echo shar: will not over-write existing file "'zminsv.m'" else cat << "SHAR_EOF" > 'zminsv.m' function [s,x] = minsv(A,tol) [m,n] = size(A); % get the dimensions of A scale = norm(A,inf); % get the magnitude of rows as scaler a = 2*rand(1,n)-1; % random initial vector a = scale*a/norm(a); % set the first row b = [scale;zeros(m,1)]; [Q,R] = qr(A); % QR decomp. of A, maybe input [T,trans] = hessqr([a;R]); % Hessenberg QR decomp. of stacked matrix z = hqrt(trans,b); % same Q on b x = backsub(T(1:n,1:n),z(1:n)); x = x/norm(x); % geting the new vector r = [scale*x';R]*x-b; ss = [norm(r(2:m+1))]; cr = []; for k = 1:3 [T,trans] = hessqr([2*scale*x';R]); z = hqrt(trans,r); u = backsub(T(1:n,1:n),z(1:n)); y = x - u; y = y / norm(y); r = [scale*y';R]*y-b; s = norm(r(2:m+1)); ss = [ss,s]; cr = [cr,norm(x-y)]; %disp(cr(k)); if k == 1 if cr(1) < tol, break; end; else if cr(k) < cr(k-1) if cr(k)^2/(cr(k-1)-cr(k))