LAPACK  3.10.0
LAPACK: Linear Algebra PACKage
ctgsna.f
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1 *> \brief \b CTGSNA
2 *
3 * =========== DOCUMENTATION ===========
4 *
5 * Online html documentation available at
6 * http://www.netlib.org/lapack/explore-html/
7 *
8 *> \htmlonly
9 *> Download CTGSNA + dependencies
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11 *> [TGZ]</a>
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13 *> [ZIP]</a>
14 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/ctgsna.f">
15 *> [TXT]</a>
16 *> \endhtmlonly
17 *
18 * Definition:
19 * ===========
20 *
21 * SUBROUTINE CTGSNA( JOB, HOWMNY, SELECT, N, A, LDA, B, LDB, VL,
22 * LDVL, VR, LDVR, S, DIF, MM, M, WORK, LWORK,
23 * IWORK, INFO )
24 *
25 * .. Scalar Arguments ..
26 * CHARACTER HOWMNY, JOB
27 * INTEGER INFO, LDA, LDB, LDVL, LDVR, LWORK, M, MM, N
28 * ..
29 * .. Array Arguments ..
30 * LOGICAL SELECT( * )
31 * INTEGER IWORK( * )
32 * REAL DIF( * ), S( * )
33 * COMPLEX A( LDA, * ), B( LDB, * ), VL( LDVL, * ),
34 * $ VR( LDVR, * ), WORK( * )
35 * ..
36 *
37 *
38 *> \par Purpose:
39 * =============
40 *>
41 *> \verbatim
42 *>
43 *> CTGSNA estimates reciprocal condition numbers for specified
44 *> eigenvalues and/or eigenvectors of a matrix pair (A, B).
45 *>
46 *> (A, B) must be in generalized Schur canonical form, that is, A and
47 *> B are both upper triangular.
48 *> \endverbatim
49 *
50 * Arguments:
51 * ==========
52 *
53 *> \param[in] JOB
54 *> \verbatim
55 *> JOB is CHARACTER*1
56 *> Specifies whether condition numbers are required for
57 *> eigenvalues (S) or eigenvectors (DIF):
58 *> = 'E': for eigenvalues only (S);
59 *> = 'V': for eigenvectors only (DIF);
60 *> = 'B': for both eigenvalues and eigenvectors (S and DIF).
61 *> \endverbatim
62 *>
63 *> \param[in] HOWMNY
64 *> \verbatim
65 *> HOWMNY is CHARACTER*1
66 *> = 'A': compute condition numbers for all eigenpairs;
67 *> = 'S': compute condition numbers for selected eigenpairs
68 *> specified by the array SELECT.
69 *> \endverbatim
70 *>
71 *> \param[in] SELECT
72 *> \verbatim
73 *> SELECT is LOGICAL array, dimension (N)
74 *> If HOWMNY = 'S', SELECT specifies the eigenpairs for which
75 *> condition numbers are required. To select condition numbers
76 *> for the corresponding j-th eigenvalue and/or eigenvector,
77 *> SELECT(j) must be set to .TRUE..
78 *> If HOWMNY = 'A', SELECT is not referenced.
79 *> \endverbatim
80 *>
81 *> \param[in] N
82 *> \verbatim
83 *> N is INTEGER
84 *> The order of the square matrix pair (A, B). N >= 0.
85 *> \endverbatim
86 *>
87 *> \param[in] A
88 *> \verbatim
89 *> A is COMPLEX array, dimension (LDA,N)
90 *> The upper triangular matrix A in the pair (A,B).
91 *> \endverbatim
92 *>
93 *> \param[in] LDA
94 *> \verbatim
95 *> LDA is INTEGER
96 *> The leading dimension of the array A. LDA >= max(1,N).
97 *> \endverbatim
98 *>
99 *> \param[in] B
100 *> \verbatim
101 *> B is COMPLEX array, dimension (LDB,N)
102 *> The upper triangular matrix B in the pair (A, B).
103 *> \endverbatim
104 *>
105 *> \param[in] LDB
106 *> \verbatim
107 *> LDB is INTEGER
108 *> The leading dimension of the array B. LDB >= max(1,N).
109 *> \endverbatim
110 *>
111 *> \param[in] VL
112 *> \verbatim
113 *> VL is COMPLEX array, dimension (LDVL,M)
114 *> IF JOB = 'E' or 'B', VL must contain left eigenvectors of
115 *> (A, B), corresponding to the eigenpairs specified by HOWMNY
116 *> and SELECT. The eigenvectors must be stored in consecutive
117 *> columns of VL, as returned by CTGEVC.
118 *> If JOB = 'V', VL is not referenced.
119 *> \endverbatim
120 *>
121 *> \param[in] LDVL
122 *> \verbatim
123 *> LDVL is INTEGER
124 *> The leading dimension of the array VL. LDVL >= 1; and
125 *> If JOB = 'E' or 'B', LDVL >= N.
126 *> \endverbatim
127 *>
128 *> \param[in] VR
129 *> \verbatim
130 *> VR is COMPLEX array, dimension (LDVR,M)
131 *> IF JOB = 'E' or 'B', VR must contain right eigenvectors of
132 *> (A, B), corresponding to the eigenpairs specified by HOWMNY
133 *> and SELECT. The eigenvectors must be stored in consecutive
134 *> columns of VR, as returned by CTGEVC.
135 *> If JOB = 'V', VR is not referenced.
136 *> \endverbatim
137 *>
138 *> \param[in] LDVR
139 *> \verbatim
140 *> LDVR is INTEGER
141 *> The leading dimension of the array VR. LDVR >= 1;
142 *> If JOB = 'E' or 'B', LDVR >= N.
143 *> \endverbatim
144 *>
145 *> \param[out] S
146 *> \verbatim
147 *> S is REAL array, dimension (MM)
148 *> If JOB = 'E' or 'B', the reciprocal condition numbers of the
149 *> selected eigenvalues, stored in consecutive elements of the
150 *> array.
151 *> If JOB = 'V', S is not referenced.
152 *> \endverbatim
153 *>
154 *> \param[out] DIF
155 *> \verbatim
156 *> DIF is REAL array, dimension (MM)
157 *> If JOB = 'V' or 'B', the estimated reciprocal condition
158 *> numbers of the selected eigenvectors, stored in consecutive
159 *> elements of the array.
160 *> If the eigenvalues cannot be reordered to compute DIF(j),
161 *> DIF(j) is set to 0; this can only occur when the true value
162 *> would be very small anyway.
163 *> For each eigenvalue/vector specified by SELECT, DIF stores
164 *> a Frobenius norm-based estimate of Difl.
165 *> If JOB = 'E', DIF is not referenced.
166 *> \endverbatim
167 *>
168 *> \param[in] MM
169 *> \verbatim
170 *> MM is INTEGER
171 *> The number of elements in the arrays S and DIF. MM >= M.
172 *> \endverbatim
173 *>
174 *> \param[out] M
175 *> \verbatim
176 *> M is INTEGER
177 *> The number of elements of the arrays S and DIF used to store
178 *> the specified condition numbers; for each selected eigenvalue
179 *> one element is used. If HOWMNY = 'A', M is set to N.
180 *> \endverbatim
181 *>
182 *> \param[out] WORK
183 *> \verbatim
184 *> WORK is COMPLEX array, dimension (MAX(1,LWORK))
185 *> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
186 *> \endverbatim
187 *>
188 *> \param[in] LWORK
189 *> \verbatim
190 *> LWORK is INTEGER
191 *> The dimension of the array WORK. LWORK >= max(1,N).
192 *> If JOB = 'V' or 'B', LWORK >= max(1,2*N*N).
193 *> \endverbatim
194 *>
195 *> \param[out] IWORK
196 *> \verbatim
197 *> IWORK is INTEGER array, dimension (N+2)
198 *> If JOB = 'E', IWORK is not referenced.
199 *> \endverbatim
200 *>
201 *> \param[out] INFO
202 *> \verbatim
203 *> INFO is INTEGER
204 *> = 0: Successful exit
205 *> < 0: If INFO = -i, the i-th argument had an illegal value
206 *> \endverbatim
207 *
208 * Authors:
209 * ========
210 *
211 *> \author Univ. of Tennessee
212 *> \author Univ. of California Berkeley
213 *> \author Univ. of Colorado Denver
214 *> \author NAG Ltd.
215 *
216 *> \ingroup complexOTHERcomputational
217 *
218 *> \par Further Details:
219 * =====================
220 *>
221 *> \verbatim
222 *>
223 *> The reciprocal of the condition number of the i-th generalized
224 *> eigenvalue w = (a, b) is defined as
225 *>
226 *> S(I) = (|v**HAu|**2 + |v**HBu|**2)**(1/2) / (norm(u)*norm(v))
227 *>
228 *> where u and v are the right and left eigenvectors of (A, B)
229 *> corresponding to w; |z| denotes the absolute value of the complex
230 *> number, and norm(u) denotes the 2-norm of the vector u. The pair
231 *> (a, b) corresponds to an eigenvalue w = a/b (= v**HAu/v**HBu) of the
232 *> matrix pair (A, B). If both a and b equal zero, then (A,B) is
233 *> singular and S(I) = -1 is returned.
234 *>
235 *> An approximate error bound on the chordal distance between the i-th
236 *> computed generalized eigenvalue w and the corresponding exact
237 *> eigenvalue lambda is
238 *>
239 *> chord(w, lambda) <= EPS * norm(A, B) / S(I),
240 *>
241 *> where EPS is the machine precision.
242 *>
243 *> The reciprocal of the condition number of the right eigenvector u
244 *> and left eigenvector v corresponding to the generalized eigenvalue w
245 *> is defined as follows. Suppose
246 *>
247 *> (A, B) = ( a * ) ( b * ) 1
248 *> ( 0 A22 ),( 0 B22 ) n-1
249 *> 1 n-1 1 n-1
250 *>
251 *> Then the reciprocal condition number DIF(I) is
252 *>
253 *> Difl[(a, b), (A22, B22)] = sigma-min( Zl )
254 *>
255 *> where sigma-min(Zl) denotes the smallest singular value of
256 *>
257 *> Zl = [ kron(a, In-1) -kron(1, A22) ]
258 *> [ kron(b, In-1) -kron(1, B22) ].
259 *>
260 *> Here In-1 is the identity matrix of size n-1 and X**H is the conjugate
261 *> transpose of X. kron(X, Y) is the Kronecker product between the
262 *> matrices X and Y.
263 *>
264 *> We approximate the smallest singular value of Zl with an upper
265 *> bound. This is done by CLATDF.
266 *>
267 *> An approximate error bound for a computed eigenvector VL(i) or
268 *> VR(i) is given by
269 *>
270 *> EPS * norm(A, B) / DIF(i).
271 *>
272 *> See ref. [2-3] for more details and further references.
273 *> \endverbatim
274 *
275 *> \par Contributors:
276 * ==================
277 *>
278 *> Bo Kagstrom and Peter Poromaa, Department of Computing Science,
279 *> Umea University, S-901 87 Umea, Sweden.
280 *
281 *> \par References:
282 * ================
283 *>
284 *> \verbatim
285 *>
286 *> [1] B. Kagstrom; A Direct Method for Reordering Eigenvalues in the
287 *> Generalized Real Schur Form of a Regular Matrix Pair (A, B), in
288 *> M.S. Moonen et al (eds), Linear Algebra for Large Scale and
289 *> Real-Time Applications, Kluwer Academic Publ. 1993, pp 195-218.
290 *>
291 *> [2] B. Kagstrom and P. Poromaa; Computing Eigenspaces with Specified
292 *> Eigenvalues of a Regular Matrix Pair (A, B) and Condition
293 *> Estimation: Theory, Algorithms and Software, Report
294 *> UMINF - 94.04, Department of Computing Science, Umea University,
295 *> S-901 87 Umea, Sweden, 1994. Also as LAPACK Working Note 87.
296 *> To appear in Numerical Algorithms, 1996.
297 *>
298 *> [3] B. Kagstrom and P. Poromaa, LAPACK-Style Algorithms and Software
299 *> for Solving the Generalized Sylvester Equation and Estimating the
300 *> Separation between Regular Matrix Pairs, Report UMINF - 93.23,
301 *> Department of Computing Science, Umea University, S-901 87 Umea,
302 *> Sweden, December 1993, Revised April 1994, Also as LAPACK Working
303 *> Note 75.
304 *> To appear in ACM Trans. on Math. Software, Vol 22, No 1, 1996.
305 *> \endverbatim
306 *>
307 * =====================================================================
308  SUBROUTINE ctgsna( JOB, HOWMNY, SELECT, N, A, LDA, B, LDB, VL,
309  $ LDVL, VR, LDVR, S, DIF, MM, M, WORK, LWORK,
310  $ IWORK, INFO )
311 *
312 * -- LAPACK computational routine --
313 * -- LAPACK is a software package provided by Univ. of Tennessee, --
314 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
315 *
316 * .. Scalar Arguments ..
317  CHARACTER HOWMNY, JOB
318  INTEGER INFO, LDA, LDB, LDVL, LDVR, LWORK, M, MM, N
319 * ..
320 * .. Array Arguments ..
321  LOGICAL SELECT( * )
322  INTEGER IWORK( * )
323  REAL DIF( * ), S( * )
324  COMPLEX A( LDA, * ), B( LDB, * ), VL( LDVL, * ),
325  $ vr( ldvr, * ), work( * )
326 * ..
327 *
328 * =====================================================================
329 *
330 * .. Parameters ..
331  REAL ZERO, ONE
332  INTEGER IDIFJB
333  parameter( zero = 0.0e+0, one = 1.0e+0, idifjb = 3 )
334 * ..
335 * .. Local Scalars ..
336  LOGICAL LQUERY, SOMCON, WANTBH, WANTDF, WANTS
337  INTEGER I, IERR, IFST, ILST, K, KS, LWMIN, N1, N2
338  REAL BIGNUM, COND, EPS, LNRM, RNRM, SCALE, SMLNUM
339  COMPLEX YHAX, YHBX
340 * ..
341 * .. Local Arrays ..
342  COMPLEX DUMMY( 1 ), DUMMY1( 1 )
343 * ..
344 * .. External Functions ..
345  LOGICAL LSAME
346  REAL SCNRM2, SLAMCH, SLAPY2
347  COMPLEX CDOTC
348  EXTERNAL lsame, scnrm2, slamch, slapy2, cdotc
349 * ..
350 * .. External Subroutines ..
351  EXTERNAL cgemv, clacpy, ctgexc, ctgsyl, slabad, xerbla
352 * ..
353 * .. Intrinsic Functions ..
354  INTRINSIC abs, cmplx, max
355 * ..
356 * .. Executable Statements ..
357 *
358 * Decode and test the input parameters
359 *
360  wantbh = lsame( job, 'B' )
361  wants = lsame( job, 'E' ) .OR. wantbh
362  wantdf = lsame( job, 'V' ) .OR. wantbh
363 *
364  somcon = lsame( howmny, 'S' )
365 *
366  info = 0
367  lquery = ( lwork.EQ.-1 )
368 *
369  IF( .NOT.wants .AND. .NOT.wantdf ) THEN
370  info = -1
371  ELSE IF( .NOT.lsame( howmny, 'A' ) .AND. .NOT.somcon ) THEN
372  info = -2
373  ELSE IF( n.LT.0 ) THEN
374  info = -4
375  ELSE IF( lda.LT.max( 1, n ) ) THEN
376  info = -6
377  ELSE IF( ldb.LT.max( 1, n ) ) THEN
378  info = -8
379  ELSE IF( wants .AND. ldvl.LT.n ) THEN
380  info = -10
381  ELSE IF( wants .AND. ldvr.LT.n ) THEN
382  info = -12
383  ELSE
384 *
385 * Set M to the number of eigenpairs for which condition numbers
386 * are required, and test MM.
387 *
388  IF( somcon ) THEN
389  m = 0
390  DO 10 k = 1, n
391  IF( SELECT( k ) )
392  $ m = m + 1
393  10 CONTINUE
394  ELSE
395  m = n
396  END IF
397 *
398  IF( n.EQ.0 ) THEN
399  lwmin = 1
400  ELSE IF( lsame( job, 'V' ) .OR. lsame( job, 'B' ) ) THEN
401  lwmin = 2*n*n
402  ELSE
403  lwmin = n
404  END IF
405  work( 1 ) = lwmin
406 *
407  IF( mm.LT.m ) THEN
408  info = -15
409  ELSE IF( lwork.LT.lwmin .AND. .NOT.lquery ) THEN
410  info = -18
411  END IF
412  END IF
413 *
414  IF( info.NE.0 ) THEN
415  CALL xerbla( 'CTGSNA', -info )
416  RETURN
417  ELSE IF( lquery ) THEN
418  RETURN
419  END IF
420 *
421 * Quick return if possible
422 *
423  IF( n.EQ.0 )
424  $ RETURN
425 *
426 * Get machine constants
427 *
428  eps = slamch( 'P' )
429  smlnum = slamch( 'S' ) / eps
430  bignum = one / smlnum
431  CALL slabad( smlnum, bignum )
432  ks = 0
433  DO 20 k = 1, n
434 *
435 * Determine whether condition numbers are required for the k-th
436 * eigenpair.
437 *
438  IF( somcon ) THEN
439  IF( .NOT.SELECT( k ) )
440  $ GO TO 20
441  END IF
442 *
443  ks = ks + 1
444 *
445  IF( wants ) THEN
446 *
447 * Compute the reciprocal condition number of the k-th
448 * eigenvalue.
449 *
450  rnrm = scnrm2( n, vr( 1, ks ), 1 )
451  lnrm = scnrm2( n, vl( 1, ks ), 1 )
452  CALL cgemv( 'N', n, n, cmplx( one, zero ), a, lda,
453  $ vr( 1, ks ), 1, cmplx( zero, zero ), work, 1 )
454  yhax = cdotc( n, work, 1, vl( 1, ks ), 1 )
455  CALL cgemv( 'N', n, n, cmplx( one, zero ), b, ldb,
456  $ vr( 1, ks ), 1, cmplx( zero, zero ), work, 1 )
457  yhbx = cdotc( n, work, 1, vl( 1, ks ), 1 )
458  cond = slapy2( abs( yhax ), abs( yhbx ) )
459  IF( cond.EQ.zero ) THEN
460  s( ks ) = -one
461  ELSE
462  s( ks ) = cond / ( rnrm*lnrm )
463  END IF
464  END IF
465 *
466  IF( wantdf ) THEN
467  IF( n.EQ.1 ) THEN
468  dif( ks ) = slapy2( abs( a( 1, 1 ) ), abs( b( 1, 1 ) ) )
469  ELSE
470 *
471 * Estimate the reciprocal condition number of the k-th
472 * eigenvectors.
473 *
474 * Copy the matrix (A, B) to the array WORK and move the
475 * (k,k)th pair to the (1,1) position.
476 *
477  CALL clacpy( 'Full', n, n, a, lda, work, n )
478  CALL clacpy( 'Full', n, n, b, ldb, work( n*n+1 ), n )
479  ifst = k
480  ilst = 1
481 *
482  CALL ctgexc( .false., .false., n, work, n, work( n*n+1 ),
483  $ n, dummy, 1, dummy1, 1, ifst, ilst, ierr )
484 *
485  IF( ierr.GT.0 ) THEN
486 *
487 * Ill-conditioned problem - swap rejected.
488 *
489  dif( ks ) = zero
490  ELSE
491 *
492 * Reordering successful, solve generalized Sylvester
493 * equation for R and L,
494 * A22 * R - L * A11 = A12
495 * B22 * R - L * B11 = B12,
496 * and compute estimate of Difl[(A11,B11), (A22, B22)].
497 *
498  n1 = 1
499  n2 = n - n1
500  i = n*n + 1
501  CALL ctgsyl( 'N', idifjb, n2, n1, work( n*n1+n1+1 ),
502  $ n, work, n, work( n1+1 ), n,
503  $ work( n*n1+n1+i ), n, work( i ), n,
504  $ work( n1+i ), n, scale, dif( ks ), dummy,
505  $ 1, iwork, ierr )
506  END IF
507  END IF
508  END IF
509 *
510  20 CONTINUE
511  work( 1 ) = lwmin
512  RETURN
513 *
514 * End of CTGSNA
515 *
516  END
subroutine slabad(SMALL, LARGE)
SLABAD
Definition: slabad.f:74
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
subroutine cgemv(TRANS, M, N, ALPHA, A, LDA, X, INCX, BETA, Y, INCY)
CGEMV
Definition: cgemv.f:158
subroutine ctgexc(WANTQ, WANTZ, N, A, LDA, B, LDB, Q, LDQ, Z, LDZ, IFST, ILST, INFO)
CTGEXC
Definition: ctgexc.f:200
subroutine clacpy(UPLO, M, N, A, LDA, B, LDB)
CLACPY copies all or part of one two-dimensional array to another.
Definition: clacpy.f:103
subroutine ctgsna(JOB, HOWMNY, SELECT, N, A, LDA, B, LDB, VL, LDVL, VR, LDVR, S, DIF, MM, M, WORK, LWORK, IWORK, INFO)
CTGSNA
Definition: ctgsna.f:311
subroutine ctgsyl(TRANS, IJOB, M, N, A, LDA, B, LDB, C, LDC, D, LDD, E, LDE, F, LDF, SCALE, DIF, WORK, LWORK, IWORK, INFO)
CTGSYL
Definition: ctgsyl.f:295