LAPACK 3.3.1
Linear Algebra PACKage

slaqr4.f

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00001       SUBROUTINE SLAQR4( WANTT, WANTZ, N, ILO, IHI, H, LDH, WR, WI,
00002      $                   ILOZ, IHIZ, Z, LDZ, WORK, LWORK, INFO )
00003 *
00004 *  -- LAPACK auxiliary routine (version 3.2) --
00005 *     Univ. of Tennessee, Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..
00006 *     November 2006
00007 *
00008 *     .. Scalar Arguments ..
00009       INTEGER            IHI, IHIZ, ILO, ILOZ, INFO, LDH, LDZ, LWORK, N
00010       LOGICAL            WANTT, WANTZ
00011 *     ..
00012 *     .. Array Arguments ..
00013       REAL               H( LDH, * ), WI( * ), WORK( * ), WR( * ),
00014      $                   Z( LDZ, * )
00015 *     ..
00016 *
00017 *     This subroutine implements one level of recursion for SLAQR0.
00018 *     It is a complete implementation of the small bulge multi-shift
00019 *     QR algorithm.  It may be called by SLAQR0 and, for large enough
00020 *     deflation window size, it may be called by SLAQR3.  This
00021 *     subroutine is identical to SLAQR0 except that it calls SLAQR2
00022 *     instead of SLAQR3.
00023 *
00024 *     Purpose
00025 *     =======
00026 *
00027 *     SLAQR4 computes the eigenvalues of a Hessenberg matrix H
00028 *     and, optionally, the matrices T and Z from the Schur decomposition
00029 *     H = Z T Z**T, where T is an upper quasi-triangular matrix (the
00030 *     Schur form), and Z is the orthogonal matrix of Schur vectors.
00031 *
00032 *     Optionally Z may be postmultiplied into an input orthogonal
00033 *     matrix Q so that this routine can give the Schur factorization
00034 *     of a matrix A which has been reduced to the Hessenberg form H
00035 *     by the orthogonal matrix Q:  A = Q*H*Q**T = (QZ)*T*(QZ)**T.
00036 *
00037 *     Arguments
00038 *     =========
00039 *
00040 *     WANTT   (input) LOGICAL
00041 *          = .TRUE. : the full Schur form T is required;
00042 *          = .FALSE.: only eigenvalues are required.
00043 *
00044 *     WANTZ   (input) LOGICAL
00045 *          = .TRUE. : the matrix of Schur vectors Z is required;
00046 *          = .FALSE.: Schur vectors are not required.
00047 *
00048 *     N     (input) INTEGER
00049 *           The order of the matrix H.  N .GE. 0.
00050 *
00051 *     ILO   (input) INTEGER
00052 *     IHI   (input) INTEGER
00053 *           It is assumed that H is already upper triangular in rows
00054 *           and columns 1:ILO-1 and IHI+1:N and, if ILO.GT.1,
00055 *           H(ILO,ILO-1) is zero. ILO and IHI are normally set by a
00056 *           previous call to SGEBAL, and then passed to SGEHRD when the
00057 *           matrix output by SGEBAL is reduced to Hessenberg form.
00058 *           Otherwise, ILO and IHI should be set to 1 and N,
00059 *           respectively.  If N.GT.0, then 1.LE.ILO.LE.IHI.LE.N.
00060 *           If N = 0, then ILO = 1 and IHI = 0.
00061 *
00062 *     H     (input/output) REAL array, dimension (LDH,N)
00063 *           On entry, the upper Hessenberg matrix H.
00064 *           On exit, if INFO = 0 and WANTT is .TRUE., then H contains
00065 *           the upper quasi-triangular matrix T from the Schur
00066 *           decomposition (the Schur form); 2-by-2 diagonal blocks
00067 *           (corresponding to complex conjugate pairs of eigenvalues)
00068 *           are returned in standard form, with H(i,i) = H(i+1,i+1)
00069 *           and H(i+1,i)*H(i,i+1).LT.0. If INFO = 0 and WANTT is
00070 *           .FALSE., then the contents of H are unspecified on exit.
00071 *           (The output value of H when INFO.GT.0 is given under the
00072 *           description of INFO below.)
00073 *
00074 *           This subroutine may explicitly set H(i,j) = 0 for i.GT.j and
00075 *           j = 1, 2, ... ILO-1 or j = IHI+1, IHI+2, ... N.
00076 *
00077 *     LDH   (input) INTEGER
00078 *           The leading dimension of the array H. LDH .GE. max(1,N).
00079 *
00080 *     WR    (output) REAL array, dimension (IHI)
00081 *     WI    (output) REAL array, dimension (IHI)
00082 *           The real and imaginary parts, respectively, of the computed
00083 *           eigenvalues of H(ILO:IHI,ILO:IHI) are stored in WR(ILO:IHI)
00084 *           and WI(ILO:IHI). If two eigenvalues are computed as a
00085 *           complex conjugate pair, they are stored in consecutive
00086 *           elements of WR and WI, say the i-th and (i+1)th, with
00087 *           WI(i) .GT. 0 and WI(i+1) .LT. 0. If WANTT is .TRUE., then
00088 *           the eigenvalues are stored in the same order as on the
00089 *           diagonal of the Schur form returned in H, with
00090 *           WR(i) = H(i,i) and, if H(i:i+1,i:i+1) is a 2-by-2 diagonal
00091 *           block, WI(i) = sqrt(-H(i+1,i)*H(i,i+1)) and
00092 *           WI(i+1) = -WI(i).
00093 *
00094 *     ILOZ     (input) INTEGER
00095 *     IHIZ     (input) INTEGER
00096 *           Specify the rows of Z to which transformations must be
00097 *           applied if WANTZ is .TRUE..
00098 *           1 .LE. ILOZ .LE. ILO; IHI .LE. IHIZ .LE. N.
00099 *
00100 *     Z     (input/output) REAL array, dimension (LDZ,IHI)
00101 *           If WANTZ is .FALSE., then Z is not referenced.
00102 *           If WANTZ is .TRUE., then Z(ILO:IHI,ILOZ:IHIZ) is
00103 *           replaced by Z(ILO:IHI,ILOZ:IHIZ)*U where U is the
00104 *           orthogonal Schur factor of H(ILO:IHI,ILO:IHI).
00105 *           (The output value of Z when INFO.GT.0 is given under
00106 *           the description of INFO below.)
00107 *
00108 *     LDZ   (input) INTEGER
00109 *           The leading dimension of the array Z.  if WANTZ is .TRUE.
00110 *           then LDZ.GE.MAX(1,IHIZ).  Otherwize, LDZ.GE.1.
00111 *
00112 *     WORK  (workspace/output) REAL array, dimension LWORK
00113 *           On exit, if LWORK = -1, WORK(1) returns an estimate of
00114 *           the optimal value for LWORK.
00115 *
00116 *     LWORK (input) INTEGER
00117 *           The dimension of the array WORK.  LWORK .GE. max(1,N)
00118 *           is sufficient, but LWORK typically as large as 6*N may
00119 *           be required for optimal performance.  A workspace query
00120 *           to determine the optimal workspace size is recommended.
00121 *
00122 *           If LWORK = -1, then SLAQR4 does a workspace query.
00123 *           In this case, SLAQR4 checks the input parameters and
00124 *           estimates the optimal workspace size for the given
00125 *           values of N, ILO and IHI.  The estimate is returned
00126 *           in WORK(1).  No error message related to LWORK is
00127 *           issued by XERBLA.  Neither H nor Z are accessed.
00128 *
00129 *
00130 *     INFO  (output) INTEGER
00131 *             =  0:  successful exit
00132 *           .GT. 0:  if INFO = i, SLAQR4 failed to compute all of
00133 *                the eigenvalues.  Elements 1:ilo-1 and i+1:n of WR
00134 *                and WI contain those eigenvalues which have been
00135 *                successfully computed.  (Failures are rare.)
00136 *
00137 *                If INFO .GT. 0 and WANT is .FALSE., then on exit,
00138 *                the remaining unconverged eigenvalues are the eigen-
00139 *                values of the upper Hessenberg matrix rows and
00140 *                columns ILO through INFO of the final, output
00141 *                value of H.
00142 *
00143 *                If INFO .GT. 0 and WANTT is .TRUE., then on exit
00144 *
00145 *           (*)  (initial value of H)*U  = U*(final value of H)
00146 *
00147 *                where U is an orthogonal matrix.  The final
00148 *                value of H is upper Hessenberg and quasi-triangular
00149 *                in rows and columns INFO+1 through IHI.
00150 *
00151 *                If INFO .GT. 0 and WANTZ is .TRUE., then on exit
00152 *
00153 *                  (final value of Z(ILO:IHI,ILOZ:IHIZ)
00154 *                   =  (initial value of Z(ILO:IHI,ILOZ:IHIZ)*U
00155 *
00156 *                where U is the orthogonal matrix in (*) (regard-
00157 *                less of the value of WANTT.)
00158 *
00159 *                If INFO .GT. 0 and WANTZ is .FALSE., then Z is not
00160 *                accessed.
00161 *
00162 *     ================================================================
00163 *     Based on contributions by
00164 *        Karen Braman and Ralph Byers, Department of Mathematics,
00165 *        University of Kansas, USA
00166 *
00167 *     ================================================================
00168 *     References:
00169 *       K. Braman, R. Byers and R. Mathias, The Multi-Shift QR
00170 *       Algorithm Part I: Maintaining Well Focused Shifts, and Level 3
00171 *       Performance, SIAM Journal of Matrix Analysis, volume 23, pages
00172 *       929--947, 2002.
00173 *
00174 *       K. Braman, R. Byers and R. Mathias, The Multi-Shift QR
00175 *       Algorithm Part II: Aggressive Early Deflation, SIAM Journal
00176 *       of Matrix Analysis, volume 23, pages 948--973, 2002.
00177 *
00178 *     ================================================================
00179 *     .. Parameters ..
00180 *
00181 *     ==== Matrices of order NTINY or smaller must be processed by
00182 *     .    SLAHQR because of insufficient subdiagonal scratch space.
00183 *     .    (This is a hard limit.) ====
00184       INTEGER            NTINY
00185       PARAMETER          ( NTINY = 11 )
00186 *
00187 *     ==== Exceptional deflation windows:  try to cure rare
00188 *     .    slow convergence by varying the size of the
00189 *     .    deflation window after KEXNW iterations. ====
00190       INTEGER            KEXNW
00191       PARAMETER          ( KEXNW = 5 )
00192 *
00193 *     ==== Exceptional shifts: try to cure rare slow convergence
00194 *     .    with ad-hoc exceptional shifts every KEXSH iterations.
00195 *     .    ====
00196       INTEGER            KEXSH
00197       PARAMETER          ( KEXSH = 6 )
00198 *
00199 *     ==== The constants WILK1 and WILK2 are used to form the
00200 *     .    exceptional shifts. ====
00201       REAL               WILK1, WILK2
00202       PARAMETER          ( WILK1 = 0.75e0, WILK2 = -0.4375e0 )
00203       REAL               ZERO, ONE
00204       PARAMETER          ( ZERO = 0.0e0, ONE = 1.0e0 )
00205 *     ..
00206 *     .. Local Scalars ..
00207       REAL               AA, BB, CC, CS, DD, SN, SS, SWAP
00208       INTEGER            I, INF, IT, ITMAX, K, KACC22, KBOT, KDU, KS,
00209      $                   KT, KTOP, KU, KV, KWH, KWTOP, KWV, LD, LS,
00210      $                   LWKOPT, NDEC, NDFL, NH, NHO, NIBBLE, NMIN, NS,
00211      $                   NSMAX, NSR, NVE, NW, NWMAX, NWR, NWUPBD
00212       LOGICAL            SORTED
00213       CHARACTER          JBCMPZ*2
00214 *     ..
00215 *     .. External Functions ..
00216       INTEGER            ILAENV
00217       EXTERNAL           ILAENV
00218 *     ..
00219 *     .. Local Arrays ..
00220       REAL               ZDUM( 1, 1 )
00221 *     ..
00222 *     .. External Subroutines ..
00223       EXTERNAL           SLACPY, SLAHQR, SLANV2, SLAQR2, SLAQR5
00224 *     ..
00225 *     .. Intrinsic Functions ..
00226       INTRINSIC          ABS, INT, MAX, MIN, MOD, REAL
00227 *     ..
00228 *     .. Executable Statements ..
00229       INFO = 0
00230 *
00231 *     ==== Quick return for N = 0: nothing to do. ====
00232 *
00233       IF( N.EQ.0 ) THEN
00234          WORK( 1 ) = ONE
00235          RETURN
00236       END IF
00237 *
00238       IF( N.LE.NTINY ) THEN
00239 *
00240 *        ==== Tiny matrices must use SLAHQR. ====
00241 *
00242          LWKOPT = 1
00243          IF( LWORK.NE.-1 )
00244      $      CALL SLAHQR( WANTT, WANTZ, N, ILO, IHI, H, LDH, WR, WI,
00245      $                   ILOZ, IHIZ, Z, LDZ, INFO )
00246       ELSE
00247 *
00248 *        ==== Use small bulge multi-shift QR with aggressive early
00249 *        .    deflation on larger-than-tiny matrices. ====
00250 *
00251 *        ==== Hope for the best. ====
00252 *
00253          INFO = 0
00254 *
00255 *        ==== Set up job flags for ILAENV. ====
00256 *
00257          IF( WANTT ) THEN
00258             JBCMPZ( 1: 1 ) = 'S'
00259          ELSE
00260             JBCMPZ( 1: 1 ) = 'E'
00261          END IF
00262          IF( WANTZ ) THEN
00263             JBCMPZ( 2: 2 ) = 'V'
00264          ELSE
00265             JBCMPZ( 2: 2 ) = 'N'
00266          END IF
00267 *
00268 *        ==== NWR = recommended deflation window size.  At this
00269 *        .    point,  N .GT. NTINY = 11, so there is enough
00270 *        .    subdiagonal workspace for NWR.GE.2 as required.
00271 *        .    (In fact, there is enough subdiagonal space for
00272 *        .    NWR.GE.3.) ====
00273 *
00274          NWR = ILAENV( 13, 'SLAQR4', JBCMPZ, N, ILO, IHI, LWORK )
00275          NWR = MAX( 2, NWR )
00276          NWR = MIN( IHI-ILO+1, ( N-1 ) / 3, NWR )
00277 *
00278 *        ==== NSR = recommended number of simultaneous shifts.
00279 *        .    At this point N .GT. NTINY = 11, so there is at
00280 *        .    enough subdiagonal workspace for NSR to be even
00281 *        .    and greater than or equal to two as required. ====
00282 *
00283          NSR = ILAENV( 15, 'SLAQR4', JBCMPZ, N, ILO, IHI, LWORK )
00284          NSR = MIN( NSR, ( N+6 ) / 9, IHI-ILO )
00285          NSR = MAX( 2, NSR-MOD( NSR, 2 ) )
00286 *
00287 *        ==== Estimate optimal workspace ====
00288 *
00289 *        ==== Workspace query call to SLAQR2 ====
00290 *
00291          CALL SLAQR2( WANTT, WANTZ, N, ILO, IHI, NWR+1, H, LDH, ILOZ,
00292      $                IHIZ, Z, LDZ, LS, LD, WR, WI, H, LDH, N, H, LDH,
00293      $                N, H, LDH, WORK, -1 )
00294 *
00295 *        ==== Optimal workspace = MAX(SLAQR5, SLAQR2) ====
00296 *
00297          LWKOPT = MAX( 3*NSR / 2, INT( WORK( 1 ) ) )
00298 *
00299 *        ==== Quick return in case of workspace query. ====
00300 *
00301          IF( LWORK.EQ.-1 ) THEN
00302             WORK( 1 ) = REAL( LWKOPT )
00303             RETURN
00304          END IF
00305 *
00306 *        ==== SLAHQR/SLAQR0 crossover point ====
00307 *
00308          NMIN = ILAENV( 12, 'SLAQR4', JBCMPZ, N, ILO, IHI, LWORK )
00309          NMIN = MAX( NTINY, NMIN )
00310 *
00311 *        ==== Nibble crossover point ====
00312 *
00313          NIBBLE = ILAENV( 14, 'SLAQR4', JBCMPZ, N, ILO, IHI, LWORK )
00314          NIBBLE = MAX( 0, NIBBLE )
00315 *
00316 *        ==== Accumulate reflections during ttswp?  Use block
00317 *        .    2-by-2 structure during matrix-matrix multiply? ====
00318 *
00319          KACC22 = ILAENV( 16, 'SLAQR4', JBCMPZ, N, ILO, IHI, LWORK )
00320          KACC22 = MAX( 0, KACC22 )
00321          KACC22 = MIN( 2, KACC22 )
00322 *
00323 *        ==== NWMAX = the largest possible deflation window for
00324 *        .    which there is sufficient workspace. ====
00325 *
00326          NWMAX = MIN( ( N-1 ) / 3, LWORK / 2 )
00327          NW = NWMAX
00328 *
00329 *        ==== NSMAX = the Largest number of simultaneous shifts
00330 *        .    for which there is sufficient workspace. ====
00331 *
00332          NSMAX = MIN( ( N+6 ) / 9, 2*LWORK / 3 )
00333          NSMAX = NSMAX - MOD( NSMAX, 2 )
00334 *
00335 *        ==== NDFL: an iteration count restarted at deflation. ====
00336 *
00337          NDFL = 1
00338 *
00339 *        ==== ITMAX = iteration limit ====
00340 *
00341          ITMAX = MAX( 30, 2*KEXSH )*MAX( 10, ( IHI-ILO+1 ) )
00342 *
00343 *        ==== Last row and column in the active block ====
00344 *
00345          KBOT = IHI
00346 *
00347 *        ==== Main Loop ====
00348 *
00349          DO 80 IT = 1, ITMAX
00350 *
00351 *           ==== Done when KBOT falls below ILO ====
00352 *
00353             IF( KBOT.LT.ILO )
00354      $         GO TO 90
00355 *
00356 *           ==== Locate active block ====
00357 *
00358             DO 10 K = KBOT, ILO + 1, -1
00359                IF( H( K, K-1 ).EQ.ZERO )
00360      $            GO TO 20
00361    10       CONTINUE
00362             K = ILO
00363    20       CONTINUE
00364             KTOP = K
00365 *
00366 *           ==== Select deflation window size:
00367 *           .    Typical Case:
00368 *           .      If possible and advisable, nibble the entire
00369 *           .      active block.  If not, use size MIN(NWR,NWMAX)
00370 *           .      or MIN(NWR+1,NWMAX) depending upon which has
00371 *           .      the smaller corresponding subdiagonal entry
00372 *           .      (a heuristic).
00373 *           .
00374 *           .    Exceptional Case:
00375 *           .      If there have been no deflations in KEXNW or
00376 *           .      more iterations, then vary the deflation window
00377 *           .      size.   At first, because, larger windows are,
00378 *           .      in general, more powerful than smaller ones,
00379 *           .      rapidly increase the window to the maximum possible.
00380 *           .      Then, gradually reduce the window size. ====
00381 *
00382             NH = KBOT - KTOP + 1
00383             NWUPBD = MIN( NH, NWMAX )
00384             IF( NDFL.LT.KEXNW ) THEN
00385                NW = MIN( NWUPBD, NWR )
00386             ELSE
00387                NW = MIN( NWUPBD, 2*NW )
00388             END IF
00389             IF( NW.LT.NWMAX ) THEN
00390                IF( NW.GE.NH-1 ) THEN
00391                   NW = NH
00392                ELSE
00393                   KWTOP = KBOT - NW + 1
00394                   IF( ABS( H( KWTOP, KWTOP-1 ) ).GT.
00395      $                ABS( H( KWTOP-1, KWTOP-2 ) ) )NW = NW + 1
00396                END IF
00397             END IF
00398             IF( NDFL.LT.KEXNW ) THEN
00399                NDEC = -1
00400             ELSE IF( NDEC.GE.0 .OR. NW.GE.NWUPBD ) THEN
00401                NDEC = NDEC + 1
00402                IF( NW-NDEC.LT.2 )
00403      $            NDEC = 0
00404                NW = NW - NDEC
00405             END IF
00406 *
00407 *           ==== Aggressive early deflation:
00408 *           .    split workspace under the subdiagonal into
00409 *           .      - an nw-by-nw work array V in the lower
00410 *           .        left-hand-corner,
00411 *           .      - an NW-by-at-least-NW-but-more-is-better
00412 *           .        (NW-by-NHO) horizontal work array along
00413 *           .        the bottom edge,
00414 *           .      - an at-least-NW-but-more-is-better (NHV-by-NW)
00415 *           .        vertical work array along the left-hand-edge.
00416 *           .        ====
00417 *
00418             KV = N - NW + 1
00419             KT = NW + 1
00420             NHO = ( N-NW-1 ) - KT + 1
00421             KWV = NW + 2
00422             NVE = ( N-NW ) - KWV + 1
00423 *
00424 *           ==== Aggressive early deflation ====
00425 *
00426             CALL SLAQR2( WANTT, WANTZ, N, KTOP, KBOT, NW, H, LDH, ILOZ,
00427      $                   IHIZ, Z, LDZ, LS, LD, WR, WI, H( KV, 1 ), LDH,
00428      $                   NHO, H( KV, KT ), LDH, NVE, H( KWV, 1 ), LDH,
00429      $                   WORK, LWORK )
00430 *
00431 *           ==== Adjust KBOT accounting for new deflations. ====
00432 *
00433             KBOT = KBOT - LD
00434 *
00435 *           ==== KS points to the shifts. ====
00436 *
00437             KS = KBOT - LS + 1
00438 *
00439 *           ==== Skip an expensive QR sweep if there is a (partly
00440 *           .    heuristic) reason to expect that many eigenvalues
00441 *           .    will deflate without it.  Here, the QR sweep is
00442 *           .    skipped if many eigenvalues have just been deflated
00443 *           .    or if the remaining active block is small.
00444 *
00445             IF( ( LD.EQ.0 ) .OR. ( ( 100*LD.LE.NW*NIBBLE ) .AND. ( KBOT-
00446      $          KTOP+1.GT.MIN( NMIN, NWMAX ) ) ) ) THEN
00447 *
00448 *              ==== NS = nominal number of simultaneous shifts.
00449 *              .    This may be lowered (slightly) if SLAQR2
00450 *              .    did not provide that many shifts. ====
00451 *
00452                NS = MIN( NSMAX, NSR, MAX( 2, KBOT-KTOP ) )
00453                NS = NS - MOD( NS, 2 )
00454 *
00455 *              ==== If there have been no deflations
00456 *              .    in a multiple of KEXSH iterations,
00457 *              .    then try exceptional shifts.
00458 *              .    Otherwise use shifts provided by
00459 *              .    SLAQR2 above or from the eigenvalues
00460 *              .    of a trailing principal submatrix. ====
00461 *
00462                IF( MOD( NDFL, KEXSH ).EQ.0 ) THEN
00463                   KS = KBOT - NS + 1
00464                   DO 30 I = KBOT, MAX( KS+1, KTOP+2 ), -2
00465                      SS = ABS( H( I, I-1 ) ) + ABS( H( I-1, I-2 ) )
00466                      AA = WILK1*SS + H( I, I )
00467                      BB = SS
00468                      CC = WILK2*SS
00469                      DD = AA
00470                      CALL SLANV2( AA, BB, CC, DD, WR( I-1 ), WI( I-1 ),
00471      $                            WR( I ), WI( I ), CS, SN )
00472    30             CONTINUE
00473                   IF( KS.EQ.KTOP ) THEN
00474                      WR( KS+1 ) = H( KS+1, KS+1 )
00475                      WI( KS+1 ) = ZERO
00476                      WR( KS ) = WR( KS+1 )
00477                      WI( KS ) = WI( KS+1 )
00478                   END IF
00479                ELSE
00480 *
00481 *                 ==== Got NS/2 or fewer shifts? Use SLAHQR
00482 *                 .    on a trailing principal submatrix to
00483 *                 .    get more. (Since NS.LE.NSMAX.LE.(N+6)/9,
00484 *                 .    there is enough space below the subdiagonal
00485 *                 .    to fit an NS-by-NS scratch array.) ====
00486 *
00487                   IF( KBOT-KS+1.LE.NS / 2 ) THEN
00488                      KS = KBOT - NS + 1
00489                      KT = N - NS + 1
00490                      CALL SLACPY( 'A', NS, NS, H( KS, KS ), LDH,
00491      $                            H( KT, 1 ), LDH )
00492                      CALL SLAHQR( .false., .false., NS, 1, NS,
00493      $                            H( KT, 1 ), LDH, WR( KS ), WI( KS ),
00494      $                            1, 1, ZDUM, 1, INF )
00495                      KS = KS + INF
00496 *
00497 *                    ==== In case of a rare QR failure use
00498 *                    .    eigenvalues of the trailing 2-by-2
00499 *                    .    principal submatrix.  ====
00500 *
00501                      IF( KS.GE.KBOT ) THEN
00502                         AA = H( KBOT-1, KBOT-1 )
00503                         CC = H( KBOT, KBOT-1 )
00504                         BB = H( KBOT-1, KBOT )
00505                         DD = H( KBOT, KBOT )
00506                         CALL SLANV2( AA, BB, CC, DD, WR( KBOT-1 ),
00507      $                               WI( KBOT-1 ), WR( KBOT ),
00508      $                               WI( KBOT ), CS, SN )
00509                         KS = KBOT - 1
00510                      END IF
00511                   END IF
00512 *
00513                   IF( KBOT-KS+1.GT.NS ) THEN
00514 *
00515 *                    ==== Sort the shifts (Helps a little)
00516 *                    .    Bubble sort keeps complex conjugate
00517 *                    .    pairs together. ====
00518 *
00519                      SORTED = .false.
00520                      DO 50 K = KBOT, KS + 1, -1
00521                         IF( SORTED )
00522      $                     GO TO 60
00523                         SORTED = .true.
00524                         DO 40 I = KS, K - 1
00525                            IF( ABS( WR( I ) )+ABS( WI( I ) ).LT.
00526      $                         ABS( WR( I+1 ) )+ABS( WI( I+1 ) ) ) THEN
00527                               SORTED = .false.
00528 *
00529                               SWAP = WR( I )
00530                               WR( I ) = WR( I+1 )
00531                               WR( I+1 ) = SWAP
00532 *
00533                               SWAP = WI( I )
00534                               WI( I ) = WI( I+1 )
00535                               WI( I+1 ) = SWAP
00536                            END IF
00537    40                   CONTINUE
00538    50                CONTINUE
00539    60                CONTINUE
00540                   END IF
00541 *
00542 *                 ==== Shuffle shifts into pairs of real shifts
00543 *                 .    and pairs of complex conjugate shifts
00544 *                 .    assuming complex conjugate shifts are
00545 *                 .    already adjacent to one another. (Yes,
00546 *                 .    they are.)  ====
00547 *
00548                   DO 70 I = KBOT, KS + 2, -2
00549                      IF( WI( I ).NE.-WI( I-1 ) ) THEN
00550 *
00551                         SWAP = WR( I )
00552                         WR( I ) = WR( I-1 )
00553                         WR( I-1 ) = WR( I-2 )
00554                         WR( I-2 ) = SWAP
00555 *
00556                         SWAP = WI( I )
00557                         WI( I ) = WI( I-1 )
00558                         WI( I-1 ) = WI( I-2 )
00559                         WI( I-2 ) = SWAP
00560                      END IF
00561    70             CONTINUE
00562                END IF
00563 *
00564 *              ==== If there are only two shifts and both are
00565 *              .    real, then use only one.  ====
00566 *
00567                IF( KBOT-KS+1.EQ.2 ) THEN
00568                   IF( WI( KBOT ).EQ.ZERO ) THEN
00569                      IF( ABS( WR( KBOT )-H( KBOT, KBOT ) ).LT.
00570      $                   ABS( WR( KBOT-1 )-H( KBOT, KBOT ) ) ) THEN
00571                         WR( KBOT-1 ) = WR( KBOT )
00572                      ELSE
00573                         WR( KBOT ) = WR( KBOT-1 )
00574                      END IF
00575                   END IF
00576                END IF
00577 *
00578 *              ==== Use up to NS of the the smallest magnatiude
00579 *              .    shifts.  If there aren't NS shifts available,
00580 *              .    then use them all, possibly dropping one to
00581 *              .    make the number of shifts even. ====
00582 *
00583                NS = MIN( NS, KBOT-KS+1 )
00584                NS = NS - MOD( NS, 2 )
00585                KS = KBOT - NS + 1
00586 *
00587 *              ==== Small-bulge multi-shift QR sweep:
00588 *              .    split workspace under the subdiagonal into
00589 *              .    - a KDU-by-KDU work array U in the lower
00590 *              .      left-hand-corner,
00591 *              .    - a KDU-by-at-least-KDU-but-more-is-better
00592 *              .      (KDU-by-NHo) horizontal work array WH along
00593 *              .      the bottom edge,
00594 *              .    - and an at-least-KDU-but-more-is-better-by-KDU
00595 *              .      (NVE-by-KDU) vertical work WV arrow along
00596 *              .      the left-hand-edge. ====
00597 *
00598                KDU = 3*NS - 3
00599                KU = N - KDU + 1
00600                KWH = KDU + 1
00601                NHO = ( N-KDU+1-4 ) - ( KDU+1 ) + 1
00602                KWV = KDU + 4
00603                NVE = N - KDU - KWV + 1
00604 *
00605 *              ==== Small-bulge multi-shift QR sweep ====
00606 *
00607                CALL SLAQR5( WANTT, WANTZ, KACC22, N, KTOP, KBOT, NS,
00608      $                      WR( KS ), WI( KS ), H, LDH, ILOZ, IHIZ, Z,
00609      $                      LDZ, WORK, 3, H( KU, 1 ), LDH, NVE,
00610      $                      H( KWV, 1 ), LDH, NHO, H( KU, KWH ), LDH )
00611             END IF
00612 *
00613 *           ==== Note progress (or the lack of it). ====
00614 *
00615             IF( LD.GT.0 ) THEN
00616                NDFL = 1
00617             ELSE
00618                NDFL = NDFL + 1
00619             END IF
00620 *
00621 *           ==== End of main loop ====
00622    80    CONTINUE
00623 *
00624 *        ==== Iteration limit exceeded.  Set INFO to show where
00625 *        .    the problem occurred and exit. ====
00626 *
00627          INFO = KBOT
00628    90    CONTINUE
00629       END IF
00630 *
00631 *     ==== Return the optimal value of LWORK. ====
00632 *
00633       WORK( 1 ) = REAL( LWKOPT )
00634 *
00635 *     ==== End of SLAQR4 ====
00636 *
00637       END
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