LAPACK  3.10.1
LAPACK: Linear Algebra PACKage
ztrsna.f
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1 *> \brief \b ZTRSNA
2 *
3 * =========== DOCUMENTATION ===========
4 *
5 * Online html documentation available at
6 * http://www.netlib.org/lapack/explore-html/
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16 *> \endhtmlonly
17 *
18 * Definition:
19 * ===========
20 *
21 * SUBROUTINE ZTRSNA( JOB, HOWMNY, SELECT, N, T, LDT, VL, LDVL, VR,
22 * LDVR, S, SEP, MM, M, WORK, LDWORK, RWORK,
23 * INFO )
24 *
25 * .. Scalar Arguments ..
26 * CHARACTER HOWMNY, JOB
27 * INTEGER INFO, LDT, LDVL, LDVR, LDWORK, M, MM, N
28 * ..
29 * .. Array Arguments ..
30 * LOGICAL SELECT( * )
31 * DOUBLE PRECISION RWORK( * ), S( * ), SEP( * )
32 * COMPLEX*16 T( LDT, * ), VL( LDVL, * ), VR( LDVR, * ),
33 * $ WORK( LDWORK, * )
34 * ..
35 *
36 *
37 *> \par Purpose:
38 * =============
39 *>
40 *> \verbatim
41 *>
42 *> ZTRSNA estimates reciprocal condition numbers for specified
43 *> eigenvalues and/or right eigenvectors of a complex upper triangular
44 *> matrix T (or of any matrix Q*T*Q**H with Q unitary).
45 *> \endverbatim
46 *
47 * Arguments:
48 * ==========
49 *
50 *> \param[in] JOB
51 *> \verbatim
52 *> JOB is CHARACTER*1
53 *> Specifies whether condition numbers are required for
54 *> eigenvalues (S) or eigenvectors (SEP):
55 *> = 'E': for eigenvalues only (S);
56 *> = 'V': for eigenvectors only (SEP);
57 *> = 'B': for both eigenvalues and eigenvectors (S and SEP).
58 *> \endverbatim
59 *>
60 *> \param[in] HOWMNY
61 *> \verbatim
62 *> HOWMNY is CHARACTER*1
63 *> = 'A': compute condition numbers for all eigenpairs;
64 *> = 'S': compute condition numbers for selected eigenpairs
65 *> specified by the array SELECT.
66 *> \endverbatim
67 *>
68 *> \param[in] SELECT
69 *> \verbatim
70 *> SELECT is LOGICAL array, dimension (N)
71 *> If HOWMNY = 'S', SELECT specifies the eigenpairs for which
72 *> condition numbers are required. To select condition numbers
73 *> for the j-th eigenpair, SELECT(j) must be set to .TRUE..
74 *> If HOWMNY = 'A', SELECT is not referenced.
75 *> \endverbatim
76 *>
77 *> \param[in] N
78 *> \verbatim
79 *> N is INTEGER
80 *> The order of the matrix T. N >= 0.
81 *> \endverbatim
82 *>
83 *> \param[in] T
84 *> \verbatim
85 *> T is COMPLEX*16 array, dimension (LDT,N)
86 *> The upper triangular matrix T.
87 *> \endverbatim
88 *>
89 *> \param[in] LDT
90 *> \verbatim
91 *> LDT is INTEGER
92 *> The leading dimension of the array T. LDT >= max(1,N).
93 *> \endverbatim
94 *>
95 *> \param[in] VL
96 *> \verbatim
97 *> VL is COMPLEX*16 array, dimension (LDVL,M)
98 *> If JOB = 'E' or 'B', VL must contain left eigenvectors of T
99 *> (or of any Q*T*Q**H with Q unitary), corresponding to the
100 *> eigenpairs specified by HOWMNY and SELECT. The eigenvectors
101 *> must be stored in consecutive columns of VL, as returned by
102 *> ZHSEIN or ZTREVC.
103 *> If JOB = 'V', VL is not referenced.
104 *> \endverbatim
105 *>
106 *> \param[in] LDVL
107 *> \verbatim
108 *> LDVL is INTEGER
109 *> The leading dimension of the array VL.
110 *> LDVL >= 1; and if JOB = 'E' or 'B', LDVL >= N.
111 *> \endverbatim
112 *>
113 *> \param[in] VR
114 *> \verbatim
115 *> VR is COMPLEX*16 array, dimension (LDVR,M)
116 *> If JOB = 'E' or 'B', VR must contain right eigenvectors of T
117 *> (or of any Q*T*Q**H with Q unitary), corresponding to the
118 *> eigenpairs specified by HOWMNY and SELECT. The eigenvectors
119 *> must be stored in consecutive columns of VR, as returned by
120 *> ZHSEIN or ZTREVC.
121 *> If JOB = 'V', VR is not referenced.
122 *> \endverbatim
123 *>
124 *> \param[in] LDVR
125 *> \verbatim
126 *> LDVR is INTEGER
127 *> The leading dimension of the array VR.
128 *> LDVR >= 1; and if JOB = 'E' or 'B', LDVR >= N.
129 *> \endverbatim
130 *>
131 *> \param[out] S
132 *> \verbatim
133 *> S is DOUBLE PRECISION array, dimension (MM)
134 *> If JOB = 'E' or 'B', the reciprocal condition numbers of the
135 *> selected eigenvalues, stored in consecutive elements of the
136 *> array. Thus S(j), SEP(j), and the j-th columns of VL and VR
137 *> all correspond to the same eigenpair (but not in general the
138 *> j-th eigenpair, unless all eigenpairs are selected).
139 *> If JOB = 'V', S is not referenced.
140 *> \endverbatim
141 *>
142 *> \param[out] SEP
143 *> \verbatim
144 *> SEP is DOUBLE PRECISION array, dimension (MM)
145 *> If JOB = 'V' or 'B', the estimated reciprocal condition
146 *> numbers of the selected eigenvectors, stored in consecutive
147 *> elements of the array.
148 *> If JOB = 'E', SEP is not referenced.
149 *> \endverbatim
150 *>
151 *> \param[in] MM
152 *> \verbatim
153 *> MM is INTEGER
154 *> The number of elements in the arrays S (if JOB = 'E' or 'B')
155 *> and/or SEP (if JOB = 'V' or 'B'). MM >= M.
156 *> \endverbatim
157 *>
158 *> \param[out] M
159 *> \verbatim
160 *> M is INTEGER
161 *> The number of elements of the arrays S and/or SEP actually
162 *> used to store the estimated condition numbers.
163 *> If HOWMNY = 'A', M is set to N.
164 *> \endverbatim
165 *>
166 *> \param[out] WORK
167 *> \verbatim
168 *> WORK is COMPLEX*16 array, dimension (LDWORK,N+6)
169 *> If JOB = 'E', WORK is not referenced.
170 *> \endverbatim
171 *>
172 *> \param[in] LDWORK
173 *> \verbatim
174 *> LDWORK is INTEGER
175 *> The leading dimension of the array WORK.
176 *> LDWORK >= 1; and if JOB = 'V' or 'B', LDWORK >= N.
177 *> \endverbatim
178 *>
179 *> \param[out] RWORK
180 *> \verbatim
181 *> RWORK is DOUBLE PRECISION array, dimension (N)
182 *> If JOB = 'E', RWORK is not referenced.
183 *> \endverbatim
184 *>
185 *> \param[out] INFO
186 *> \verbatim
187 *> INFO is INTEGER
188 *> = 0: successful exit
189 *> < 0: if INFO = -i, the i-th argument had an illegal value
190 *> \endverbatim
191 *
192 * Authors:
193 * ========
194 *
195 *> \author Univ. of Tennessee
196 *> \author Univ. of California Berkeley
197 *> \author Univ. of Colorado Denver
198 *> \author NAG Ltd.
199 *
200 *> \ingroup complex16OTHERcomputational
201 *
202 *> \par Further Details:
203 * =====================
204 *>
205 *> \verbatim
206 *>
207 *> The reciprocal of the condition number of an eigenvalue lambda is
208 *> defined as
209 *>
210 *> S(lambda) = |v**H*u| / (norm(u)*norm(v))
211 *>
212 *> where u and v are the right and left eigenvectors of T corresponding
213 *> to lambda; v**H denotes the conjugate transpose of v, and norm(u)
214 *> denotes the Euclidean norm. These reciprocal condition numbers always
215 *> lie between zero (very badly conditioned) and one (very well
216 *> conditioned). If n = 1, S(lambda) is defined to be 1.
217 *>
218 *> An approximate error bound for a computed eigenvalue W(i) is given by
219 *>
220 *> EPS * norm(T) / S(i)
221 *>
222 *> where EPS is the machine precision.
223 *>
224 *> The reciprocal of the condition number of the right eigenvector u
225 *> corresponding to lambda is defined as follows. Suppose
226 *>
227 *> T = ( lambda c )
228 *> ( 0 T22 )
229 *>
230 *> Then the reciprocal condition number is
231 *>
232 *> SEP( lambda, T22 ) = sigma-min( T22 - lambda*I )
233 *>
234 *> where sigma-min denotes the smallest singular value. We approximate
235 *> the smallest singular value by the reciprocal of an estimate of the
236 *> one-norm of the inverse of T22 - lambda*I. If n = 1, SEP(1) is
237 *> defined to be abs(T(1,1)).
238 *>
239 *> An approximate error bound for a computed right eigenvector VR(i)
240 *> is given by
241 *>
242 *> EPS * norm(T) / SEP(i)
243 *> \endverbatim
244 *>
245 * =====================================================================
246  SUBROUTINE ztrsna( JOB, HOWMNY, SELECT, N, T, LDT, VL, LDVL, VR,
247  $ LDVR, S, SEP, MM, M, WORK, LDWORK, RWORK,
248  $ INFO )
249 *
250 * -- LAPACK computational routine --
251 * -- LAPACK is a software package provided by Univ. of Tennessee, --
252 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
253 *
254 * .. Scalar Arguments ..
255  CHARACTER HOWMNY, JOB
256  INTEGER INFO, LDT, LDVL, LDVR, LDWORK, M, MM, N
257 * ..
258 * .. Array Arguments ..
259  LOGICAL SELECT( * )
260  DOUBLE PRECISION RWORK( * ), S( * ), SEP( * )
261  COMPLEX*16 T( LDT, * ), VL( LDVL, * ), VR( LDVR, * ),
262  $ work( ldwork, * )
263 * ..
264 *
265 * =====================================================================
266 *
267 * .. Parameters ..
268  DOUBLE PRECISION ZERO, ONE
269  PARAMETER ( ZERO = 0.0d+0, one = 1.0d0+0 )
270 * ..
271 * .. Local Scalars ..
272  LOGICAL SOMCON, WANTBH, WANTS, WANTSP
273  CHARACTER NORMIN
274  INTEGER I, IERR, IX, J, K, KASE, KS
275  DOUBLE PRECISION BIGNUM, EPS, EST, LNRM, RNRM, SCALE, SMLNUM,
276  $ xnorm
277  COMPLEX*16 CDUM, PROD
278 * ..
279 * .. Local Arrays ..
280  INTEGER ISAVE( 3 )
281  COMPLEX*16 DUMMY( 1 )
282 * ..
283 * .. External Functions ..
284  LOGICAL LSAME
285  INTEGER IZAMAX
286  DOUBLE PRECISION DLAMCH, DZNRM2
287  COMPLEX*16 ZDOTC
288  EXTERNAL lsame, izamax, dlamch, dznrm2, zdotc
289 * ..
290 * .. External Subroutines ..
291  EXTERNAL xerbla, zdrscl, zlacn2, zlacpy, zlatrs, ztrexc,
292  $ dlabad
293 * ..
294 * .. Intrinsic Functions ..
295  INTRINSIC abs, dble, dimag, max
296 * ..
297 * .. Statement Functions ..
298  DOUBLE PRECISION CABS1
299 * ..
300 * .. Statement Function definitions ..
301  cabs1( cdum ) = abs( dble( cdum ) ) + abs( dimag( cdum ) )
302 * ..
303 * .. Executable Statements ..
304 *
305 * Decode and test the input parameters
306 *
307  wantbh = lsame( job, 'B' )
308  wants = lsame( job, 'E' ) .OR. wantbh
309  wantsp = lsame( job, 'V' ) .OR. wantbh
310 *
311  somcon = lsame( howmny, 'S' )
312 *
313 * Set M to the number of eigenpairs for which condition numbers are
314 * to be computed.
315 *
316  IF( somcon ) THEN
317  m = 0
318  DO 10 j = 1, n
319  IF( SELECT( j ) )
320  $ m = m + 1
321  10 CONTINUE
322  ELSE
323  m = n
324  END IF
325 *
326  info = 0
327  IF( .NOT.wants .AND. .NOT.wantsp ) THEN
328  info = -1
329  ELSE IF( .NOT.lsame( howmny, 'A' ) .AND. .NOT.somcon ) THEN
330  info = -2
331  ELSE IF( n.LT.0 ) THEN
332  info = -4
333  ELSE IF( ldt.LT.max( 1, n ) ) THEN
334  info = -6
335  ELSE IF( ldvl.LT.1 .OR. ( wants .AND. ldvl.LT.n ) ) THEN
336  info = -8
337  ELSE IF( ldvr.LT.1 .OR. ( wants .AND. ldvr.LT.n ) ) THEN
338  info = -10
339  ELSE IF( mm.LT.m ) THEN
340  info = -13
341  ELSE IF( ldwork.LT.1 .OR. ( wantsp .AND. ldwork.LT.n ) ) THEN
342  info = -16
343  END IF
344  IF( info.NE.0 ) THEN
345  CALL xerbla( 'ZTRSNA', -info )
346  RETURN
347  END IF
348 *
349 * Quick return if possible
350 *
351  IF( n.EQ.0 )
352  $ RETURN
353 *
354  IF( n.EQ.1 ) THEN
355  IF( somcon ) THEN
356  IF( .NOT.SELECT( 1 ) )
357  $ RETURN
358  END IF
359  IF( wants )
360  $ s( 1 ) = one
361  IF( wantsp )
362  $ sep( 1 ) = abs( t( 1, 1 ) )
363  RETURN
364  END IF
365 *
366 * Get machine constants
367 *
368  eps = dlamch( 'P' )
369  smlnum = dlamch( 'S' ) / eps
370  bignum = one / smlnum
371  CALL dlabad( smlnum, bignum )
372 *
373  ks = 1
374  DO 50 k = 1, n
375 *
376  IF( somcon ) THEN
377  IF( .NOT.SELECT( k ) )
378  $ GO TO 50
379  END IF
380 *
381  IF( wants ) THEN
382 *
383 * Compute the reciprocal condition number of the k-th
384 * eigenvalue.
385 *
386  prod = zdotc( n, vr( 1, ks ), 1, vl( 1, ks ), 1 )
387  rnrm = dznrm2( n, vr( 1, ks ), 1 )
388  lnrm = dznrm2( n, vl( 1, ks ), 1 )
389  s( ks ) = abs( prod ) / ( rnrm*lnrm )
390 *
391  END IF
392 *
393  IF( wantsp ) THEN
394 *
395 * Estimate the reciprocal condition number of the k-th
396 * eigenvector.
397 *
398 * Copy the matrix T to the array WORK and swap the k-th
399 * diagonal element to the (1,1) position.
400 *
401  CALL zlacpy( 'Full', n, n, t, ldt, work, ldwork )
402  CALL ztrexc( 'No Q', n, work, ldwork, dummy, 1, k, 1, ierr )
403 *
404 * Form C = T22 - lambda*I in WORK(2:N,2:N).
405 *
406  DO 20 i = 2, n
407  work( i, i ) = work( i, i ) - work( 1, 1 )
408  20 CONTINUE
409 *
410 * Estimate a lower bound for the 1-norm of inv(C**H). The 1st
411 * and (N+1)th columns of WORK are used to store work vectors.
412 *
413  sep( ks ) = zero
414  est = zero
415  kase = 0
416  normin = 'N'
417  30 CONTINUE
418  CALL zlacn2( n-1, work( 1, n+1 ), work, est, kase, isave )
419 *
420  IF( kase.NE.0 ) THEN
421  IF( kase.EQ.1 ) THEN
422 *
423 * Solve C**H*x = scale*b
424 *
425  CALL zlatrs( 'Upper', 'Conjugate transpose',
426  $ 'Nonunit', normin, n-1, work( 2, 2 ),
427  $ ldwork, work, scale, rwork, ierr )
428  ELSE
429 *
430 * Solve C*x = scale*b
431 *
432  CALL zlatrs( 'Upper', 'No transpose', 'Nonunit',
433  $ normin, n-1, work( 2, 2 ), ldwork, work,
434  $ scale, rwork, ierr )
435  END IF
436  normin = 'Y'
437  IF( scale.NE.one ) THEN
438 *
439 * Multiply by 1/SCALE if doing so will not cause
440 * overflow.
441 *
442  ix = izamax( n-1, work, 1 )
443  xnorm = cabs1( work( ix, 1 ) )
444  IF( scale.LT.xnorm*smlnum .OR. scale.EQ.zero )
445  $ GO TO 40
446  CALL zdrscl( n, scale, work, 1 )
447  END IF
448  GO TO 30
449  END IF
450 *
451  sep( ks ) = one / max( est, smlnum )
452  END IF
453 *
454  40 CONTINUE
455  ks = ks + 1
456  50 CONTINUE
457  RETURN
458 *
459 * End of ZTRSNA
460 *
461  END
subroutine dlabad(SMALL, LARGE)
DLABAD
Definition: dlabad.f:74
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
subroutine zlacn2(N, V, X, EST, KASE, ISAVE)
ZLACN2 estimates the 1-norm of a square matrix, using reverse communication for evaluating matrix-vec...
Definition: zlacn2.f:133
subroutine zlacpy(UPLO, M, N, A, LDA, B, LDB)
ZLACPY copies all or part of one two-dimensional array to another.
Definition: zlacpy.f:103
subroutine zlatrs(UPLO, TRANS, DIAG, NORMIN, N, A, LDA, X, SCALE, CNORM, INFO)
ZLATRS solves a triangular system of equations with the scale factor set to prevent overflow.
Definition: zlatrs.f:239
subroutine zdrscl(N, SA, SX, INCX)
ZDRSCL multiplies a vector by the reciprocal of a real scalar.
Definition: zdrscl.f:84
subroutine ztrexc(COMPQ, N, T, LDT, Q, LDQ, IFST, ILST, INFO)
ZTREXC
Definition: ztrexc.f:126
subroutine ztrsna(JOB, HOWMNY, SELECT, N, T, LDT, VL, LDVL, VR, LDVR, S, SEP, MM, M, WORK, LDWORK, RWORK, INFO)
ZTRSNA
Definition: ztrsna.f:249