LAPACK 3.12.0 LAPACK: Linear Algebra PACKage
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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
10*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/ctgsna.f">
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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 tgsna
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, SROUNDUP_LWORK
347 COMPLEX CDOTC
348 EXTERNAL lsame, scnrm2, slamch, slapy2, sroundup_lwork,
349 \$ cdotc
350* ..
351* .. External Subroutines ..
352 EXTERNAL cgemv, clacpy, ctgexc, ctgsyl, xerbla
353* ..
354* .. Intrinsic Functions ..
355 INTRINSIC abs, cmplx, max
356* ..
357* .. Executable Statements ..
358*
359* Decode and test the input parameters
360*
361 wantbh = lsame( job, 'B' )
362 wants = lsame( job, 'E' ) .OR. wantbh
363 wantdf = lsame( job, 'V' ) .OR. wantbh
364*
365 somcon = lsame( howmny, 'S' )
366*
367 info = 0
368 lquery = ( lwork.EQ.-1 )
369*
370 IF( .NOT.wants .AND. .NOT.wantdf ) THEN
371 info = -1
372 ELSE IF( .NOT.lsame( howmny, 'A' ) .AND. .NOT.somcon ) THEN
373 info = -2
374 ELSE IF( n.LT.0 ) THEN
375 info = -4
376 ELSE IF( lda.LT.max( 1, n ) ) THEN
377 info = -6
378 ELSE IF( ldb.LT.max( 1, n ) ) THEN
379 info = -8
380 ELSE IF( wants .AND. ldvl.LT.n ) THEN
381 info = -10
382 ELSE IF( wants .AND. ldvr.LT.n ) THEN
383 info = -12
384 ELSE
385*
386* Set M to the number of eigenpairs for which condition numbers
387* are required, and test MM.
388*
389 IF( somcon ) THEN
390 m = 0
391 DO 10 k = 1, n
392 IF( SELECT( k ) )
393 \$ m = m + 1
394 10 CONTINUE
395 ELSE
396 m = n
397 END IF
398*
399 IF( n.EQ.0 ) THEN
400 lwmin = 1
401 ELSE IF( lsame( job, 'V' ) .OR. lsame( job, 'B' ) ) THEN
402 lwmin = 2*n*n
403 ELSE
404 lwmin = n
405 END IF
406 work( 1 ) = sroundup_lwork(lwmin)
407*
408 IF( mm.LT.m ) THEN
409 info = -15
410 ELSE IF( lwork.LT.lwmin .AND. .NOT.lquery ) THEN
411 info = -18
412 END IF
413 END IF
414*
415 IF( info.NE.0 ) THEN
416 CALL xerbla( 'CTGSNA', -info )
417 RETURN
418 ELSE IF( lquery ) THEN
419 RETURN
420 END IF
421*
422* Quick return if possible
423*
424 IF( n.EQ.0 )
425 \$ RETURN
426*
427* Get machine constants
428*
429 eps = slamch( 'P' )
430 smlnum = slamch( 'S' ) / eps
431 bignum = one / smlnum
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 ) = sroundup_lwork(lwmin)
512 RETURN
513*
514* End of CTGSNA
515*
516 END
subroutine xerbla(srname, info)
Definition cblat2.f:3285
subroutine cgemv(trans, m, n, alpha, a, lda, x, incx, beta, y, incy)
CGEMV
Definition cgemv.f:160
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 ctgexc(wantq, wantz, n, a, lda, b, ldb, q, ldq, z, ldz, ifst, ilst, info)
CTGEXC
Definition ctgexc.f:200
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