 LAPACK  3.8.0 LAPACK: Linear Algebra PACKage

◆ sgges3()

 subroutine sgges3 ( character JOBVSL, character JOBVSR, character SORT, external SELCTG, integer N, real, dimension( lda, * ) A, integer LDA, real, dimension( ldb, * ) B, integer LDB, integer SDIM, real, dimension( * ) ALPHAR, real, dimension( * ) ALPHAI, real, dimension( * ) BETA, real, dimension( ldvsl, * ) VSL, integer LDVSL, real, dimension( ldvsr, * ) VSR, integer LDVSR, real, dimension( * ) WORK, integer LWORK, logical, dimension( * ) BWORK, integer INFO )

SGGES3 computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices (blocked algorithm)

Purpose:
``` SGGES3 computes for a pair of N-by-N real nonsymmetric matrices (A,B),
the generalized eigenvalues, the generalized real Schur form (S,T),
optionally, the left and/or right matrices of Schur vectors (VSL and
VSR). This gives the generalized Schur factorization

(A,B) = ( (VSL)*S*(VSR)**T, (VSL)*T*(VSR)**T )

Optionally, it also orders the eigenvalues so that a selected cluster
of eigenvalues appears in the leading diagonal blocks of the upper
quasi-triangular matrix S and the upper triangular matrix T.The
leading columns of VSL and VSR then form an orthonormal basis for the
corresponding left and right eigenspaces (deflating subspaces).

(If only the generalized eigenvalues are needed, use the driver

A generalized eigenvalue for a pair of matrices (A,B) is a scalar w
or a ratio alpha/beta = w, such that  A - w*B is singular.  It is
usually represented as the pair (alpha,beta), as there is a
reasonable interpretation for beta=0 or both being zero.

A pair of matrices (S,T) is in generalized real Schur form if T is
upper triangular with non-negative diagonal and S is block upper
triangular with 1-by-1 and 2-by-2 blocks.  1-by-1 blocks correspond
to real generalized eigenvalues, while 2-by-2 blocks of S will be
"standardized" by making the corresponding elements of T have the
form:
[  a  0  ]
[  0  b  ]

and the pair of corresponding 2-by-2 blocks in S and T will have a
complex conjugate pair of generalized eigenvalues.```
Parameters
 [in] JOBVSL ``` JOBVSL is CHARACTER*1 = 'N': do not compute the left Schur vectors; = 'V': compute the left Schur vectors.``` [in] JOBVSR ``` JOBVSR is CHARACTER*1 = 'N': do not compute the right Schur vectors; = 'V': compute the right Schur vectors.``` [in] SORT ``` SORT is CHARACTER*1 Specifies whether or not to order the eigenvalues on the diagonal of the generalized Schur form. = 'N': Eigenvalues are not ordered; = 'S': Eigenvalues are ordered (see SELCTG);``` [in] SELCTG ``` SELCTG is a LOGICAL FUNCTION of three REAL arguments SELCTG must be declared EXTERNAL in the calling subroutine. If SORT = 'N', SELCTG is not referenced. If SORT = 'S', SELCTG is used to select eigenvalues to sort to the top left of the Schur form. An eigenvalue (ALPHAR(j)+ALPHAI(j))/BETA(j) is selected if SELCTG(ALPHAR(j),ALPHAI(j),BETA(j)) is true; i.e. if either one of a complex conjugate pair of eigenvalues is selected, then both complex eigenvalues are selected. Note that in the ill-conditioned case, a selected complex eigenvalue may no longer satisfy SELCTG(ALPHAR(j),ALPHAI(j), BETA(j)) = .TRUE. after ordering. INFO is to be set to N+2 in this case.``` [in] N ``` N is INTEGER The order of the matrices A, B, VSL, and VSR. N >= 0.``` [in,out] A ``` A is REAL array, dimension (LDA, N) On entry, the first of the pair of matrices. On exit, A has been overwritten by its generalized Schur form S.``` [in] LDA ``` LDA is INTEGER The leading dimension of A. LDA >= max(1,N).``` [in,out] B ``` B is REAL array, dimension (LDB, N) On entry, the second of the pair of matrices. On exit, B has been overwritten by its generalized Schur form T.``` [in] LDB ``` LDB is INTEGER The leading dimension of B. LDB >= max(1,N).``` [out] SDIM ``` SDIM is INTEGER If SORT = 'N', SDIM = 0. If SORT = 'S', SDIM = number of eigenvalues (after sorting) for which SELCTG is true. (Complex conjugate pairs for which SELCTG is true for either eigenvalue count as 2.)``` [out] ALPHAR ` ALPHAR is REAL array, dimension (N)` [out] ALPHAI ` ALPHAI is REAL array, dimension (N)` [out] BETA ``` BETA is REAL array, dimension (N) On exit, (ALPHAR(j) + ALPHAI(j)*i)/BETA(j), j=1,...,N, will be the generalized eigenvalues. ALPHAR(j) + ALPHAI(j)*i, and BETA(j),j=1,...,N are the diagonals of the complex Schur form (S,T) that would result if the 2-by-2 diagonal blocks of the real Schur form of (A,B) were further reduced to triangular form using 2-by-2 complex unitary transformations. If ALPHAI(j) is zero, then the j-th eigenvalue is real; if positive, then the j-th and (j+1)-st eigenvalues are a complex conjugate pair, with ALPHAI(j+1) negative. Note: the quotients ALPHAR(j)/BETA(j) and ALPHAI(j)/BETA(j) may easily over- or underflow, and BETA(j) may even be zero. Thus, the user should avoid naively computing the ratio. However, ALPHAR and ALPHAI will be always less than and usually comparable with norm(A) in magnitude, and BETA always less than and usually comparable with norm(B).``` [out] VSL ``` VSL is REAL array, dimension (LDVSL,N) If JOBVSL = 'V', VSL will contain the left Schur vectors. Not referenced if JOBVSL = 'N'.``` [in] LDVSL ``` LDVSL is INTEGER The leading dimension of the matrix VSL. LDVSL >=1, and if JOBVSL = 'V', LDVSL >= N.``` [out] VSR ``` VSR is REAL array, dimension (LDVSR,N) If JOBVSR = 'V', VSR will contain the right Schur vectors. Not referenced if JOBVSR = 'N'.``` [in] LDVSR ``` LDVSR is INTEGER The leading dimension of the matrix VSR. LDVSR >= 1, and if JOBVSR = 'V', LDVSR >= N.``` [out] WORK ``` WORK is REAL array, dimension (MAX(1,LWORK)) On exit, if INFO = 0, WORK(1) returns the optimal LWORK.``` [in] LWORK ``` LWORK is INTEGER The dimension of the array WORK. If LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the WORK array, returns this value as the first entry of the WORK array, and no error message related to LWORK is issued by XERBLA.``` [out] BWORK ``` BWORK is LOGICAL array, dimension (N) Not referenced if SORT = 'N'.``` [out] INFO ``` INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value. = 1,...,N: The QZ iteration failed. (A,B) are not in Schur form, but ALPHAR(j), ALPHAI(j), and BETA(j) should be correct for j=INFO+1,...,N. > N: =N+1: other than QZ iteration failed in SHGEQZ. =N+2: after reordering, roundoff changed values of some complex eigenvalues so that leading eigenvalues in the Generalized Schur form no longer satisfy SELCTG=.TRUE. This could also be caused due to scaling. =N+3: reordering failed in STGSEN.```
Date
January 2015

Definition at line 284 of file sgges3.f.

284 *
285 * -- LAPACK driver routine (version 3.6.0) --
286 * -- LAPACK is a software package provided by Univ. of Tennessee, --
287 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
288 * January 2015
289 *
290 * .. Scalar Arguments ..
291  CHARACTER jobvsl, jobvsr, sort
292  INTEGER info, lda, ldb, ldvsl, ldvsr, lwork, n, sdim
293 * ..
294 * .. Array Arguments ..
295  LOGICAL bwork( * )
296  REAL a( lda, * ), alphai( * ), alphar( * ),
297  \$ b( ldb, * ), beta( * ), vsl( ldvsl, * ),
298  \$ vsr( ldvsr, * ), work( * )
299 * ..
300 * .. Function Arguments ..
301  LOGICAL selctg
302  EXTERNAL selctg
303 * ..
304 *
305 * =====================================================================
306 *
307 * .. Parameters ..
308  REAL zero, one
309  parameter( zero = 0.0e+0, one = 1.0e+0 )
310 * ..
311 * .. Local Scalars ..
312  LOGICAL cursl, ilascl, ilbscl, ilvsl, ilvsr, lastsl,
313  \$ lquery, lst2sl, wantst
314  INTEGER i, icols, ierr, ihi, ijobvl, ijobvr, ileft,
315  \$ ilo, ip, iright, irows, itau, iwrk, lwkopt
316  REAL anrm, anrmto, bignum, bnrm, bnrmto, eps, pvsl,
317  \$ pvsr, safmax, safmin, smlnum
318 * ..
319 * .. Local Arrays ..
320  INTEGER idum( 1 )
321  REAL dif( 2 )
322 * ..
323 * .. External Subroutines ..
324  EXTERNAL sgeqrf, sggbak, sggbal, sgghd3, shgeqz, slabad,
326  \$ xerbla
327 * ..
328 * .. External Functions ..
329  LOGICAL lsame
330  REAL slamch, slange
331  EXTERNAL lsame, slamch, slange
332 * ..
333 * .. Intrinsic Functions ..
334  INTRINSIC abs, max, sqrt
335 * ..
336 * .. Executable Statements ..
337 *
338 * Decode the input arguments
339 *
340  IF( lsame( jobvsl, 'N' ) ) THEN
341  ijobvl = 1
342  ilvsl = .false.
343  ELSE IF( lsame( jobvsl, 'V' ) ) THEN
344  ijobvl = 2
345  ilvsl = .true.
346  ELSE
347  ijobvl = -1
348  ilvsl = .false.
349  END IF
350 *
351  IF( lsame( jobvsr, 'N' ) ) THEN
352  ijobvr = 1
353  ilvsr = .false.
354  ELSE IF( lsame( jobvsr, 'V' ) ) THEN
355  ijobvr = 2
356  ilvsr = .true.
357  ELSE
358  ijobvr = -1
359  ilvsr = .false.
360  END IF
361 *
362  wantst = lsame( sort, 'S' )
363 *
364 * Test the input arguments
365 *
366  info = 0
367  lquery = ( lwork.EQ.-1 )
368  IF( ijobvl.LE.0 ) THEN
369  info = -1
370  ELSE IF( ijobvr.LE.0 ) THEN
371  info = -2
372  ELSE IF( ( .NOT.wantst ) .AND. ( .NOT.lsame( sort, 'N' ) ) ) THEN
373  info = -3
374  ELSE IF( n.LT.0 ) THEN
375  info = -5
376  ELSE IF( lda.LT.max( 1, n ) ) THEN
377  info = -7
378  ELSE IF( ldb.LT.max( 1, n ) ) THEN
379  info = -9
380  ELSE IF( ldvsl.LT.1 .OR. ( ilvsl .AND. ldvsl.LT.n ) ) THEN
381  info = -15
382  ELSE IF( ldvsr.LT.1 .OR. ( ilvsr .AND. ldvsr.LT.n ) ) THEN
383  info = -17
384  ELSE IF( lwork.LT.6*n+16 .AND. .NOT.lquery ) THEN
385  info = -19
386  END IF
387 *
388 * Compute workspace
389 *
390  IF( info.EQ.0 ) THEN
391  CALL sgeqrf( n, n, b, ldb, work, work, -1, ierr )
392  lwkopt = max( 6*n+16, 3*n+int( work( 1 ) ) )
393  CALL sormqr( 'L', 'T', n, n, n, b, ldb, work, a, lda, work,
394  \$ -1, ierr )
395  lwkopt = max( lwkopt, 3*n+int( work( 1 ) ) )
396  IF( ilvsl ) THEN
397  CALL sorgqr( n, n, n, vsl, ldvsl, work, work, -1, ierr )
398  lwkopt = max( lwkopt, 3*n+int( work( 1 ) ) )
399  END IF
400  CALL sgghd3( jobvsl, jobvsr, n, 1, n, a, lda, b, ldb, vsl,
401  \$ ldvsl, vsr, ldvsr, work, -1, ierr )
402  lwkopt = max( lwkopt, 3*n+int( work( 1 ) ) )
403  CALL shgeqz( 'S', jobvsl, jobvsr, n, 1, n, a, lda, b, ldb,
404  \$ alphar, alphai, beta, vsl, ldvsl, vsr, ldvsr,
405  \$ work, -1, ierr )
406  lwkopt = max( lwkopt, 2*n+int( work( 1 ) ) )
407  IF( wantst ) THEN
408  CALL stgsen( 0, ilvsl, ilvsr, bwork, n, a, lda, b, ldb,
409  \$ alphar, alphai, beta, vsl, ldvsl, vsr, ldvsr,
410  \$ sdim, pvsl, pvsr, dif, work, -1, idum, 1,
411  \$ ierr )
412  lwkopt = max( lwkopt, 2*n+int( work( 1 ) ) )
413  END IF
414  work( 1 ) = lwkopt
415  END IF
416 *
417  IF( info.NE.0 ) THEN
418  CALL xerbla( 'SGGES3 ', -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 ) THEN
427  sdim = 0
428  RETURN
429  END IF
430 *
431 * Get machine constants
432 *
433  eps = slamch( 'P' )
434  safmin = slamch( 'S' )
435  safmax = one / safmin
436  CALL slabad( safmin, safmax )
437  smlnum = sqrt( safmin ) / eps
438  bignum = one / smlnum
439 *
440 * Scale A if max element outside range [SMLNUM,BIGNUM]
441 *
442  anrm = slange( 'M', n, n, a, lda, work )
443  ilascl = .false.
444  IF( anrm.GT.zero .AND. anrm.LT.smlnum ) THEN
445  anrmto = smlnum
446  ilascl = .true.
447  ELSE IF( anrm.GT.bignum ) THEN
448  anrmto = bignum
449  ilascl = .true.
450  END IF
451  IF( ilascl )
452  \$ CALL slascl( 'G', 0, 0, anrm, anrmto, n, n, a, lda, ierr )
453 *
454 * Scale B if max element outside range [SMLNUM,BIGNUM]
455 *
456  bnrm = slange( 'M', n, n, b, ldb, work )
457  ilbscl = .false.
458  IF( bnrm.GT.zero .AND. bnrm.LT.smlnum ) THEN
459  bnrmto = smlnum
460  ilbscl = .true.
461  ELSE IF( bnrm.GT.bignum ) THEN
462  bnrmto = bignum
463  ilbscl = .true.
464  END IF
465  IF( ilbscl )
466  \$ CALL slascl( 'G', 0, 0, bnrm, bnrmto, n, n, b, ldb, ierr )
467 *
468 * Permute the matrix to make it more nearly triangular
469 *
470  ileft = 1
471  iright = n + 1
472  iwrk = iright + n
473  CALL sggbal( 'P', n, a, lda, b, ldb, ilo, ihi, work( ileft ),
474  \$ work( iright ), work( iwrk ), ierr )
475 *
476 * Reduce B to triangular form (QR decomposition of B)
477 *
478  irows = ihi + 1 - ilo
479  icols = n + 1 - ilo
480  itau = iwrk
481  iwrk = itau + irows
482  CALL sgeqrf( irows, icols, b( ilo, ilo ), ldb, work( itau ),
483  \$ work( iwrk ), lwork+1-iwrk, ierr )
484 *
485 * Apply the orthogonal transformation to matrix A
486 *
487  CALL sormqr( 'L', 'T', irows, icols, irows, b( ilo, ilo ), ldb,
488  \$ work( itau ), a( ilo, ilo ), lda, work( iwrk ),
489  \$ lwork+1-iwrk, ierr )
490 *
491 * Initialize VSL
492 *
493  IF( ilvsl ) THEN
494  CALL slaset( 'Full', n, n, zero, one, vsl, ldvsl )
495  IF( irows.GT.1 ) THEN
496  CALL slacpy( 'L', irows-1, irows-1, b( ilo+1, ilo ), ldb,
497  \$ vsl( ilo+1, ilo ), ldvsl )
498  END IF
499  CALL sorgqr( irows, irows, irows, vsl( ilo, ilo ), ldvsl,
500  \$ work( itau ), work( iwrk ), lwork+1-iwrk, ierr )
501  END IF
502 *
503 * Initialize VSR
504 *
505  IF( ilvsr )
506  \$ CALL slaset( 'Full', n, n, zero, one, vsr, ldvsr )
507 *
508 * Reduce to generalized Hessenberg form
509 *
510  CALL sgghd3( jobvsl, jobvsr, n, ilo, ihi, a, lda, b, ldb, vsl,
511  \$ ldvsl, vsr, ldvsr, work( iwrk ), lwork+1-iwrk, ierr )
512 *
513 * Perform QZ algorithm, computing Schur vectors if desired
514 *
515  iwrk = itau
516  CALL shgeqz( 'S', jobvsl, jobvsr, n, ilo, ihi, a, lda, b, ldb,
517  \$ alphar, alphai, beta, vsl, ldvsl, vsr, ldvsr,
518  \$ work( iwrk ), lwork+1-iwrk, ierr )
519  IF( ierr.NE.0 ) THEN
520  IF( ierr.GT.0 .AND. ierr.LE.n ) THEN
521  info = ierr
522  ELSE IF( ierr.GT.n .AND. ierr.LE.2*n ) THEN
523  info = ierr - n
524  ELSE
525  info = n + 1
526  END IF
527  GO TO 40
528  END IF
529 *
530 * Sort eigenvalues ALPHA/BETA if desired
531 *
532  sdim = 0
533  IF( wantst ) THEN
534 *
535 * Undo scaling on eigenvalues before SELCTGing
536 *
537  IF( ilascl ) THEN
538  CALL slascl( 'G', 0, 0, anrmto, anrm, n, 1, alphar, n,
539  \$ ierr )
540  CALL slascl( 'G', 0, 0, anrmto, anrm, n, 1, alphai, n,
541  \$ ierr )
542  END IF
543  IF( ilbscl )
544  \$ CALL slascl( 'G', 0, 0, bnrmto, bnrm, n, 1, beta, n, ierr )
545 *
546 * Select eigenvalues
547 *
548  DO 10 i = 1, n
549  bwork( i ) = selctg( alphar( i ), alphai( i ), beta( i ) )
550  10 CONTINUE
551 *
552  CALL stgsen( 0, ilvsl, ilvsr, bwork, n, a, lda, b, ldb, alphar,
553  \$ alphai, beta, vsl, ldvsl, vsr, ldvsr, sdim, pvsl,
554  \$ pvsr, dif, work( iwrk ), lwork-iwrk+1, idum, 1,
555  \$ ierr )
556  IF( ierr.EQ.1 )
557  \$ info = n + 3
558 *
559  END IF
560 *
561 * Apply back-permutation to VSL and VSR
562 *
563  IF( ilvsl )
564  \$ CALL sggbak( 'P', 'L', n, ilo, ihi, work( ileft ),
565  \$ work( iright ), n, vsl, ldvsl, ierr )
566 *
567  IF( ilvsr )
568  \$ CALL sggbak( 'P', 'R', n, ilo, ihi, work( ileft ),
569  \$ work( iright ), n, vsr, ldvsr, ierr )
570 *
571 * Check if unscaling would cause over/underflow, if so, rescale
572 * (ALPHAR(I),ALPHAI(I),BETA(I)) so BETA(I) is on the order of
573 * B(I,I) and ALPHAR(I) and ALPHAI(I) are on the order of A(I,I)
574 *
575  IF( ilascl )THEN
576  DO 50 i = 1, n
577  IF( alphai( i ).NE.zero ) THEN
578  IF( ( alphar( i )/safmax ).GT.( anrmto/anrm ) .OR.
579  \$ ( safmin/alphar( i ) ).GT.( anrm/anrmto ) ) THEN
580  work( 1 ) = abs( a( i, i )/alphar( i ) )
581  beta( i ) = beta( i )*work( 1 )
582  alphar( i ) = alphar( i )*work( 1 )
583  alphai( i ) = alphai( i )*work( 1 )
584  ELSE IF( ( alphai( i )/safmax ).GT.( anrmto/anrm ) .OR.
585  \$ ( safmin/alphai( i ) ).GT.( anrm/anrmto ) ) THEN
586  work( 1 ) = abs( a( i, i+1 )/alphai( i ) )
587  beta( i ) = beta( i )*work( 1 )
588  alphar( i ) = alphar( i )*work( 1 )
589  alphai( i ) = alphai( i )*work( 1 )
590  END IF
591  END IF
592  50 CONTINUE
593  END IF
594 *
595  IF( ilbscl )THEN
596  DO 60 i = 1, n
597  IF( alphai( i ).NE.zero ) THEN
598  IF( ( beta( i )/safmax ).GT.( bnrmto/bnrm ) .OR.
599  \$ ( safmin/beta( i ) ).GT.( bnrm/bnrmto ) ) THEN
600  work( 1 ) = abs(b( i, i )/beta( i ))
601  beta( i ) = beta( i )*work( 1 )
602  alphar( i ) = alphar( i )*work( 1 )
603  alphai( i ) = alphai( i )*work( 1 )
604  END IF
605  END IF
606  60 CONTINUE
607  END IF
608 *
609 * Undo scaling
610 *
611  IF( ilascl ) THEN
612  CALL slascl( 'H', 0, 0, anrmto, anrm, n, n, a, lda, ierr )
613  CALL slascl( 'G', 0, 0, anrmto, anrm, n, 1, alphar, n, ierr )
614  CALL slascl( 'G', 0, 0, anrmto, anrm, n, 1, alphai, n, ierr )
615  END IF
616 *
617  IF( ilbscl ) THEN
618  CALL slascl( 'U', 0, 0, bnrmto, bnrm, n, n, b, ldb, ierr )
619  CALL slascl( 'G', 0, 0, bnrmto, bnrm, n, 1, beta, n, ierr )
620  END IF
621 *
622  IF( wantst ) THEN
623 *
624 * Check if reordering is correct
625 *
626  lastsl = .true.
627  lst2sl = .true.
628  sdim = 0
629  ip = 0
630  DO 30 i = 1, n
631  cursl = selctg( alphar( i ), alphai( i ), beta( i ) )
632  IF( alphai( i ).EQ.zero ) THEN
633  IF( cursl )
634  \$ sdim = sdim + 1
635  ip = 0
636  IF( cursl .AND. .NOT.lastsl )
637  \$ info = n + 2
638  ELSE
639  IF( ip.EQ.1 ) THEN
640 *
641 * Last eigenvalue of conjugate pair
642 *
643  cursl = cursl .OR. lastsl
644  lastsl = cursl
645  IF( cursl )
646  \$ sdim = sdim + 2
647  ip = -1
648  IF( cursl .AND. .NOT.lst2sl )
649  \$ info = n + 2
650  ELSE
651 *
652 * First eigenvalue of conjugate pair
653 *
654  ip = 1
655  END IF
656  END IF
657  lst2sl = lastsl
658  lastsl = cursl
659  30 CONTINUE
660 *
661  END IF
662 *
663  40 CONTINUE
664 *
665  work( 1 ) = lwkopt
666 *
667  RETURN
668 *
669 * End of SGGES3
670 *
subroutine sggbal(JOB, N, A, LDA, B, LDB, ILO, IHI, LSCALE, RSCALE, WORK, INFO)
SGGBAL
Definition: sggbal.f:179
subroutine sgeqrf(M, N, A, LDA, TAU, WORK, LWORK, INFO)
SGEQRF
Definition: sgeqrf.f:138
real function slamch(CMACH)
SLAMCH
Definition: slamch.f:69
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 stgsen(IJOB, WANTQ, WANTZ, SELECT, N, A, LDA, B, LDB, ALPHAR, ALPHAI, BETA, Q, LDQ, Z, LDZ, M, PL, PR, DIF, WORK, LWORK, IWORK, LIWORK, INFO)
STGSEN
Definition: stgsen.f:453
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:55
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
real function slange(NORM, M, N, A, LDA, WORK)
SLANGE returns the value of the 1-norm, Frobenius norm, infinity-norm, or the largest absolute value ...
Definition: slange.f:116
subroutine sorgqr(M, N, K, A, LDA, TAU, WORK, LWORK, INFO)
SORGQR
Definition: sorgqr.f:130
subroutine sormqr(SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC, WORK, LWORK, INFO)
SORMQR
Definition: sormqr.f:170
subroutine sgghd3(COMPQ, COMPZ, N, ILO, IHI, A, LDA, B, LDB, Q, LDQ, Z, LDZ, WORK, LWORK, INFO)
SGGHD3
Definition: sgghd3.f:232
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:62