 LAPACK 3.11.0 LAPACK: Linear Algebra PACKage
Searching...
No Matches

## ◆ sggesx()

 subroutine sggesx ( character JOBVSL, character JOBVSR, character SORT, external SELCTG, character SENSE, 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( 2 ) RCONDE, real, dimension( 2 ) RCONDV, real, dimension( * ) WORK, integer LWORK, integer, dimension( * ) IWORK, integer LIWORK, logical, dimension( * ) BWORK, integer INFO )

SGGESX computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices

Purpose:
``` SGGESX computes for a pair of N-by-N real nonsymmetric matrices
(A,B), the generalized eigenvalues, the real Schur form (S,T), and,
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; computes
a reciprocal condition number for the average of the selected
eigenvalues (RCONDE); and computes a reciprocal condition number for
the right and left deflating subspaces corresponding to the selected
eigenvalues (RCONDV). The leading columns of VSL and VSR then form
an orthonormal basis for the corresponding left and right eigenspaces
(deflating subspaces).

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 for 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 a selected complex eigenvalue may no longer satisfy SELCTG(ALPHAR(j),ALPHAI(j),BETA(j)) = .TRUE. after ordering, since ordering may change the value of complex eigenvalues (especially if the eigenvalue is ill-conditioned), in this case INFO is set to N+3.``` [in] SENSE ``` SENSE is CHARACTER*1 Determines which reciprocal condition numbers are computed. = 'N': None are computed; = 'E': Computed for average of selected eigenvalues only; = 'V': Computed for selected deflating subspaces only; = 'B': Computed for both. If SENSE = 'E', 'V', or 'B', SORT must equal 'S'.``` [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] RCONDE ``` RCONDE is REAL array, dimension ( 2 ) If SENSE = 'E' or 'B', RCONDE(1) and RCONDE(2) contain the reciprocal condition numbers for the average of the selected eigenvalues. Not referenced if SENSE = 'N' or 'V'.``` [out] RCONDV ``` RCONDV is REAL array, dimension ( 2 ) If SENSE = 'V' or 'B', RCONDV(1) and RCONDV(2) contain the reciprocal condition numbers for the selected deflating subspaces. Not referenced if SENSE = 'N' or 'E'.``` [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 N = 0, LWORK >= 1, else if SENSE = 'E', 'V', or 'B', LWORK >= max( 8*N, 6*N+16, 2*SDIM*(N-SDIM) ), else LWORK >= max( 8*N, 6*N+16 ). Note that 2*SDIM*(N-SDIM) <= N*N/2. Note also that an error is only returned if LWORK < max( 8*N, 6*N+16), but if SENSE = 'E' or 'V' or 'B' this may not be large enough. If LWORK = -1, then a workspace query is assumed; the routine only calculates the bound on the optimal size of the WORK array and the minimum size of the IWORK array, returns these values as the first entries of the WORK and IWORK arrays, and no error message related to LWORK or LIWORK is issued by XERBLA.``` [out] IWORK ``` IWORK is INTEGER array, dimension (MAX(1,LIWORK)) On exit, if INFO = 0, IWORK(1) returns the minimum LIWORK.``` [in] LIWORK ``` LIWORK is INTEGER The dimension of the array IWORK. If SENSE = 'N' or N = 0, LIWORK >= 1, otherwise LIWORK >= N+6. If LIWORK = -1, then a workspace query is assumed; the routine only calculates the bound on the optimal size of the WORK array and the minimum size of the IWORK array, returns these values as the first entries of the WORK and IWORK arrays, and no error message related to LWORK or LIWORK 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.```
Further Details:
```  An approximate (asymptotic) bound on the average absolute error of
the selected eigenvalues is

EPS * norm((A, B)) / RCONDE( 1 ).

An approximate (asymptotic) bound on the maximum angular error in
the computed deflating subspaces is

EPS * norm((A, B)) / RCONDV( 2 ).

Definition at line 361 of file sggesx.f.

365*
366* -- LAPACK driver routine --
367* -- LAPACK is a software package provided by Univ. of Tennessee, --
368* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
369*
370* .. Scalar Arguments ..
371 CHARACTER JOBVSL, JOBVSR, SENSE, SORT
372 INTEGER INFO, LDA, LDB, LDVSL, LDVSR, LIWORK, LWORK, N,
373 \$ SDIM
374* ..
375* .. Array Arguments ..
376 LOGICAL BWORK( * )
377 INTEGER IWORK( * )
378 REAL A( LDA, * ), ALPHAI( * ), ALPHAR( * ),
379 \$ B( LDB, * ), BETA( * ), RCONDE( 2 ),
380 \$ RCONDV( 2 ), VSL( LDVSL, * ), VSR( LDVSR, * ),
381 \$ WORK( * )
382* ..
383* .. Function Arguments ..
384 LOGICAL SELCTG
385 EXTERNAL selctg
386* ..
387*
388* =====================================================================
389*
390* .. Parameters ..
391 REAL ZERO, ONE
392 parameter( zero = 0.0e+0, one = 1.0e+0 )
393* ..
394* .. Local Scalars ..
395 LOGICAL CURSL, ILASCL, ILBSCL, ILVSL, ILVSR, LASTSL,
396 \$ LQUERY, LST2SL, WANTSB, WANTSE, WANTSN, WANTST,
397 \$ WANTSV
398 INTEGER I, ICOLS, IERR, IHI, IJOB, IJOBVL, IJOBVR,
399 \$ ILEFT, ILO, IP, IRIGHT, IROWS, ITAU, IWRK,
400 \$ LIWMIN, LWRK, MAXWRK, MINWRK
401 REAL ANRM, ANRMTO, BIGNUM, BNRM, BNRMTO, EPS, PL,
402 \$ PR, SAFMAX, SAFMIN, SMLNUM
403* ..
404* .. Local Arrays ..
405 REAL DIF( 2 )
406* ..
407* .. External Subroutines ..
408 EXTERNAL sgeqrf, sggbak, sggbal, sgghrd, shgeqz, slabad,
410 \$ xerbla
411* ..
412* .. External Functions ..
413 LOGICAL LSAME
414 INTEGER ILAENV
415 REAL SLAMCH, SLANGE
416 EXTERNAL lsame, ilaenv, slamch, slange
417* ..
418* .. Intrinsic Functions ..
419 INTRINSIC abs, max, sqrt
420* ..
421* .. Executable Statements ..
422*
423* Decode the input arguments
424*
425 IF( lsame( jobvsl, 'N' ) ) THEN
426 ijobvl = 1
427 ilvsl = .false.
428 ELSE IF( lsame( jobvsl, 'V' ) ) THEN
429 ijobvl = 2
430 ilvsl = .true.
431 ELSE
432 ijobvl = -1
433 ilvsl = .false.
434 END IF
435*
436 IF( lsame( jobvsr, 'N' ) ) THEN
437 ijobvr = 1
438 ilvsr = .false.
439 ELSE IF( lsame( jobvsr, 'V' ) ) THEN
440 ijobvr = 2
441 ilvsr = .true.
442 ELSE
443 ijobvr = -1
444 ilvsr = .false.
445 END IF
446*
447 wantst = lsame( sort, 'S' )
448 wantsn = lsame( sense, 'N' )
449 wantse = lsame( sense, 'E' )
450 wantsv = lsame( sense, 'V' )
451 wantsb = lsame( sense, 'B' )
452 lquery = ( lwork.EQ.-1 .OR. liwork.EQ.-1 )
453 IF( wantsn ) THEN
454 ijob = 0
455 ELSE IF( wantse ) THEN
456 ijob = 1
457 ELSE IF( wantsv ) THEN
458 ijob = 2
459 ELSE IF( wantsb ) THEN
460 ijob = 4
461 END IF
462*
463* Test the input arguments
464*
465 info = 0
466 IF( ijobvl.LE.0 ) THEN
467 info = -1
468 ELSE IF( ijobvr.LE.0 ) THEN
469 info = -2
470 ELSE IF( ( .NOT.wantst ) .AND. ( .NOT.lsame( sort, 'N' ) ) ) THEN
471 info = -3
472 ELSE IF( .NOT.( wantsn .OR. wantse .OR. wantsv .OR. wantsb ) .OR.
473 \$ ( .NOT.wantst .AND. .NOT.wantsn ) ) THEN
474 info = -5
475 ELSE IF( n.LT.0 ) THEN
476 info = -6
477 ELSE IF( lda.LT.max( 1, n ) ) THEN
478 info = -8
479 ELSE IF( ldb.LT.max( 1, n ) ) THEN
480 info = -10
481 ELSE IF( ldvsl.LT.1 .OR. ( ilvsl .AND. ldvsl.LT.n ) ) THEN
482 info = -16
483 ELSE IF( ldvsr.LT.1 .OR. ( ilvsr .AND. ldvsr.LT.n ) ) THEN
484 info = -18
485 END IF
486*
487* Compute workspace
488* (Note: Comments in the code beginning "Workspace:" describe the
489* minimal amount of workspace needed at that point in the code,
490* as well as the preferred amount for good performance.
491* NB refers to the optimal block size for the immediately
492* following subroutine, as returned by ILAENV.)
493*
494 IF( info.EQ.0 ) THEN
495 IF( n.GT.0) THEN
496 minwrk = max( 8*n, 6*n + 16 )
497 maxwrk = minwrk - n +
498 \$ n*ilaenv( 1, 'SGEQRF', ' ', n, 1, n, 0 )
499 maxwrk = max( maxwrk, minwrk - n +
500 \$ n*ilaenv( 1, 'SORMQR', ' ', n, 1, n, -1 ) )
501 IF( ilvsl ) THEN
502 maxwrk = max( maxwrk, minwrk - n +
503 \$ n*ilaenv( 1, 'SORGQR', ' ', n, 1, n, -1 ) )
504 END IF
505 lwrk = maxwrk
506 IF( ijob.GE.1 )
507 \$ lwrk = max( lwrk, n*n/2 )
508 ELSE
509 minwrk = 1
510 maxwrk = 1
511 lwrk = 1
512 END IF
513 work( 1 ) = lwrk
514 IF( wantsn .OR. n.EQ.0 ) THEN
515 liwmin = 1
516 ELSE
517 liwmin = n + 6
518 END IF
519 iwork( 1 ) = liwmin
520*
521 IF( lwork.LT.minwrk .AND. .NOT.lquery ) THEN
522 info = -22
523 ELSE IF( liwork.LT.liwmin .AND. .NOT.lquery ) THEN
524 info = -24
525 END IF
526 END IF
527*
528 IF( info.NE.0 ) THEN
529 CALL xerbla( 'SGGESX', -info )
530 RETURN
531 ELSE IF (lquery) THEN
532 RETURN
533 END IF
534*
535* Quick return if possible
536*
537 IF( n.EQ.0 ) THEN
538 sdim = 0
539 RETURN
540 END IF
541*
542* Get machine constants
543*
544 eps = slamch( 'P' )
545 safmin = slamch( 'S' )
546 safmax = one / safmin
547 CALL slabad( safmin, safmax )
548 smlnum = sqrt( safmin ) / eps
549 bignum = one / smlnum
550*
551* Scale A if max element outside range [SMLNUM,BIGNUM]
552*
553 anrm = slange( 'M', n, n, a, lda, work )
554 ilascl = .false.
555 IF( anrm.GT.zero .AND. anrm.LT.smlnum ) THEN
556 anrmto = smlnum
557 ilascl = .true.
558 ELSE IF( anrm.GT.bignum ) THEN
559 anrmto = bignum
560 ilascl = .true.
561 END IF
562 IF( ilascl )
563 \$ CALL slascl( 'G', 0, 0, anrm, anrmto, n, n, a, lda, ierr )
564*
565* Scale B if max element outside range [SMLNUM,BIGNUM]
566*
567 bnrm = slange( 'M', n, n, b, ldb, work )
568 ilbscl = .false.
569 IF( bnrm.GT.zero .AND. bnrm.LT.smlnum ) THEN
570 bnrmto = smlnum
571 ilbscl = .true.
572 ELSE IF( bnrm.GT.bignum ) THEN
573 bnrmto = bignum
574 ilbscl = .true.
575 END IF
576 IF( ilbscl )
577 \$ CALL slascl( 'G', 0, 0, bnrm, bnrmto, n, n, b, ldb, ierr )
578*
579* Permute the matrix to make it more nearly triangular
580* (Workspace: need 6*N + 2*N for permutation parameters)
581*
582 ileft = 1
583 iright = n + 1
584 iwrk = iright + n
585 CALL sggbal( 'P', n, a, lda, b, ldb, ilo, ihi, work( ileft ),
586 \$ work( iright ), work( iwrk ), ierr )
587*
588* Reduce B to triangular form (QR decomposition of B)
589* (Workspace: need N, prefer N*NB)
590*
591 irows = ihi + 1 - ilo
592 icols = n + 1 - ilo
593 itau = iwrk
594 iwrk = itau + irows
595 CALL sgeqrf( irows, icols, b( ilo, ilo ), ldb, work( itau ),
596 \$ work( iwrk ), lwork+1-iwrk, ierr )
597*
598* Apply the orthogonal transformation to matrix A
599* (Workspace: need N, prefer N*NB)
600*
601 CALL sormqr( 'L', 'T', irows, icols, irows, b( ilo, ilo ), ldb,
602 \$ work( itau ), a( ilo, ilo ), lda, work( iwrk ),
603 \$ lwork+1-iwrk, ierr )
604*
605* Initialize VSL
606* (Workspace: need N, prefer N*NB)
607*
608 IF( ilvsl ) THEN
609 CALL slaset( 'Full', n, n, zero, one, vsl, ldvsl )
610 IF( irows.GT.1 ) THEN
611 CALL slacpy( 'L', irows-1, irows-1, b( ilo+1, ilo ), ldb,
612 \$ vsl( ilo+1, ilo ), ldvsl )
613 END IF
614 CALL sorgqr( irows, irows, irows, vsl( ilo, ilo ), ldvsl,
615 \$ work( itau ), work( iwrk ), lwork+1-iwrk, ierr )
616 END IF
617*
618* Initialize VSR
619*
620 IF( ilvsr )
621 \$ CALL slaset( 'Full', n, n, zero, one, vsr, ldvsr )
622*
623* Reduce to generalized Hessenberg form
624* (Workspace: none needed)
625*
626 CALL sgghrd( jobvsl, jobvsr, n, ilo, ihi, a, lda, b, ldb, vsl,
627 \$ ldvsl, vsr, ldvsr, ierr )
628*
629 sdim = 0
630*
631* Perform QZ algorithm, computing Schur vectors if desired
632* (Workspace: need N)
633*
634 iwrk = itau
635 CALL shgeqz( 'S', jobvsl, jobvsr, n, ilo, ihi, a, lda, b, ldb,
636 \$ alphar, alphai, beta, vsl, ldvsl, vsr, ldvsr,
637 \$ work( iwrk ), lwork+1-iwrk, ierr )
638 IF( ierr.NE.0 ) THEN
639 IF( ierr.GT.0 .AND. ierr.LE.n ) THEN
640 info = ierr
641 ELSE IF( ierr.GT.n .AND. ierr.LE.2*n ) THEN
642 info = ierr - n
643 ELSE
644 info = n + 1
645 END IF
646 GO TO 50
647 END IF
648*
649* Sort eigenvalues ALPHA/BETA and compute the reciprocal of
650* condition number(s)
651* (Workspace: If IJOB >= 1, need MAX( 8*(N+1), 2*SDIM*(N-SDIM) )
652* otherwise, need 8*(N+1) )
653*
654 IF( wantst ) THEN
655*
656* Undo scaling on eigenvalues before SELCTGing
657*
658 IF( ilascl ) THEN
659 CALL slascl( 'G', 0, 0, anrmto, anrm, n, 1, alphar, n,
660 \$ ierr )
661 CALL slascl( 'G', 0, 0, anrmto, anrm, n, 1, alphai, n,
662 \$ ierr )
663 END IF
664 IF( ilbscl )
665 \$ CALL slascl( 'G', 0, 0, bnrmto, bnrm, n, 1, beta, n, ierr )
666*
667* Select eigenvalues
668*
669 DO 10 i = 1, n
670 bwork( i ) = selctg( alphar( i ), alphai( i ), beta( i ) )
671 10 CONTINUE
672*
673* Reorder eigenvalues, transform Generalized Schur vectors, and
674* compute reciprocal condition numbers
675*
676 CALL stgsen( ijob, ilvsl, ilvsr, bwork, n, a, lda, b, ldb,
677 \$ alphar, alphai, beta, vsl, ldvsl, vsr, ldvsr,
678 \$ sdim, pl, pr, dif, work( iwrk ), lwork-iwrk+1,
679 \$ iwork, liwork, ierr )
680*
681 IF( ijob.GE.1 )
682 \$ maxwrk = max( maxwrk, 2*sdim*( n-sdim ) )
683 IF( ierr.EQ.-22 ) THEN
684*
685* not enough real workspace
686*
687 info = -22
688 ELSE
689 IF( ijob.EQ.1 .OR. ijob.EQ.4 ) THEN
690 rconde( 1 ) = pl
691 rconde( 2 ) = pr
692 END IF
693 IF( ijob.EQ.2 .OR. ijob.EQ.4 ) THEN
694 rcondv( 1 ) = dif( 1 )
695 rcondv( 2 ) = dif( 2 )
696 END IF
697 IF( ierr.EQ.1 )
698 \$ info = n + 3
699 END IF
700*
701 END IF
702*
703* Apply permutation to VSL and VSR
704* (Workspace: none needed)
705*
706 IF( ilvsl )
707 \$ CALL sggbak( 'P', 'L', n, ilo, ihi, work( ileft ),
708 \$ work( iright ), n, vsl, ldvsl, ierr )
709*
710 IF( ilvsr )
711 \$ CALL sggbak( 'P', 'R', n, ilo, ihi, work( ileft ),
712 \$ work( iright ), n, vsr, ldvsr, ierr )
713*
714* Check if unscaling would cause over/underflow, if so, rescale
715* (ALPHAR(I),ALPHAI(I),BETA(I)) so BETA(I) is on the order of
716* B(I,I) and ALPHAR(I) and ALPHAI(I) are on the order of A(I,I)
717*
718 IF( ilascl ) THEN
719 DO 20 i = 1, n
720 IF( alphai( i ).NE.zero ) THEN
721 IF( ( alphar( i ) / safmax ).GT.( anrmto / anrm ) .OR.
722 \$ ( safmin / alphar( i ) ).GT.( anrm / anrmto ) )
723 \$ THEN
724 work( 1 ) = abs( a( i, i ) / alphar( i ) )
725 beta( i ) = beta( i )*work( 1 )
726 alphar( i ) = alphar( i )*work( 1 )
727 alphai( i ) = alphai( i )*work( 1 )
728 ELSE IF( ( alphai( i ) / safmax ).GT.( anrmto / anrm )
729 \$ .OR. ( safmin / alphai( i ) ).GT.( anrm / anrmto ) )
730 \$ THEN
731 work( 1 ) = abs( a( i, i+1 ) / alphai( i ) )
732 beta( i ) = beta( i )*work( 1 )
733 alphar( i ) = alphar( i )*work( 1 )
734 alphai( i ) = alphai( i )*work( 1 )
735 END IF
736 END IF
737 20 CONTINUE
738 END IF
739*
740 IF( ilbscl ) THEN
741 DO 25 i = 1, n
742 IF( alphai( i ).NE.zero ) THEN
743 IF( ( beta( i ) / safmax ).GT.( bnrmto / bnrm ) .OR.
744 \$ ( safmin / beta( i ) ).GT.( bnrm / bnrmto ) ) THEN
745 work( 1 ) = abs( b( i, i ) / beta( i ) )
746 beta( i ) = beta( i )*work( 1 )
747 alphar( i ) = alphar( i )*work( 1 )
748 alphai( i ) = alphai( i )*work( 1 )
749 END IF
750 END IF
751 25 CONTINUE
752 END IF
753*
754* Undo scaling
755*
756 IF( ilascl ) THEN
757 CALL slascl( 'H', 0, 0, anrmto, anrm, n, n, a, lda, ierr )
758 CALL slascl( 'G', 0, 0, anrmto, anrm, n, 1, alphar, n, ierr )
759 CALL slascl( 'G', 0, 0, anrmto, anrm, n, 1, alphai, n, ierr )
760 END IF
761*
762 IF( ilbscl ) THEN
763 CALL slascl( 'U', 0, 0, bnrmto, bnrm, n, n, b, ldb, ierr )
764 CALL slascl( 'G', 0, 0, bnrmto, bnrm, n, 1, beta, n, ierr )
765 END IF
766*
767 IF( wantst ) THEN
768*
769* Check if reordering is correct
770*
771 lastsl = .true.
772 lst2sl = .true.
773 sdim = 0
774 ip = 0
775 DO 40 i = 1, n
776 cursl = selctg( alphar( i ), alphai( i ), beta( i ) )
777 IF( alphai( i ).EQ.zero ) THEN
778 IF( cursl )
779 \$ sdim = sdim + 1
780 ip = 0
781 IF( cursl .AND. .NOT.lastsl )
782 \$ info = n + 2
783 ELSE
784 IF( ip.EQ.1 ) THEN
785*
786* Last eigenvalue of conjugate pair
787*
788 cursl = cursl .OR. lastsl
789 lastsl = cursl
790 IF( cursl )
791 \$ sdim = sdim + 2
792 ip = -1
793 IF( cursl .AND. .NOT.lst2sl )
794 \$ info = n + 2
795 ELSE
796*
797* First eigenvalue of conjugate pair
798*
799 ip = 1
800 END IF
801 END IF
802 lst2sl = lastsl
803 lastsl = cursl
804 40 CONTINUE
805*
806 END IF
807*
808 50 CONTINUE
809*
810 work( 1 ) = maxwrk
811 iwork( 1 ) = liwmin
812*
813 RETURN
814*
815* End of SGGESX
816*
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:143
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:110
subroutine slacpy(UPLO, M, N, A, LDA, B, LDB)
SLACPY copies all or part of one two-dimensional array to another.
Definition: slacpy.f:103
integer function ilaenv(ISPEC, NAME, OPTS, N1, N2, N3, N4)
ILAENV
Definition: ilaenv.f:162
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:53
subroutine sggbak(JOB, SIDE, N, ILO, IHI, LSCALE, RSCALE, M, V, LDV, INFO)
SGGBAK
Definition: sggbak.f:147
subroutine sggbal(JOB, N, A, LDA, B, LDB, ILO, IHI, LSCALE, RSCALE, WORK, INFO)
SGGBAL
Definition: sggbal.f:177
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:114
subroutine sgeqrf(M, N, A, LDA, TAU, WORK, LWORK, INFO)
SGEQRF
Definition: sgeqrf.f:146
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:304
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:451
subroutine sorgqr(M, N, K, A, LDA, TAU, WORK, LWORK, INFO)
SORGQR
Definition: sorgqr.f:128
subroutine sormqr(SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC, WORK, LWORK, INFO)
SORMQR
Definition: sormqr.f:168
subroutine sgghrd(COMPQ, COMPZ, N, ILO, IHI, A, LDA, B, LDB, Q, LDQ, Z, LDZ, INFO)
SGGHRD
Definition: sgghrd.f:207
real function slamch(CMACH)
SLAMCH
Definition: slamch.f:68
Here is the call graph for this function:
Here is the caller graph for this function: