LAPACK  3.10.0
LAPACK: Linear Algebra PACKage
slarrr.f
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1 *> \brief \b SLARRR performs tests to decide whether the symmetric tridiagonal matrix T warrants expensive computations which guarantee high relative accuracy in the eigenvalues.
2 *
3 * =========== DOCUMENTATION ===========
4 *
5 * Online html documentation available at
6 * http://www.netlib.org/lapack/explore-html/
7 *
8 *> \htmlonly
9 *> Download SLARRR + dependencies
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11 *> [TGZ]</a>
12 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/slarrr.f">
13 *> [ZIP]</a>
14 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/slarrr.f">
15 *> [TXT]</a>
16 *> \endhtmlonly
17 *
18 * Definition:
19 * ===========
20 *
21 * SUBROUTINE SLARRR( N, D, E, INFO )
22 *
23 * .. Scalar Arguments ..
24 * INTEGER N, INFO
25 * ..
26 * .. Array Arguments ..
27 * REAL D( * ), E( * )
28 * ..
29 *
30 *
31 *
32 *> \par Purpose:
33 * =============
34 *>
35 *> \verbatim
36 *>
37 *> Perform tests to decide whether the symmetric tridiagonal matrix T
38 *> warrants expensive computations which guarantee high relative accuracy
39 *> in the eigenvalues.
40 *> \endverbatim
41 *
42 * Arguments:
43 * ==========
44 *
45 *> \param[in] N
46 *> \verbatim
47 *> N is INTEGER
48 *> The order of the matrix. N > 0.
49 *> \endverbatim
50 *>
51 *> \param[in] D
52 *> \verbatim
53 *> D is REAL array, dimension (N)
54 *> The N diagonal elements of the tridiagonal matrix T.
55 *> \endverbatim
56 *>
57 *> \param[in,out] E
58 *> \verbatim
59 *> E is REAL array, dimension (N)
60 *> On entry, the first (N-1) entries contain the subdiagonal
61 *> elements of the tridiagonal matrix T; E(N) is set to ZERO.
62 *> \endverbatim
63 *>
64 *> \param[out] INFO
65 *> \verbatim
66 *> INFO is INTEGER
67 *> INFO = 0(default) : the matrix warrants computations preserving
68 *> relative accuracy.
69 *> INFO = 1 : the matrix warrants computations guaranteeing
70 *> only absolute accuracy.
71 *> \endverbatim
72 *
73 * Authors:
74 * ========
75 *
76 *> \author Univ. of Tennessee
77 *> \author Univ. of California Berkeley
78 *> \author Univ. of Colorado Denver
79 *> \author NAG Ltd.
80 *
81 *> \ingroup OTHERauxiliary
82 *
83 *> \par Contributors:
84 * ==================
85 *>
86 *> Beresford Parlett, University of California, Berkeley, USA \n
87 *> Jim Demmel, University of California, Berkeley, USA \n
88 *> Inderjit Dhillon, University of Texas, Austin, USA \n
89 *> Osni Marques, LBNL/NERSC, USA \n
90 *> Christof Voemel, University of California, Berkeley, USA
91 *
92 * =====================================================================
93  SUBROUTINE slarrr( N, D, E, INFO )
94 *
95 * -- LAPACK auxiliary routine --
96 * -- LAPACK is a software package provided by Univ. of Tennessee, --
97 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
98 *
99 * .. Scalar Arguments ..
100  INTEGER N, INFO
101 * ..
102 * .. Array Arguments ..
103  REAL D( * ), E( * )
104 * ..
105 *
106 *
107 * =====================================================================
108 *
109 * .. Parameters ..
110  REAL ZERO, RELCOND
111  parameter( zero = 0.0e0,
112  $ relcond = 0.999e0 )
113 * ..
114 * .. Local Scalars ..
115  INTEGER I
116  LOGICAL YESREL
117  REAL EPS, SAFMIN, SMLNUM, RMIN, TMP, TMP2,
118  $ OFFDIG, OFFDIG2
119 
120 * ..
121 * .. External Functions ..
122  REAL SLAMCH
123  EXTERNAL slamch
124 * ..
125 * .. Intrinsic Functions ..
126  INTRINSIC abs
127 * ..
128 * .. Executable Statements ..
129 *
130 * Quick return if possible
131 *
132  IF( n.LE.0 ) THEN
133  info = 0
134  RETURN
135  END IF
136 *
137 * As a default, do NOT go for relative-accuracy preserving computations.
138  info = 1
139 
140  safmin = slamch( 'Safe minimum' )
141  eps = slamch( 'Precision' )
142  smlnum = safmin / eps
143  rmin = sqrt( smlnum )
144 
145 * Tests for relative accuracy
146 *
147 * Test for scaled diagonal dominance
148 * Scale the diagonal entries to one and check whether the sum of the
149 * off-diagonals is less than one
150 *
151 * The sdd relative error bounds have a 1/(1- 2*x) factor in them,
152 * x = max(OFFDIG + OFFDIG2), so when x is close to 1/2, no relative
153 * accuracy is promised. In the notation of the code fragment below,
154 * 1/(1 - (OFFDIG + OFFDIG2)) is the condition number.
155 * We don't think it is worth going into "sdd mode" unless the relative
156 * condition number is reasonable, not 1/macheps.
157 * The threshold should be compatible with other thresholds used in the
158 * code. We set OFFDIG + OFFDIG2 <= .999 =: RELCOND, it corresponds
159 * to losing at most 3 decimal digits: 1 / (1 - (OFFDIG + OFFDIG2)) <= 1000
160 * instead of the current OFFDIG + OFFDIG2 < 1
161 *
162  yesrel = .true.
163  offdig = zero
164  tmp = sqrt(abs(d(1)))
165  IF (tmp.LT.rmin) yesrel = .false.
166  IF(.NOT.yesrel) GOTO 11
167  DO 10 i = 2, n
168  tmp2 = sqrt(abs(d(i)))
169  IF (tmp2.LT.rmin) yesrel = .false.
170  IF(.NOT.yesrel) GOTO 11
171  offdig2 = abs(e(i-1))/(tmp*tmp2)
172  IF(offdig+offdig2.GE.relcond) yesrel = .false.
173  IF(.NOT.yesrel) GOTO 11
174  tmp = tmp2
175  offdig = offdig2
176  10 CONTINUE
177  11 CONTINUE
178 
179  IF( yesrel ) THEN
180  info = 0
181  RETURN
182  ELSE
183  ENDIF
184 *
185 
186 *
187 * *** MORE TO BE IMPLEMENTED ***
188 *
189 
190 *
191 * Test if the lower bidiagonal matrix L from T = L D L^T
192 * (zero shift facto) is well conditioned
193 *
194 
195 *
196 * Test if the upper bidiagonal matrix U from T = U D U^T
197 * (zero shift facto) is well conditioned.
198 * In this case, the matrix needs to be flipped and, at the end
199 * of the eigenvector computation, the flip needs to be applied
200 * to the computed eigenvectors (and the support)
201 *
202 
203 *
204  RETURN
205 *
206 * End of SLARRR
207 *
208  END
subroutine slarrr(N, D, E, INFO)
SLARRR performs tests to decide whether the symmetric tridiagonal matrix T warrants expensive computa...
Definition: slarrr.f:94