LAPACK  3.4.2 LAPACK: Linear Algebra PACKage
ztgsna.f
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1 *> \brief \b ZTGSNA
2 *
3 * =========== DOCUMENTATION ===========
4 *
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17 *
18 * Definition:
19 * ===========
20 *
21 * SUBROUTINE ZTGSNA( 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 * DOUBLE PRECISION DIF( * ), S( * )
33 * COMPLEX*16 A( LDA, * ), B( LDB, * ), VL( LDVL, * ),
34 * \$ VR( LDVR, * ), WORK( * )
35 * ..
36 *
37 *
38 *> \par Purpose:
39 * =============
40 *>
41 *> \verbatim
42 *>
43 *> ZTGSNA 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*16 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*16 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*16 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 ZTGEVC.
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*16 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 ZTGEVC.
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 DOUBLE PRECISION 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 DOUBLE PRECISION 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*16 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 *> \date November 2011
217 *
218 *> \ingroup complex16OTHERcomputational
219 *
220 *> \par Further Details:
221 * =====================
222 *>
223 *> \verbatim
224 *>
225 *> The reciprocal of the condition number of the i-th generalized
226 *> eigenvalue w = (a, b) is defined as
227 *>
228 *> S(I) = (|v**HAu|**2 + |v**HBu|**2)**(1/2) / (norm(u)*norm(v))
229 *>
230 *> where u and v are the right and left eigenvectors of (A, B)
231 *> corresponding to w; |z| denotes the absolute value of the complex
232 *> number, and norm(u) denotes the 2-norm of the vector u. The pair
233 *> (a, b) corresponds to an eigenvalue w = a/b (= v**HAu/v**HBu) of the
234 *> matrix pair (A, B). If both a and b equal zero, then (A,B) is
235 *> singular and S(I) = -1 is returned.
236 *>
237 *> An approximate error bound on the chordal distance between the i-th
238 *> computed generalized eigenvalue w and the corresponding exact
239 *> eigenvalue lambda is
240 *>
241 *> chord(w, lambda) <= EPS * norm(A, B) / S(I),
242 *>
243 *> where EPS is the machine precision.
244 *>
245 *> The reciprocal of the condition number of the right eigenvector u
246 *> and left eigenvector v corresponding to the generalized eigenvalue w
247 *> is defined as follows. Suppose
248 *>
249 *> (A, B) = ( a * ) ( b * ) 1
250 *> ( 0 A22 ),( 0 B22 ) n-1
251 *> 1 n-1 1 n-1
252 *>
253 *> Then the reciprocal condition number DIF(I) is
254 *>
255 *> Difl[(a, b), (A22, B22)] = sigma-min( Zl )
256 *>
257 *> where sigma-min(Zl) denotes the smallest singular value of
258 *>
259 *> Zl = [ kron(a, In-1) -kron(1, A22) ]
260 *> [ kron(b, In-1) -kron(1, B22) ].
261 *>
262 *> Here In-1 is the identity matrix of size n-1 and X**H is the conjugate
263 *> transpose of X. kron(X, Y) is the Kronecker product between the
264 *> matrices X and Y.
265 *>
266 *> We approximate the smallest singular value of Zl with an upper
267 *> bound. This is done by ZLATDF.
268 *>
269 *> An approximate error bound for a computed eigenvector VL(i) or
270 *> VR(i) is given by
271 *>
272 *> EPS * norm(A, B) / DIF(i).
273 *>
274 *> See ref. [2-3] for more details and further references.
275 *> \endverbatim
276 *
277 *> \par Contributors:
278 * ==================
279 *>
280 *> Bo Kagstrom and Peter Poromaa, Department of Computing Science,
281 *> Umea University, S-901 87 Umea, Sweden.
282 *
283 *> \par References:
284 * ================
285 *>
286 *> \verbatim
287 *>
288 *> [1] B. Kagstrom; A Direct Method for Reordering Eigenvalues in the
289 *> Generalized Real Schur Form of a Regular Matrix Pair (A, B), in
290 *> M.S. Moonen et al (eds), Linear Algebra for Large Scale and
291 *> Real-Time Applications, Kluwer Academic Publ. 1993, pp 195-218.
292 *>
293 *> [2] B. Kagstrom and P. Poromaa; Computing Eigenspaces with Specified
294 *> Eigenvalues of a Regular Matrix Pair (A, B) and Condition
295 *> Estimation: Theory, Algorithms and Software, Report
296 *> UMINF - 94.04, Department of Computing Science, Umea University,
297 *> S-901 87 Umea, Sweden, 1994. Also as LAPACK Working Note 87.
298 *> To appear in Numerical Algorithms, 1996.
299 *>
300 *> [3] B. Kagstrom and P. Poromaa, LAPACK-Style Algorithms and Software
301 *> for Solving the Generalized Sylvester Equation and Estimating the
302 *> Separation between Regular Matrix Pairs, Report UMINF - 93.23,
303 *> Department of Computing Science, Umea University, S-901 87 Umea,
304 *> Sweden, December 1993, Revised April 1994, Also as LAPACK Working
305 *> Note 75.
306 *> To appear in ACM Trans. on Math. Software, Vol 22, No 1, 1996.
307 *> \endverbatim
308 *>
309 * =====================================================================
310  SUBROUTINE ztgsna( JOB, HOWMNY, SELECT, N, A, LDA, B, LDB, VL,
311  \$ ldvl, vr, ldvr, s, dif, mm, m, work, lwork,
312  \$ iwork, info )
313 *
314 * -- LAPACK computational routine (version 3.4.0) --
315 * -- LAPACK is a software package provided by Univ. of Tennessee, --
316 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
317 * November 2011
318 *
319 * .. Scalar Arguments ..
320  CHARACTER howmny, job
321  INTEGER info, lda, ldb, ldvl, ldvr, lwork, m, mm, n
322 * ..
323 * .. Array Arguments ..
324  LOGICAL select( * )
325  INTEGER iwork( * )
326  DOUBLE PRECISION dif( * ), s( * )
327  COMPLEX*16 a( lda, * ), b( ldb, * ), vl( ldvl, * ),
328  \$ vr( ldvr, * ), work( * )
329 * ..
330 *
331 * =====================================================================
332 *
333 * .. Parameters ..
334  DOUBLE PRECISION zero, one
335  INTEGER idifjb
336  parameter( zero = 0.0d+0, one = 1.0d+0, idifjb = 3 )
337 * ..
338 * .. Local Scalars ..
339  LOGICAL lquery, somcon, wantbh, wantdf, wants
340  INTEGER i, ierr, ifst, ilst, k, ks, lwmin, n1, n2
341  DOUBLE PRECISION bignum, cond, eps, lnrm, rnrm, scale, smlnum
342  COMPLEX*16 yhax, yhbx
343 * ..
344 * .. Local Arrays ..
345  COMPLEX*16 dummy( 1 ), dummy1( 1 )
346 * ..
347 * .. External Functions ..
348  LOGICAL lsame
349  DOUBLE PRECISION dlamch, dlapy2, dznrm2
350  COMPLEX*16 zdotc
351  EXTERNAL lsame, dlamch, dlapy2, dznrm2, zdotc
352 * ..
353 * .. External Subroutines ..
354  EXTERNAL dlabad, xerbla, zgemv, zlacpy, ztgexc, ztgsyl
355 * ..
356 * .. Intrinsic Functions ..
357  INTRINSIC abs, dcmplx, max
358 * ..
359 * .. Executable Statements ..
360 *
361 * Decode and test the input parameters
362 *
363  wantbh = lsame( job, 'B' )
364  wants = lsame( job, 'E' ) .OR. wantbh
365  wantdf = lsame( job, 'V' ) .OR. wantbh
366 *
367  somcon = lsame( howmny, 'S' )
368 *
369  info = 0
370  lquery = ( lwork.EQ.-1 )
371 *
372  IF( .NOT.wants .AND. .NOT.wantdf ) THEN
373  info = -1
374  ELSE IF( .NOT.lsame( howmny, 'A' ) .AND. .NOT.somcon ) THEN
375  info = -2
376  ELSE IF( n.LT.0 ) THEN
377  info = -4
378  ELSE IF( lda.LT.max( 1, n ) ) THEN
379  info = -6
380  ELSE IF( ldb.LT.max( 1, n ) ) THEN
381  info = -8
382  ELSE IF( wants .AND. ldvl.LT.n ) THEN
383  info = -10
384  ELSE IF( wants .AND. ldvr.LT.n ) THEN
385  info = -12
386  ELSE
387 *
388 * Set M to the number of eigenpairs for which condition numbers
389 * are required, and test MM.
390 *
391  IF( somcon ) THEN
392  m = 0
393  DO 10 k = 1, n
394  IF( SELECT( k ) )
395  \$ m = m + 1
396  10 continue
397  ELSE
398  m = n
399  END IF
400 *
401  IF( n.EQ.0 ) THEN
402  lwmin = 1
403  ELSE IF( lsame( job, 'V' ) .OR. lsame( job, 'B' ) ) THEN
404  lwmin = 2*n*n
405  ELSE
406  lwmin = n
407  END IF
408  work( 1 ) = lwmin
409 *
410  IF( mm.LT.m ) THEN
411  info = -15
412  ELSE IF( lwork.LT.lwmin .AND. .NOT.lquery ) THEN
413  info = -18
414  END IF
415  END IF
416 *
417  IF( info.NE.0 ) THEN
418  CALL xerbla( 'ZTGSNA', -info )
419  return
420  ELSE IF( lquery ) THEN
421  return
422  END IF
423 *
424 * Quick return if possible
425 *
426  IF( n.EQ.0 )
427  \$ return
428 *
429 * Get machine constants
430 *
431  eps = dlamch( 'P' )
432  smlnum = dlamch( 'S' ) / eps
433  bignum = one / smlnum
434  CALL dlabad( smlnum, bignum )
435  ks = 0
436  DO 20 k = 1, n
437 *
438 * Determine whether condition numbers are required for the k-th
439 * eigenpair.
440 *
441  IF( somcon ) THEN
442  IF( .NOT.SELECT( k ) )
443  \$ go to 20
444  END IF
445 *
446  ks = ks + 1
447 *
448  IF( wants ) THEN
449 *
450 * Compute the reciprocal condition number of the k-th
451 * eigenvalue.
452 *
453  rnrm = dznrm2( n, vr( 1, ks ), 1 )
454  lnrm = dznrm2( n, vl( 1, ks ), 1 )
455  CALL zgemv( 'N', n, n, dcmplx( one, zero ), a, lda,
456  \$ vr( 1, ks ), 1, dcmplx( zero, zero ), work, 1 )
457  yhax = zdotc( n, work, 1, vl( 1, ks ), 1 )
458  CALL zgemv( 'N', n, n, dcmplx( one, zero ), b, ldb,
459  \$ vr( 1, ks ), 1, dcmplx( zero, zero ), work, 1 )
460  yhbx = zdotc( n, work, 1, vl( 1, ks ), 1 )
461  cond = dlapy2( abs( yhax ), abs( yhbx ) )
462  IF( cond.EQ.zero ) THEN
463  s( ks ) = -one
464  ELSE
465  s( ks ) = cond / ( rnrm*lnrm )
466  END IF
467  END IF
468 *
469  IF( wantdf ) THEN
470  IF( n.EQ.1 ) THEN
471  dif( ks ) = dlapy2( abs( a( 1, 1 ) ), abs( b( 1, 1 ) ) )
472  ELSE
473 *
474 * Estimate the reciprocal condition number of the k-th
475 * eigenvectors.
476 *
477 * Copy the matrix (A, B) to the array WORK and move the
478 * (k,k)th pair to the (1,1) position.
479 *
480  CALL zlacpy( 'Full', n, n, a, lda, work, n )
481  CALL zlacpy( 'Full', n, n, b, ldb, work( n*n+1 ), n )
482  ifst = k
483  ilst = 1
484 *
485  CALL ztgexc( .false., .false., n, work, n, work( n*n+1 ),
486  \$ n, dummy, 1, dummy1, 1, ifst, ilst, ierr )
487 *
488  IF( ierr.GT.0 ) THEN
489 *
490 * Ill-conditioned problem - swap rejected.
491 *
492  dif( ks ) = zero
493  ELSE
494 *
495 * Reordering successful, solve generalized Sylvester
496 * equation for R and L,
497 * A22 * R - L * A11 = A12
498 * B22 * R - L * B11 = B12,
499 * and compute estimate of Difl[(A11,B11), (A22, B22)].
500 *
501  n1 = 1
502  n2 = n - n1
503  i = n*n + 1
504  CALL ztgsyl( 'N', idifjb, n2, n1, work( n*n1+n1+1 ),
505  \$ n, work, n, work( n1+1 ), n,
506  \$ work( n*n1+n1+i ), n, work( i ), n,
507  \$ work( n1+i ), n, scale, dif( ks ), dummy,
508  \$ 1, iwork, ierr )
509  END IF
510  END IF
511  END IF
512 *
513  20 continue
514  work( 1 ) = lwmin
515  return
516 *
517 * End of ZTGSNA
518 *
519  END