LAPACK  3.6.1 LAPACK: Linear Algebra PACKage
sggev3.f
Go to the documentation of this file.
1 *> \brief <b> SGGEV3 computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices (blocked algorithm)</b>
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/sggev3.f">
11 *> [TGZ]</a>
12 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/sggev3.f">
13 *> [ZIP]</a>
14 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/sggev3.f">
15 *> [TXT]</a>
16 *> \endhtmlonly
17 *
18 * Definition:
19 * ===========
20 *
21 * SUBROUTINE SGGEV3( JOBVL, JOBVR, N, A, LDA, B, LDB, ALPHAR,
22 * \$ ALPHAI, BETA, VL, LDVL, VR, LDVR, WORK, LWORK,
23 * \$ INFO )
24 *
25 * .. Scalar Arguments ..
26 * CHARACTER JOBVL, JOBVR
27 * INTEGER INFO, LDA, LDB, LDVL, LDVR, LWORK, N
28 * ..
29 * .. Array Arguments ..
30 * REAL A( LDA, * ), ALPHAI( * ), ALPHAR( * ),
31 * \$ B( LDB, * ), BETA( * ), VL( LDVL, * ),
32 * \$ VR( LDVR, * ), WORK( * )
33 * ..
34 *
35 *
36 *> \par Purpose:
37 * =============
38 *>
39 *> \verbatim
40 *>
41 *> SGGEV3 computes for a pair of N-by-N real nonsymmetric matrices (A,B)
42 *> the generalized eigenvalues, and optionally, the left and/or right
43 *> generalized eigenvectors.
44 *>
45 *> A generalized eigenvalue for a pair of matrices (A,B) is a scalar
46 *> lambda or a ratio alpha/beta = lambda, such that A - lambda*B is
47 *> singular. It is usually represented as the pair (alpha,beta), as
48 *> there is a reasonable interpretation for beta=0, and even for both
49 *> being zero.
50 *>
51 *> The right eigenvector v(j) corresponding to the eigenvalue lambda(j)
52 *> of (A,B) satisfies
53 *>
54 *> A * v(j) = lambda(j) * B * v(j).
55 *>
56 *> The left eigenvector u(j) corresponding to the eigenvalue lambda(j)
57 *> of (A,B) satisfies
58 *>
59 *> u(j)**H * A = lambda(j) * u(j)**H * B .
60 *>
61 *> where u(j)**H is the conjugate-transpose of u(j).
62 *>
63 *> \endverbatim
64 *
65 * Arguments:
66 * ==========
67 *
68 *> \param[in] JOBVL
69 *> \verbatim
70 *> JOBVL is CHARACTER*1
71 *> = 'N': do not compute the left generalized eigenvectors;
72 *> = 'V': compute the left generalized eigenvectors.
73 *> \endverbatim
74 *>
75 *> \param[in] JOBVR
76 *> \verbatim
77 *> JOBVR is CHARACTER*1
78 *> = 'N': do not compute the right generalized eigenvectors;
79 *> = 'V': compute the right generalized eigenvectors.
80 *> \endverbatim
81 *>
82 *> \param[in] N
83 *> \verbatim
84 *> N is INTEGER
85 *> The order of the matrices A, B, VL, and VR. N >= 0.
86 *> \endverbatim
87 *>
88 *> \param[in,out] A
89 *> \verbatim
90 *> A is REAL array, dimension (LDA, N)
91 *> On entry, the matrix A in the pair (A,B).
92 *> On exit, A has been overwritten.
93 *> \endverbatim
94 *>
95 *> \param[in] LDA
96 *> \verbatim
97 *> LDA is INTEGER
98 *> The leading dimension of A. LDA >= max(1,N).
99 *> \endverbatim
100 *>
101 *> \param[in,out] B
102 *> \verbatim
103 *> B is REAL array, dimension (LDB, N)
104 *> On entry, the matrix B in the pair (A,B).
105 *> On exit, B has been overwritten.
106 *> \endverbatim
107 *>
108 *> \param[in] LDB
109 *> \verbatim
110 *> LDB is INTEGER
111 *> The leading dimension of B. LDB >= max(1,N).
112 *> \endverbatim
113 *>
114 *> \param[out] ALPHAR
115 *> \verbatim
116 *> ALPHAR is REAL array, dimension (N)
117 *> \endverbatim
118 *>
119 *> \param[out] ALPHAI
120 *> \verbatim
121 *> ALPHAI is REAL array, dimension (N)
122 *> \endverbatim
123 *>
124 *> \param[out] BETA
125 *> \verbatim
126 *> BETA is REAL array, dimension (N)
127 *> On exit, (ALPHAR(j) + ALPHAI(j)*i)/BETA(j), j=1,...,N, will
128 *> be the generalized eigenvalues. If ALPHAI(j) is zero, then
129 *> the j-th eigenvalue is real; if positive, then the j-th and
130 *> (j+1)-st eigenvalues are a complex conjugate pair, with
131 *> ALPHAI(j+1) negative.
132 *>
133 *> Note: the quotients ALPHAR(j)/BETA(j) and ALPHAI(j)/BETA(j)
134 *> may easily over- or underflow, and BETA(j) may even be zero.
135 *> Thus, the user should avoid naively computing the ratio
136 *> alpha/beta. However, ALPHAR and ALPHAI will be always less
137 *> than and usually comparable with norm(A) in magnitude, and
138 *> BETA always less than and usually comparable with norm(B).
139 *> \endverbatim
140 *>
141 *> \param[out] VL
142 *> \verbatim
143 *> VL is REAL array, dimension (LDVL,N)
144 *> If JOBVL = 'V', the left eigenvectors u(j) are stored one
145 *> after another in the columns of VL, in the same order as
146 *> their eigenvalues. If the j-th eigenvalue is real, then
147 *> u(j) = VL(:,j), the j-th column of VL. If the j-th and
148 *> (j+1)-th eigenvalues form a complex conjugate pair, then
149 *> u(j) = VL(:,j)+i*VL(:,j+1) and u(j+1) = VL(:,j)-i*VL(:,j+1).
150 *> Each eigenvector is scaled so the largest component has
151 *> abs(real part)+abs(imag. part)=1.
152 *> Not referenced if JOBVL = 'N'.
153 *> \endverbatim
154 *>
155 *> \param[in] LDVL
156 *> \verbatim
157 *> LDVL is INTEGER
158 *> The leading dimension of the matrix VL. LDVL >= 1, and
159 *> if JOBVL = 'V', LDVL >= N.
160 *> \endverbatim
161 *>
162 *> \param[out] VR
163 *> \verbatim
164 *> VR is REAL array, dimension (LDVR,N)
165 *> If JOBVR = 'V', the right eigenvectors v(j) are stored one
166 *> after another in the columns of VR, in the same order as
167 *> their eigenvalues. If the j-th eigenvalue is real, then
168 *> v(j) = VR(:,j), the j-th column of VR. If the j-th and
169 *> (j+1)-th eigenvalues form a complex conjugate pair, then
170 *> v(j) = VR(:,j)+i*VR(:,j+1) and v(j+1) = VR(:,j)-i*VR(:,j+1).
171 *> Each eigenvector is scaled so the largest component has
172 *> abs(real part)+abs(imag. part)=1.
173 *> Not referenced if JOBVR = 'N'.
174 *> \endverbatim
175 *>
176 *> \param[in] LDVR
177 *> \verbatim
178 *> LDVR is INTEGER
179 *> The leading dimension of the matrix VR. LDVR >= 1, and
180 *> if JOBVR = 'V', LDVR >= N.
181 *> \endverbatim
182 *>
183 *> \param[out] WORK
184 *> \verbatim
185 *> WORK is REAL array, dimension (MAX(1,LWORK))
186 *> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
187 *> \endverbatim
188 *>
189 *> \param[in] LWORK
190 *> \verbatim
191 *> LWORK is INTEGER
192 *>
193 *> If LWORK = -1, then a workspace query is assumed; the routine
194 *> only calculates the optimal size of the WORK array, returns
195 *> this value as the first entry of the WORK array, and no error
196 *> message related to LWORK is issued by XERBLA.
197 *> \endverbatim
198 *>
199 *> \param[out] INFO
200 *> \verbatim
201 *> INFO is INTEGER
202 *> = 0: successful exit
203 *> < 0: if INFO = -i, the i-th argument had an illegal value.
204 *> = 1,...,N:
205 *> The QZ iteration failed. No eigenvectors have been
206 *> calculated, but ALPHAR(j), ALPHAI(j), and BETA(j)
207 *> should be correct for j=INFO+1,...,N.
208 *> > N: =N+1: other than QZ iteration failed in SHGEQZ.
209 *> =N+2: error return from STGEVC.
210 *> \endverbatim
211 *
212 * Authors:
213 * ========
214 *
215 *> \author Univ. of Tennessee
216 *> \author Univ. of California Berkeley
217 *> \author Univ. of Colorado Denver
218 *> \author NAG Ltd.
219 *
220 *> \date January 2015
221 *
222 *> \ingroup realGEeigen
223 *
224 * =====================================================================
225  SUBROUTINE sggev3( JOBVL, JOBVR, N, A, LDA, B, LDB, ALPHAR,
226  \$ alphai, beta, vl, ldvl, vr, ldvr, work, lwork,
227  \$ info )
228 *
229 * -- LAPACK driver routine (version 3.6.0) --
230 * -- LAPACK is a software package provided by Univ. of Tennessee, --
231 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
232 * January 2015
233 *
234 * .. Scalar Arguments ..
235  CHARACTER JOBVL, JOBVR
236  INTEGER INFO, LDA, LDB, LDVL, LDVR, LWORK, N
237 * ..
238 * .. Array Arguments ..
239  REAL A( lda, * ), ALPHAI( * ), ALPHAR( * ),
240  \$ b( ldb, * ), beta( * ), vl( ldvl, * ),
241  \$ vr( ldvr, * ), work( * )
242 * ..
243 *
244 * =====================================================================
245 *
246 * .. Parameters ..
247  REAL ZERO, ONE
248  parameter ( zero = 0.0e+0, one = 1.0e+0 )
249 * ..
250 * .. Local Scalars ..
251  LOGICAL ILASCL, ILBSCL, ILV, ILVL, ILVR, LQUERY
252  CHARACTER CHTEMP
253  INTEGER ICOLS, IERR, IHI, IJOBVL, IJOBVR, ILEFT, ILO,
254  \$ in, iright, irows, itau, iwrk, jc, jr, lwkopt
255  REAL ANRM, ANRMTO, BIGNUM, BNRM, BNRMTO, EPS,
256  \$ smlnum, temp
257 * ..
258 * .. Local Arrays ..
259  LOGICAL LDUMMA( 1 )
260 * ..
261 * .. External Subroutines ..
262  EXTERNAL sgeqrf, sggbak, sggbal, sgghd3, shgeqz, slabad,
264  \$ xerbla
265 * ..
266 * .. External Functions ..
267  LOGICAL LSAME
268  REAL SLAMCH, SLANGE
269  EXTERNAL lsame, slamch, slange
270 * ..
271 * .. Intrinsic Functions ..
272  INTRINSIC abs, max, sqrt
273 * ..
274 * .. Executable Statements ..
275 *
276 * Decode the input arguments
277 *
278  IF( lsame( jobvl, 'N' ) ) THEN
279  ijobvl = 1
280  ilvl = .false.
281  ELSE IF( lsame( jobvl, 'V' ) ) THEN
282  ijobvl = 2
283  ilvl = .true.
284  ELSE
285  ijobvl = -1
286  ilvl = .false.
287  END IF
288 *
289  IF( lsame( jobvr, 'N' ) ) THEN
290  ijobvr = 1
291  ilvr = .false.
292  ELSE IF( lsame( jobvr, 'V' ) ) THEN
293  ijobvr = 2
294  ilvr = .true.
295  ELSE
296  ijobvr = -1
297  ilvr = .false.
298  END IF
299  ilv = ilvl .OR. ilvr
300 *
301 * Test the input arguments
302 *
303  info = 0
304  lquery = ( lwork.EQ.-1 )
305  IF( ijobvl.LE.0 ) THEN
306  info = -1
307  ELSE IF( ijobvr.LE.0 ) THEN
308  info = -2
309  ELSE IF( n.LT.0 ) THEN
310  info = -3
311  ELSE IF( lda.LT.max( 1, n ) ) THEN
312  info = -5
313  ELSE IF( ldb.LT.max( 1, n ) ) THEN
314  info = -7
315  ELSE IF( ldvl.LT.1 .OR. ( ilvl .AND. ldvl.LT.n ) ) THEN
316  info = -12
317  ELSE IF( ldvr.LT.1 .OR. ( ilvr .AND. ldvr.LT.n ) ) THEN
318  info = -14
319  ELSE IF( lwork.LT.max( 1, 8*n ) .AND. .NOT.lquery ) THEN
320  info = -16
321  END IF
322 *
323 * Compute workspace
324 *
325  IF( info.EQ.0 ) THEN
326  CALL sgeqrf( n, n, b, ldb, work, work, -1, ierr )
327  lwkopt = max( 1, 8*n, 3*n+int( work( 1 ) ) )
328  CALL sormqr( 'L', 'T', n, n, n, b, ldb, work, a, lda, work,
329  \$ -1, ierr )
330  lwkopt = max( lwkopt, 3*n+int( work( 1 ) ) )
331  CALL sgghd3( jobvl, jobvr, n, 1, n, a, lda, b, ldb, vl, ldvl,
332  \$ vr, ldvr, work, -1, ierr )
333  lwkopt = max( lwkopt, 3*n+int( work( 1 ) ) )
334  IF( ilvl ) THEN
335  CALL sorgqr( n, n, n, vl, ldvl, work, work, -1, ierr )
336  lwkopt = max( lwkopt, 3*n+int( work( 1 ) ) )
337  CALL shgeqz( 'S', jobvl, jobvr, n, 1, n, a, lda, b, ldb,
338  \$ alphar, alphai, beta, vl, ldvl, vr, ldvr,
339  \$ work, -1, ierr )
340  lwkopt = max( lwkopt, 2*n+int( work( 1 ) ) )
341  ELSE
342  CALL shgeqz( 'E', jobvl, jobvr, n, 1, n, a, lda, b, ldb,
343  \$ alphar, alphai, beta, vl, ldvl, vr, ldvr,
344  \$ work, -1, ierr )
345  lwkopt = max( lwkopt, 2*n+int( work( 1 ) ) )
346  END IF
347  work( 1 ) = REAL( lwkopt )
348 *
349  END IF
350 *
351  IF( info.NE.0 ) THEN
352  CALL xerbla( 'SGGEV3 ', -info )
353  RETURN
354  ELSE IF( lquery ) THEN
355  RETURN
356  END IF
357 *
358 * Quick return if possible
359 *
360  IF( n.EQ.0 )
361  \$ RETURN
362 *
363 * Get machine constants
364 *
365  eps = slamch( 'P' )
366  smlnum = slamch( 'S' )
367  bignum = one / smlnum
368  CALL slabad( smlnum, bignum )
369  smlnum = sqrt( smlnum ) / eps
370  bignum = one / smlnum
371 *
372 * Scale A if max element outside range [SMLNUM,BIGNUM]
373 *
374  anrm = slange( 'M', n, n, a, lda, work )
375  ilascl = .false.
376  IF( anrm.GT.zero .AND. anrm.LT.smlnum ) THEN
377  anrmto = smlnum
378  ilascl = .true.
379  ELSE IF( anrm.GT.bignum ) THEN
380  anrmto = bignum
381  ilascl = .true.
382  END IF
383  IF( ilascl )
384  \$ CALL slascl( 'G', 0, 0, anrm, anrmto, n, n, a, lda, ierr )
385 *
386 * Scale B if max element outside range [SMLNUM,BIGNUM]
387 *
388  bnrm = slange( 'M', n, n, b, ldb, work )
389  ilbscl = .false.
390  IF( bnrm.GT.zero .AND. bnrm.LT.smlnum ) THEN
391  bnrmto = smlnum
392  ilbscl = .true.
393  ELSE IF( bnrm.GT.bignum ) THEN
394  bnrmto = bignum
395  ilbscl = .true.
396  END IF
397  IF( ilbscl )
398  \$ CALL slascl( 'G', 0, 0, bnrm, bnrmto, n, n, b, ldb, ierr )
399 *
400 * Permute the matrices A, B to isolate eigenvalues if possible
401 *
402  ileft = 1
403  iright = n + 1
404  iwrk = iright + n
405  CALL sggbal( 'P', n, a, lda, b, ldb, ilo, ihi, work( ileft ),
406  \$ work( iright ), work( iwrk ), ierr )
407 *
408 * Reduce B to triangular form (QR decomposition of B)
409 *
410  irows = ihi + 1 - ilo
411  IF( ilv ) THEN
412  icols = n + 1 - ilo
413  ELSE
414  icols = irows
415  END IF
416  itau = iwrk
417  iwrk = itau + irows
418  CALL sgeqrf( irows, icols, b( ilo, ilo ), ldb, work( itau ),
419  \$ work( iwrk ), lwork+1-iwrk, ierr )
420 *
421 * Apply the orthogonal transformation to matrix A
422 *
423  CALL sormqr( 'L', 'T', irows, icols, irows, b( ilo, ilo ), ldb,
424  \$ work( itau ), a( ilo, ilo ), lda, work( iwrk ),
425  \$ lwork+1-iwrk, ierr )
426 *
427 * Initialize VL
428 *
429  IF( ilvl ) THEN
430  CALL slaset( 'Full', n, n, zero, one, vl, ldvl )
431  IF( irows.GT.1 ) THEN
432  CALL slacpy( 'L', irows-1, irows-1, b( ilo+1, ilo ), ldb,
433  \$ vl( ilo+1, ilo ), ldvl )
434  END IF
435  CALL sorgqr( irows, irows, irows, vl( ilo, ilo ), ldvl,
436  \$ work( itau ), work( iwrk ), lwork+1-iwrk, ierr )
437  END IF
438 *
439 * Initialize VR
440 *
441  IF( ilvr )
442  \$ CALL slaset( 'Full', n, n, zero, one, vr, ldvr )
443 *
444 * Reduce to generalized Hessenberg form
445 *
446  IF( ilv ) THEN
447 *
448 * Eigenvectors requested -- work on whole matrix.
449 *
450  CALL sgghd3( jobvl, jobvr, n, ilo, ihi, a, lda, b, ldb, vl,
451  \$ ldvl, vr, ldvr, work( iwrk ), lwork+1-iwrk, ierr )
452  ELSE
453  CALL sgghd3( 'N', 'N', irows, 1, irows, a( ilo, ilo ), lda,
454  \$ b( ilo, ilo ), ldb, vl, ldvl, vr, ldvr,
455  \$ work( iwrk ), lwork+1-iwrk, ierr )
456  END IF
457 *
458 * Perform QZ algorithm (Compute eigenvalues, and optionally, the
459 * Schur forms and Schur vectors)
460 *
461  iwrk = itau
462  IF( ilv ) THEN
463  chtemp = 'S'
464  ELSE
465  chtemp = 'E'
466  END IF
467  CALL shgeqz( chtemp, jobvl, jobvr, n, ilo, ihi, a, lda, b, ldb,
468  \$ alphar, alphai, beta, vl, ldvl, vr, ldvr,
469  \$ work( iwrk ), lwork+1-iwrk, ierr )
470  IF( ierr.NE.0 ) THEN
471  IF( ierr.GT.0 .AND. ierr.LE.n ) THEN
472  info = ierr
473  ELSE IF( ierr.GT.n .AND. ierr.LE.2*n ) THEN
474  info = ierr - n
475  ELSE
476  info = n + 1
477  END IF
478  GO TO 110
479  END IF
480 *
481 * Compute Eigenvectors
482 *
483  IF( ilv ) THEN
484  IF( ilvl ) THEN
485  IF( ilvr ) THEN
486  chtemp = 'B'
487  ELSE
488  chtemp = 'L'
489  END IF
490  ELSE
491  chtemp = 'R'
492  END IF
493  CALL stgevc( chtemp, 'B', ldumma, n, a, lda, b, ldb, vl, ldvl,
494  \$ vr, ldvr, n, in, work( iwrk ), ierr )
495  IF( ierr.NE.0 ) THEN
496  info = n + 2
497  GO TO 110
498  END IF
499 *
500 * Undo balancing on VL and VR and normalization
501 *
502  IF( ilvl ) THEN
503  CALL sggbak( 'P', 'L', n, ilo, ihi, work( ileft ),
504  \$ work( iright ), n, vl, ldvl, ierr )
505  DO 50 jc = 1, n
506  IF( alphai( jc ).LT.zero )
507  \$ GO TO 50
508  temp = zero
509  IF( alphai( jc ).EQ.zero ) THEN
510  DO 10 jr = 1, n
511  temp = max( temp, abs( vl( jr, jc ) ) )
512  10 CONTINUE
513  ELSE
514  DO 20 jr = 1, n
515  temp = max( temp, abs( vl( jr, jc ) )+
516  \$ abs( vl( jr, jc+1 ) ) )
517  20 CONTINUE
518  END IF
519  IF( temp.LT.smlnum )
520  \$ GO TO 50
521  temp = one / temp
522  IF( alphai( jc ).EQ.zero ) THEN
523  DO 30 jr = 1, n
524  vl( jr, jc ) = vl( jr, jc )*temp
525  30 CONTINUE
526  ELSE
527  DO 40 jr = 1, n
528  vl( jr, jc ) = vl( jr, jc )*temp
529  vl( jr, jc+1 ) = vl( jr, jc+1 )*temp
530  40 CONTINUE
531  END IF
532  50 CONTINUE
533  END IF
534  IF( ilvr ) THEN
535  CALL sggbak( 'P', 'R', n, ilo, ihi, work( ileft ),
536  \$ work( iright ), n, vr, ldvr, ierr )
537  DO 100 jc = 1, n
538  IF( alphai( jc ).LT.zero )
539  \$ GO TO 100
540  temp = zero
541  IF( alphai( jc ).EQ.zero ) THEN
542  DO 60 jr = 1, n
543  temp = max( temp, abs( vr( jr, jc ) ) )
544  60 CONTINUE
545  ELSE
546  DO 70 jr = 1, n
547  temp = max( temp, abs( vr( jr, jc ) )+
548  \$ abs( vr( jr, jc+1 ) ) )
549  70 CONTINUE
550  END IF
551  IF( temp.LT.smlnum )
552  \$ GO TO 100
553  temp = one / temp
554  IF( alphai( jc ).EQ.zero ) THEN
555  DO 80 jr = 1, n
556  vr( jr, jc ) = vr( jr, jc )*temp
557  80 CONTINUE
558  ELSE
559  DO 90 jr = 1, n
560  vr( jr, jc ) = vr( jr, jc )*temp
561  vr( jr, jc+1 ) = vr( jr, jc+1 )*temp
562  90 CONTINUE
563  END IF
564  100 CONTINUE
565  END IF
566 *
567 * End of eigenvector calculation
568 *
569  END IF
570 *
571 * Undo scaling if necessary
572 *
573  110 CONTINUE
574 *
575  IF( ilascl ) THEN
576  CALL slascl( 'G', 0, 0, anrmto, anrm, n, 1, alphar, n, ierr )
577  CALL slascl( 'G', 0, 0, anrmto, anrm, n, 1, alphai, n, ierr )
578  END IF
579 *
580  IF( ilbscl ) THEN
581  CALL slascl( 'G', 0, 0, bnrmto, bnrm, n, 1, beta, n, ierr )
582  END IF
583 *
584  work( 1 ) = REAL( lwkopt )
585  RETURN
586 *
587 * End of SGGEV3
588 *
589  END
subroutine sggbal(JOB, N, A, LDA, B, LDB, ILO, IHI, LSCALE, RSCALE, WORK, INFO)
SGGBAL
Definition: sggbal.f:179
subroutine sormqr(SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC, WORK, LWORK, INFO)
SORMQR
Definition: sormqr.f:170
subroutine slascl(TYPE, KL, KU, CFROM, CTO, M, N, A, LDA, INFO)
SLASCL multiplies a general rectangular matrix by a real scalar defined as cto/cfrom.
Definition: slascl.f:145
subroutine stgevc(SIDE, HOWMNY, SELECT, N, S, LDS, P, LDP, VL, LDVL, VR, LDVR, MM, M, WORK, INFO)
STGEVC
Definition: stgevc.f:297
subroutine sggev3(JOBVL, JOBVR, N, A, LDA, B, LDB, ALPHAR, ALPHAI, BETA, VL, LDVL, VR, LDVR, WORK, LWORK, INFO)
SGGEV3 computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices ...
Definition: sggev3.f:228
subroutine sgeqrf(M, N, A, LDA, TAU, WORK, LWORK, INFO)
SGEQRF
Definition: sgeqrf.f:138
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:62
subroutine slacpy(UPLO, M, N, A, LDA, B, LDB)
SLACPY copies all or part of one two-dimensional array to another.
Definition: slacpy.f:105
subroutine sgghd3(COMPQ, COMPZ, N, ILO, IHI, A, LDA, B, LDB, Q, LDQ, Z, LDZ, WORK, LWORK, INFO)
SGGHD3
Definition: sgghd3.f:232
subroutine shgeqz(JOB, COMPQ, COMPZ, N, ILO, IHI, H, LDH, T, LDT, ALPHAR, ALPHAI, BETA, Q, LDQ, Z, LDZ, WORK, LWORK, INFO)
SHGEQZ
Definition: shgeqz.f:306
subroutine slaset(UPLO, M, N, ALPHA, BETA, A, LDA)
SLASET initializes the off-diagonal elements and the diagonal elements of a matrix to given values...
Definition: slaset.f:112
subroutine sggbak(JOB, SIDE, N, ILO, IHI, LSCALE, RSCALE, M, V, LDV, INFO)
SGGBAK
Definition: sggbak.f:149
subroutine sorgqr(M, N, K, A, LDA, TAU, WORK, LWORK, INFO)
SORGQR
Definition: sorgqr.f:130