LAPACK 3.12.0 LAPACK: Linear Algebra PACKage
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## ◆ dggesx()

 subroutine dggesx ( character jobvsl, character jobvsr, character sort, external selctg, character sense, integer n, double precision, dimension( lda, * ) a, integer lda, double precision, dimension( ldb, * ) b, integer ldb, integer sdim, double precision, dimension( * ) alphar, double precision, dimension( * ) alphai, double precision, dimension( * ) beta, double precision, dimension( ldvsl, * ) vsl, integer ldvsl, double precision, dimension( ldvsr, * ) vsr, integer ldvsr, double precision, dimension( 2 ) rconde, double precision, dimension( 2 ) rcondv, double precision, dimension( * ) work, integer lwork, integer, dimension( * ) iwork, integer liwork, logical, dimension( * ) bwork, integer info )

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

Purpose:
DGGESX 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION array, dimension (N) [out] ALPHAI ALPHAI is DOUBLE PRECISION array, dimension (N) [out] BETA BETA is DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DHGEQZ =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 DTGSEN.
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 dggesx.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 DOUBLE PRECISION 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 DOUBLE PRECISION ZERO, ONE
392 parameter( zero = 0.0d+0, one = 1.0d+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 DOUBLE PRECISION ANRM, ANRMTO, BIGNUM, BNRM, BNRMTO, EPS, PL,
402 \$ PR, SAFMAX, SAFMIN, SMLNUM
403* ..
404* .. Local Arrays ..
405 DOUBLE PRECISION DIF( 2 )
406* ..
407* .. External Subroutines ..
408 EXTERNAL dgeqrf, dggbak, dggbal, dgghrd, dhgeqz, dlacpy,
410* ..
411* .. External Functions ..
412 LOGICAL LSAME
413 INTEGER ILAENV
414 DOUBLE PRECISION DLAMCH, DLANGE
415 EXTERNAL lsame, ilaenv, dlamch, dlange
416* ..
417* .. Intrinsic Functions ..
418 INTRINSIC abs, max, sqrt
419* ..
420* .. Executable Statements ..
421*
422* Decode the input arguments
423*
424 IF( lsame( jobvsl, 'N' ) ) THEN
425 ijobvl = 1
426 ilvsl = .false.
427 ELSE IF( lsame( jobvsl, 'V' ) ) THEN
428 ijobvl = 2
429 ilvsl = .true.
430 ELSE
431 ijobvl = -1
432 ilvsl = .false.
433 END IF
434*
435 IF( lsame( jobvsr, 'N' ) ) THEN
436 ijobvr = 1
437 ilvsr = .false.
438 ELSE IF( lsame( jobvsr, 'V' ) ) THEN
439 ijobvr = 2
440 ilvsr = .true.
441 ELSE
442 ijobvr = -1
443 ilvsr = .false.
444 END IF
445*
446 wantst = lsame( sort, 'S' )
447 wantsn = lsame( sense, 'N' )
448 wantse = lsame( sense, 'E' )
449 wantsv = lsame( sense, 'V' )
450 wantsb = lsame( sense, 'B' )
451 lquery = ( lwork.EQ.-1 .OR. liwork.EQ.-1 )
452 IF( wantsn ) THEN
453 ijob = 0
454 ELSE IF( wantse ) THEN
455 ijob = 1
456 ELSE IF( wantsv ) THEN
457 ijob = 2
458 ELSE IF( wantsb ) THEN
459 ijob = 4
460 END IF
461*
462* Test the input arguments
463*
464 info = 0
465 IF( ijobvl.LE.0 ) THEN
466 info = -1
467 ELSE IF( ijobvr.LE.0 ) THEN
468 info = -2
469 ELSE IF( ( .NOT.wantst ) .AND. ( .NOT.lsame( sort, 'N' ) ) ) THEN
470 info = -3
471 ELSE IF( .NOT.( wantsn .OR. wantse .OR. wantsv .OR. wantsb ) .OR.
472 \$ ( .NOT.wantst .AND. .NOT.wantsn ) ) THEN
473 info = -5
474 ELSE IF( n.LT.0 ) THEN
475 info = -6
476 ELSE IF( lda.LT.max( 1, n ) ) THEN
477 info = -8
478 ELSE IF( ldb.LT.max( 1, n ) ) THEN
479 info = -10
480 ELSE IF( ldvsl.LT.1 .OR. ( ilvsl .AND. ldvsl.LT.n ) ) THEN
481 info = -16
482 ELSE IF( ldvsr.LT.1 .OR. ( ilvsr .AND. ldvsr.LT.n ) ) THEN
483 info = -18
484 END IF
485*
486* Compute workspace
487* (Note: Comments in the code beginning "Workspace:" describe the
488* minimal amount of workspace needed at that point in the code,
489* as well as the preferred amount for good performance.
490* NB refers to the optimal block size for the immediately
491* following subroutine, as returned by ILAENV.)
492*
493 IF( info.EQ.0 ) THEN
494 IF( n.GT.0) THEN
495 minwrk = max( 8*n, 6*n + 16 )
496 maxwrk = minwrk - n +
497 \$ n*ilaenv( 1, 'DGEQRF', ' ', n, 1, n, 0 )
498 maxwrk = max( maxwrk, minwrk - n +
499 \$ n*ilaenv( 1, 'DORMQR', ' ', n, 1, n, -1 ) )
500 IF( ilvsl ) THEN
501 maxwrk = max( maxwrk, minwrk - n +
502 \$ n*ilaenv( 1, 'DORGQR', ' ', n, 1, n, -1 ) )
503 END IF
504 lwrk = maxwrk
505 IF( ijob.GE.1 )
506 \$ lwrk = max( lwrk, n*n/2 )
507 ELSE
508 minwrk = 1
509 maxwrk = 1
510 lwrk = 1
511 END IF
512 work( 1 ) = lwrk
513 IF( wantsn .OR. n.EQ.0 ) THEN
514 liwmin = 1
515 ELSE
516 liwmin = n + 6
517 END IF
518 iwork( 1 ) = liwmin
519*
520 IF( lwork.LT.minwrk .AND. .NOT.lquery ) THEN
521 info = -22
522 ELSE IF( liwork.LT.liwmin .AND. .NOT.lquery ) THEN
523 info = -24
524 END IF
525 END IF
526*
527 IF( info.NE.0 ) THEN
528 CALL xerbla( 'DGGESX', -info )
529 RETURN
530 ELSE IF (lquery) THEN
531 RETURN
532 END IF
533*
534* Quick return if possible
535*
536 IF( n.EQ.0 ) THEN
537 sdim = 0
538 RETURN
539 END IF
540*
541* Get machine constants
542*
543 eps = dlamch( 'P' )
544 safmin = dlamch( 'S' )
545 safmax = one / safmin
546 smlnum = sqrt( safmin ) / eps
547 bignum = one / smlnum
548*
549* Scale A if max element outside range [SMLNUM,BIGNUM]
550*
551 anrm = dlange( 'M', n, n, a, lda, work )
552 ilascl = .false.
553 IF( anrm.GT.zero .AND. anrm.LT.smlnum ) THEN
554 anrmto = smlnum
555 ilascl = .true.
556 ELSE IF( anrm.GT.bignum ) THEN
557 anrmto = bignum
558 ilascl = .true.
559 END IF
560 IF( ilascl )
561 \$ CALL dlascl( 'G', 0, 0, anrm, anrmto, n, n, a, lda, ierr )
562*
563* Scale B if max element outside range [SMLNUM,BIGNUM]
564*
565 bnrm = dlange( 'M', n, n, b, ldb, work )
566 ilbscl = .false.
567 IF( bnrm.GT.zero .AND. bnrm.LT.smlnum ) THEN
568 bnrmto = smlnum
569 ilbscl = .true.
570 ELSE IF( bnrm.GT.bignum ) THEN
571 bnrmto = bignum
572 ilbscl = .true.
573 END IF
574 IF( ilbscl )
575 \$ CALL dlascl( 'G', 0, 0, bnrm, bnrmto, n, n, b, ldb, ierr )
576*
577* Permute the matrix to make it more nearly triangular
578* (Workspace: need 6*N + 2*N for permutation parameters)
579*
580 ileft = 1
581 iright = n + 1
582 iwrk = iright + n
583 CALL dggbal( 'P', n, a, lda, b, ldb, ilo, ihi, work( ileft ),
584 \$ work( iright ), work( iwrk ), ierr )
585*
586* Reduce B to triangular form (QR decomposition of B)
587* (Workspace: need N, prefer N*NB)
588*
589 irows = ihi + 1 - ilo
590 icols = n + 1 - ilo
591 itau = iwrk
592 iwrk = itau + irows
593 CALL dgeqrf( irows, icols, b( ilo, ilo ), ldb, work( itau ),
594 \$ work( iwrk ), lwork+1-iwrk, ierr )
595*
596* Apply the orthogonal transformation to matrix A
597* (Workspace: need N, prefer N*NB)
598*
599 CALL dormqr( 'L', 'T', irows, icols, irows, b( ilo, ilo ), ldb,
600 \$ work( itau ), a( ilo, ilo ), lda, work( iwrk ),
601 \$ lwork+1-iwrk, ierr )
602*
603* Initialize VSL
604* (Workspace: need N, prefer N*NB)
605*
606 IF( ilvsl ) THEN
607 CALL dlaset( 'Full', n, n, zero, one, vsl, ldvsl )
608 IF( irows.GT.1 ) THEN
609 CALL dlacpy( 'L', irows-1, irows-1, b( ilo+1, ilo ), ldb,
610 \$ vsl( ilo+1, ilo ), ldvsl )
611 END IF
612 CALL dorgqr( irows, irows, irows, vsl( ilo, ilo ), ldvsl,
613 \$ work( itau ), work( iwrk ), lwork+1-iwrk, ierr )
614 END IF
615*
616* Initialize VSR
617*
618 IF( ilvsr )
619 \$ CALL dlaset( 'Full', n, n, zero, one, vsr, ldvsr )
620*
621* Reduce to generalized Hessenberg form
622* (Workspace: none needed)
623*
624 CALL dgghrd( jobvsl, jobvsr, n, ilo, ihi, a, lda, b, ldb, vsl,
625 \$ ldvsl, vsr, ldvsr, ierr )
626*
627 sdim = 0
628*
629* Perform QZ algorithm, computing Schur vectors if desired
630* (Workspace: need N)
631*
632 iwrk = itau
633 CALL dhgeqz( 'S', jobvsl, jobvsr, n, ilo, ihi, a, lda, b, ldb,
634 \$ alphar, alphai, beta, vsl, ldvsl, vsr, ldvsr,
635 \$ work( iwrk ), lwork+1-iwrk, ierr )
636 IF( ierr.NE.0 ) THEN
637 IF( ierr.GT.0 .AND. ierr.LE.n ) THEN
638 info = ierr
639 ELSE IF( ierr.GT.n .AND. ierr.LE.2*n ) THEN
640 info = ierr - n
641 ELSE
642 info = n + 1
643 END IF
644 GO TO 60
645 END IF
646*
647* Sort eigenvalues ALPHA/BETA and compute the reciprocal of
648* condition number(s)
649* (Workspace: If IJOB >= 1, need MAX( 8*(N+1), 2*SDIM*(N-SDIM) )
650* otherwise, need 8*(N+1) )
651*
652 IF( wantst ) THEN
653*
654* Undo scaling on eigenvalues before SELCTGing
655*
656 IF( ilascl ) THEN
657 CALL dlascl( 'G', 0, 0, anrmto, anrm, n, 1, alphar, n,
658 \$ ierr )
659 CALL dlascl( 'G', 0, 0, anrmto, anrm, n, 1, alphai, n,
660 \$ ierr )
661 END IF
662 IF( ilbscl )
663 \$ CALL dlascl( 'G', 0, 0, bnrmto, bnrm, n, 1, beta, n, ierr )
664*
665* Select eigenvalues
666*
667 DO 10 i = 1, n
668 bwork( i ) = selctg( alphar( i ), alphai( i ), beta( i ) )
669 10 CONTINUE
670*
671* Reorder eigenvalues, transform Generalized Schur vectors, and
672* compute reciprocal condition numbers
673*
674 CALL dtgsen( ijob, ilvsl, ilvsr, bwork, n, a, lda, b, ldb,
675 \$ alphar, alphai, beta, vsl, ldvsl, vsr, ldvsr,
676 \$ sdim, pl, pr, dif, work( iwrk ), lwork-iwrk+1,
677 \$ iwork, liwork, ierr )
678*
679 IF( ijob.GE.1 )
680 \$ maxwrk = max( maxwrk, 2*sdim*( n-sdim ) )
681 IF( ierr.EQ.-22 ) THEN
682*
683* not enough real workspace
684*
685 info = -22
686 ELSE
687 IF( ijob.EQ.1 .OR. ijob.EQ.4 ) THEN
688 rconde( 1 ) = pl
689 rconde( 2 ) = pr
690 END IF
691 IF( ijob.EQ.2 .OR. ijob.EQ.4 ) THEN
692 rcondv( 1 ) = dif( 1 )
693 rcondv( 2 ) = dif( 2 )
694 END IF
695 IF( ierr.EQ.1 )
696 \$ info = n + 3
697 END IF
698*
699 END IF
700*
701* Apply permutation to VSL and VSR
702* (Workspace: none needed)
703*
704 IF( ilvsl )
705 \$ CALL dggbak( 'P', 'L', n, ilo, ihi, work( ileft ),
706 \$ work( iright ), n, vsl, ldvsl, ierr )
707*
708 IF( ilvsr )
709 \$ CALL dggbak( 'P', 'R', n, ilo, ihi, work( ileft ),
710 \$ work( iright ), n, vsr, ldvsr, ierr )
711*
712* Check if unscaling would cause over/underflow, if so, rescale
713* (ALPHAR(I),ALPHAI(I),BETA(I)) so BETA(I) is on the order of
714* B(I,I) and ALPHAR(I) and ALPHAI(I) are on the order of A(I,I)
715*
716 IF( ilascl ) THEN
717 DO 20 i = 1, n
718 IF( alphai( i ).NE.zero ) THEN
719 IF( ( alphar( i ) / safmax ).GT.( anrmto / anrm ) .OR.
720 \$ ( safmin / alphar( i ) ).GT.( anrm / anrmto ) ) THEN
721 work( 1 ) = abs( a( i, i ) / alphar( i ) )
722 beta( i ) = beta( i )*work( 1 )
723 alphar( i ) = alphar( i )*work( 1 )
724 alphai( i ) = alphai( i )*work( 1 )
725 ELSE IF( ( alphai( i ) / safmax ).GT.
726 \$ ( anrmto / anrm ) .OR.
727 \$ ( safmin / alphai( i ) ).GT.( anrm / anrmto ) )
728 \$ THEN
729 work( 1 ) = abs( a( i, i+1 ) / alphai( i ) )
730 beta( i ) = beta( i )*work( 1 )
731 alphar( i ) = alphar( i )*work( 1 )
732 alphai( i ) = alphai( i )*work( 1 )
733 END IF
734 END IF
735 20 CONTINUE
736 END IF
737*
738 IF( ilbscl ) THEN
739 DO 30 i = 1, n
740 IF( alphai( i ).NE.zero ) THEN
741 IF( ( beta( i ) / safmax ).GT.( bnrmto / bnrm ) .OR.
742 \$ ( safmin / beta( i ) ).GT.( bnrm / bnrmto ) ) THEN
743 work( 1 ) = abs( b( i, i ) / beta( i ) )
744 beta( i ) = beta( i )*work( 1 )
745 alphar( i ) = alphar( i )*work( 1 )
746 alphai( i ) = alphai( i )*work( 1 )
747 END IF
748 END IF
749 30 CONTINUE
750 END IF
751*
752* Undo scaling
753*
754 IF( ilascl ) THEN
755 CALL dlascl( 'H', 0, 0, anrmto, anrm, n, n, a, lda, ierr )
756 CALL dlascl( 'G', 0, 0, anrmto, anrm, n, 1, alphar, n, ierr )
757 CALL dlascl( 'G', 0, 0, anrmto, anrm, n, 1, alphai, n, ierr )
758 END IF
759*
760 IF( ilbscl ) THEN
761 CALL dlascl( 'U', 0, 0, bnrmto, bnrm, n, n, b, ldb, ierr )
762 CALL dlascl( 'G', 0, 0, bnrmto, bnrm, n, 1, beta, n, ierr )
763 END IF
764*
765 IF( wantst ) THEN
766*
767* Check if reordering is correct
768*
769 lastsl = .true.
770 lst2sl = .true.
771 sdim = 0
772 ip = 0
773 DO 50 i = 1, n
774 cursl = selctg( alphar( i ), alphai( i ), beta( i ) )
775 IF( alphai( i ).EQ.zero ) THEN
776 IF( cursl )
777 \$ sdim = sdim + 1
778 ip = 0
779 IF( cursl .AND. .NOT.lastsl )
780 \$ info = n + 2
781 ELSE
782 IF( ip.EQ.1 ) THEN
783*
784* Last eigenvalue of conjugate pair
785*
786 cursl = cursl .OR. lastsl
787 lastsl = cursl
788 IF( cursl )
789 \$ sdim = sdim + 2
790 ip = -1
791 IF( cursl .AND. .NOT.lst2sl )
792 \$ info = n + 2
793 ELSE
794*
795* First eigenvalue of conjugate pair
796*
797 ip = 1
798 END IF
799 END IF
800 lst2sl = lastsl
801 lastsl = cursl
802 50 CONTINUE
803*
804 END IF
805*
806 60 CONTINUE
807*
808 work( 1 ) = maxwrk
809 iwork( 1 ) = liwmin
810*
811 RETURN
812*
813* End of DGGESX
814*
subroutine xerbla(srname, info)
Definition cblat2.f:3285
subroutine dgeqrf(m, n, a, lda, tau, work, lwork, info)
DGEQRF
Definition dgeqrf.f:146
subroutine dggbak(job, side, n, ilo, ihi, lscale, rscale, m, v, ldv, info)
DGGBAK
Definition dggbak.f:147
subroutine dggbal(job, n, a, lda, b, ldb, ilo, ihi, lscale, rscale, work, info)
DGGBAL
Definition dggbal.f:177
subroutine dgghrd(compq, compz, n, ilo, ihi, a, lda, b, ldb, q, ldq, z, ldz, info)
DGGHRD
Definition dgghrd.f:207
subroutine dhgeqz(job, compq, compz, n, ilo, ihi, h, ldh, t, ldt, alphar, alphai, beta, q, ldq, z, ldz, work, lwork, info)
DHGEQZ
Definition dhgeqz.f:304
integer function ilaenv(ispec, name, opts, n1, n2, n3, n4)
ILAENV
Definition ilaenv.f:162
subroutine dlacpy(uplo, m, n, a, lda, b, ldb)
DLACPY copies all or part of one two-dimensional array to another.
Definition dlacpy.f:103
double precision function dlamch(cmach)
DLAMCH
Definition dlamch.f:69
double precision function dlange(norm, m, n, a, lda, work)
DLANGE returns the value of the 1-norm, Frobenius norm, infinity-norm, or the largest absolute value ...
Definition dlange.f:114
subroutine dlascl(type, kl, ku, cfrom, cto, m, n, a, lda, info)
DLASCL multiplies a general rectangular matrix by a real scalar defined as cto/cfrom.
Definition dlascl.f:143
subroutine dlaset(uplo, m, n, alpha, beta, a, lda)
DLASET initializes the off-diagonal elements and the diagonal elements of a matrix to given values.
Definition dlaset.f:110
logical function lsame(ca, cb)
LSAME
Definition lsame.f:48
subroutine dtgsen(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)
DTGSEN
Definition dtgsen.f:451
subroutine dorgqr(m, n, k, a, lda, tau, work, lwork, info)
DORGQR
Definition dorgqr.f:128
subroutine dormqr(side, trans, m, n, k, a, lda, tau, c, ldc, work, lwork, info)
DORMQR
Definition dormqr.f:167
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