 LAPACK  3.10.0 LAPACK: Linear Algebra PACKage

◆ zggevx()

 subroutine zggevx ( character BALANC, character JOBVL, character JOBVR, character SENSE, integer N, complex*16, dimension( lda, * ) A, integer LDA, complex*16, dimension( ldb, * ) B, integer LDB, complex*16, dimension( * ) ALPHA, complex*16, dimension( * ) BETA, complex*16, dimension( ldvl, * ) VL, integer LDVL, complex*16, dimension( ldvr, * ) VR, integer LDVR, integer ILO, integer IHI, double precision, dimension( * ) LSCALE, double precision, dimension( * ) RSCALE, double precision ABNRM, double precision BBNRM, double precision, dimension( * ) RCONDE, double precision, dimension( * ) RCONDV, complex*16, dimension( * ) WORK, integer LWORK, double precision, dimension( * ) RWORK, integer, dimension( * ) IWORK, logical, dimension( * ) BWORK, integer INFO )

ZGGEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices

Purpose:
ZGGEVX computes for a pair of N-by-N complex nonsymmetric matrices
(A,B) the generalized eigenvalues, and optionally, the left and/or
right generalized eigenvectors.

Optionally, it also computes a balancing transformation to improve
the conditioning of the eigenvalues and eigenvectors (ILO, IHI,
LSCALE, RSCALE, ABNRM, and BBNRM), reciprocal condition numbers for
the eigenvalues (RCONDE), and reciprocal condition numbers for the
right eigenvectors (RCONDV).

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

The right eigenvector v(j) corresponding to the eigenvalue lambda(j)
of (A,B) satisfies
A * v(j) = lambda(j) * B * v(j) .
The left eigenvector u(j) corresponding to the eigenvalue lambda(j)
of (A,B) satisfies
u(j)**H * A  = lambda(j) * u(j)**H * B.
where u(j)**H is the conjugate-transpose of u(j).
Parameters
 [in] BALANC BALANC is CHARACTER*1 Specifies the balance option to be performed: = 'N': do not diagonally scale or permute; = 'P': permute only; = 'S': scale only; = 'B': both permute and scale. Computed reciprocal condition numbers will be for the matrices after permuting and/or balancing. Permuting does not change condition numbers (in exact arithmetic), but balancing does. [in] JOBVL JOBVL is CHARACTER*1 = 'N': do not compute the left generalized eigenvectors; = 'V': compute the left generalized eigenvectors. [in] JOBVR JOBVR is CHARACTER*1 = 'N': do not compute the right generalized eigenvectors; = 'V': compute the right generalized eigenvectors. [in] SENSE SENSE is CHARACTER*1 Determines which reciprocal condition numbers are computed. = 'N': none are computed; = 'E': computed for eigenvalues only; = 'V': computed for eigenvectors only; = 'B': computed for eigenvalues and eigenvectors. [in] N N is INTEGER The order of the matrices A, B, VL, and VR. N >= 0. [in,out] A A is COMPLEX*16 array, dimension (LDA, N) On entry, the matrix A in the pair (A,B). On exit, A has been overwritten. If JOBVL='V' or JOBVR='V' or both, then A contains the first part of the complex Schur form of the "balanced" versions of the input A and B. [in] LDA LDA is INTEGER The leading dimension of A. LDA >= max(1,N). [in,out] B B is COMPLEX*16 array, dimension (LDB, N) On entry, the matrix B in the pair (A,B). On exit, B has been overwritten. If JOBVL='V' or JOBVR='V' or both, then B contains the second part of the complex Schur form of the "balanced" versions of the input A and B. [in] LDB LDB is INTEGER The leading dimension of B. LDB >= max(1,N). [out] ALPHA ALPHA is COMPLEX*16 array, dimension (N) [out] BETA BETA is COMPLEX*16 array, dimension (N) On exit, ALPHA(j)/BETA(j), j=1,...,N, will be the generalized eigenvalues. Note: the quotient ALPHA(j)/BETA(j) ) may easily over- or underflow, and BETA(j) may even be zero. Thus, the user should avoid naively computing the ratio ALPHA/BETA. However, ALPHA 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] VL VL is COMPLEX*16 array, dimension (LDVL,N) If JOBVL = 'V', the left generalized eigenvectors u(j) are stored one after another in the columns of VL, in the same order as their eigenvalues. Each eigenvector will be scaled so the largest component will have abs(real part) + abs(imag. part) = 1. Not referenced if JOBVL = 'N'. [in] LDVL LDVL is INTEGER The leading dimension of the matrix VL. LDVL >= 1, and if JOBVL = 'V', LDVL >= N. [out] VR VR is COMPLEX*16 array, dimension (LDVR,N) If JOBVR = 'V', the right generalized eigenvectors v(j) are stored one after another in the columns of VR, in the same order as their eigenvalues. Each eigenvector will be scaled so the largest component will have abs(real part) + abs(imag. part) = 1. Not referenced if JOBVR = 'N'. [in] LDVR LDVR is INTEGER The leading dimension of the matrix VR. LDVR >= 1, and if JOBVR = 'V', LDVR >= N. [out] ILO ILO is INTEGER [out] IHI IHI is INTEGER ILO and IHI are integer values such that on exit A(i,j) = 0 and B(i,j) = 0 if i > j and j = 1,...,ILO-1 or i = IHI+1,...,N. If BALANC = 'N' or 'S', ILO = 1 and IHI = N. [out] LSCALE LSCALE is DOUBLE PRECISION array, dimension (N) Details of the permutations and scaling factors applied to the left side of A and B. If PL(j) is the index of the row interchanged with row j, and DL(j) is the scaling factor applied to row j, then LSCALE(j) = PL(j) for j = 1,...,ILO-1 = DL(j) for j = ILO,...,IHI = PL(j) for j = IHI+1,...,N. The order in which the interchanges are made is N to IHI+1, then 1 to ILO-1. [out] RSCALE RSCALE is DOUBLE PRECISION array, dimension (N) Details of the permutations and scaling factors applied to the right side of A and B. If PR(j) is the index of the column interchanged with column j, and DR(j) is the scaling factor applied to column j, then RSCALE(j) = PR(j) for j = 1,...,ILO-1 = DR(j) for j = ILO,...,IHI = PR(j) for j = IHI+1,...,N The order in which the interchanges are made is N to IHI+1, then 1 to ILO-1. [out] ABNRM ABNRM is DOUBLE PRECISION The one-norm of the balanced matrix A. [out] BBNRM BBNRM is DOUBLE PRECISION The one-norm of the balanced matrix B. [out] RCONDE RCONDE is DOUBLE PRECISION array, dimension (N) If SENSE = 'E' or 'B', the reciprocal condition numbers of the eigenvalues, stored in consecutive elements of the array. If SENSE = 'N' or 'V', RCONDE is not referenced. [out] RCONDV RCONDV is DOUBLE PRECISION array, dimension (N) If JOB = 'V' or 'B', the estimated reciprocal condition numbers of the eigenvectors, stored in consecutive elements of the array. If the eigenvalues cannot be reordered to compute RCONDV(j), RCONDV(j) is set to 0; this can only occur when the true value would be very small anyway. If SENSE = 'N' or 'E', RCONDV is not referenced. [out] WORK WORK is COMPLEX*16 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. LWORK >= max(1,2*N). If SENSE = 'E', LWORK >= max(1,4*N). If SENSE = 'V' or 'B', LWORK >= max(1,2*N*N+2*N). 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] RWORK RWORK is DOUBLE PRECISION array, dimension (lrwork) lrwork must be at least max(1,6*N) if BALANC = 'S' or 'B', and at least max(1,2*N) otherwise. Real workspace. [out] IWORK IWORK is INTEGER array, dimension (N+2) If SENSE = 'E', IWORK is not referenced. [out] BWORK BWORK is LOGICAL array, dimension (N) If SENSE = 'N', BWORK is not referenced. [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. No eigenvectors have been calculated, but ALPHA(j) and BETA(j) should be correct for j=INFO+1,...,N. > N: =N+1: other than QZ iteration failed in ZHGEQZ. =N+2: error return from ZTGEVC.
Further Details:
Balancing a matrix pair (A,B) includes, first, permuting rows and
columns to isolate eigenvalues, second, applying diagonal similarity
transformation to the rows and columns to make the rows and columns
as close in norm as possible. The computed reciprocal condition
numbers correspond to the balanced matrix. Permuting rows and columns
will not change the condition numbers (in exact arithmetic) but
diagonal scaling will.  For further explanation of balancing, see
section 4.11.1.2 of LAPACK Users' Guide.

An approximate error bound on the chordal distance between the i-th
computed generalized eigenvalue w and the corresponding exact
eigenvalue lambda is

chord(w, lambda) <= EPS * norm(ABNRM, BBNRM) / RCONDE(I)

An approximate error bound for the angle between the i-th computed
eigenvector VL(i) or VR(i) is given by

EPS * norm(ABNRM, BBNRM) / DIF(i).

For further explanation of the reciprocal condition numbers RCONDE
and RCONDV, see section 4.11 of LAPACK User's Guide.

Definition at line 370 of file zggevx.f.

374 *
375 * -- LAPACK driver routine --
376 * -- LAPACK is a software package provided by Univ. of Tennessee, --
377 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
378 *
379 * .. Scalar Arguments ..
380  CHARACTER BALANC, JOBVL, JOBVR, SENSE
381  INTEGER IHI, ILO, INFO, LDA, LDB, LDVL, LDVR, LWORK, N
382  DOUBLE PRECISION ABNRM, BBNRM
383 * ..
384 * .. Array Arguments ..
385  LOGICAL BWORK( * )
386  INTEGER IWORK( * )
387  DOUBLE PRECISION LSCALE( * ), RCONDE( * ), RCONDV( * ),
388  \$ RSCALE( * ), RWORK( * )
389  COMPLEX*16 A( LDA, * ), ALPHA( * ), B( LDB, * ),
390  \$ BETA( * ), VL( LDVL, * ), VR( LDVR, * ),
391  \$ WORK( * )
392 * ..
393 *
394 * =====================================================================
395 *
396 * .. Parameters ..
397  DOUBLE PRECISION ZERO, ONE
398  parameter( zero = 0.0d+0, one = 1.0d+0 )
399  COMPLEX*16 CZERO, CONE
400  parameter( czero = ( 0.0d+0, 0.0d+0 ),
401  \$ cone = ( 1.0d+0, 0.0d+0 ) )
402 * ..
403 * .. Local Scalars ..
404  LOGICAL ILASCL, ILBSCL, ILV, ILVL, ILVR, LQUERY, NOSCL,
405  \$ WANTSB, WANTSE, WANTSN, WANTSV
406  CHARACTER CHTEMP
407  INTEGER I, ICOLS, IERR, IJOBVL, IJOBVR, IN, IROWS,
408  \$ ITAU, IWRK, IWRK1, J, JC, JR, M, MAXWRK, MINWRK
409  DOUBLE PRECISION ANRM, ANRMTO, BIGNUM, BNRM, BNRMTO, EPS,
410  \$ SMLNUM, TEMP
411  COMPLEX*16 X
412 * ..
413 * .. Local Arrays ..
414  LOGICAL LDUMMA( 1 )
415 * ..
416 * .. External Subroutines ..
417  EXTERNAL dlabad, dlascl, xerbla, zgeqrf, zggbak, zggbal,
419  \$ ztgsna, zungqr, zunmqr
420 * ..
421 * .. External Functions ..
422  LOGICAL LSAME
423  INTEGER ILAENV
424  DOUBLE PRECISION DLAMCH, ZLANGE
425  EXTERNAL lsame, ilaenv, dlamch, zlange
426 * ..
427 * .. Intrinsic Functions ..
428  INTRINSIC abs, dble, dimag, max, sqrt
429 * ..
430 * .. Statement Functions ..
431  DOUBLE PRECISION ABS1
432 * ..
433 * .. Statement Function definitions ..
434  abs1( x ) = abs( dble( x ) ) + abs( dimag( x ) )
435 * ..
436 * .. Executable Statements ..
437 *
438 * Decode the input arguments
439 *
440  IF( lsame( jobvl, 'N' ) ) THEN
441  ijobvl = 1
442  ilvl = .false.
443  ELSE IF( lsame( jobvl, 'V' ) ) THEN
444  ijobvl = 2
445  ilvl = .true.
446  ELSE
447  ijobvl = -1
448  ilvl = .false.
449  END IF
450 *
451  IF( lsame( jobvr, 'N' ) ) THEN
452  ijobvr = 1
453  ilvr = .false.
454  ELSE IF( lsame( jobvr, 'V' ) ) THEN
455  ijobvr = 2
456  ilvr = .true.
457  ELSE
458  ijobvr = -1
459  ilvr = .false.
460  END IF
461  ilv = ilvl .OR. ilvr
462 *
463  noscl = lsame( balanc, 'N' ) .OR. lsame( balanc, 'P' )
464  wantsn = lsame( sense, 'N' )
465  wantse = lsame( sense, 'E' )
466  wantsv = lsame( sense, 'V' )
467  wantsb = lsame( sense, 'B' )
468 *
469 * Test the input arguments
470 *
471  info = 0
472  lquery = ( lwork.EQ.-1 )
473  IF( .NOT.( noscl .OR. lsame( balanc,'S' ) .OR.
474  \$ lsame( balanc, 'B' ) ) ) THEN
475  info = -1
476  ELSE IF( ijobvl.LE.0 ) THEN
477  info = -2
478  ELSE IF( ijobvr.LE.0 ) THEN
479  info = -3
480  ELSE IF( .NOT.( wantsn .OR. wantse .OR. wantsb .OR. wantsv ) )
481  \$ THEN
482  info = -4
483  ELSE IF( n.LT.0 ) THEN
484  info = -5
485  ELSE IF( lda.LT.max( 1, n ) ) THEN
486  info = -7
487  ELSE IF( ldb.LT.max( 1, n ) ) THEN
488  info = -9
489  ELSE IF( ldvl.LT.1 .OR. ( ilvl .AND. ldvl.LT.n ) ) THEN
490  info = -13
491  ELSE IF( ldvr.LT.1 .OR. ( ilvr .AND. ldvr.LT.n ) ) THEN
492  info = -15
493  END IF
494 *
495 * Compute workspace
496 * (Note: Comments in the code beginning "Workspace:" describe the
497 * minimal amount of workspace needed at that point in the code,
498 * as well as the preferred amount for good performance.
499 * NB refers to the optimal block size for the immediately
500 * following subroutine, as returned by ILAENV. The workspace is
501 * computed assuming ILO = 1 and IHI = N, the worst case.)
502 *
503  IF( info.EQ.0 ) THEN
504  IF( n.EQ.0 ) THEN
505  minwrk = 1
506  maxwrk = 1
507  ELSE
508  minwrk = 2*n
509  IF( wantse ) THEN
510  minwrk = 4*n
511  ELSE IF( wantsv .OR. wantsb ) THEN
512  minwrk = 2*n*( n + 1)
513  END IF
514  maxwrk = minwrk
515  maxwrk = max( maxwrk,
516  \$ n + n*ilaenv( 1, 'ZGEQRF', ' ', n, 1, n, 0 ) )
517  maxwrk = max( maxwrk,
518  \$ n + n*ilaenv( 1, 'ZUNMQR', ' ', n, 1, n, 0 ) )
519  IF( ilvl ) THEN
520  maxwrk = max( maxwrk, n +
521  \$ n*ilaenv( 1, 'ZUNGQR', ' ', n, 1, n, 0 ) )
522  END IF
523  END IF
524  work( 1 ) = maxwrk
525 *
526  IF( lwork.LT.minwrk .AND. .NOT.lquery ) THEN
527  info = -25
528  END IF
529  END IF
530 *
531  IF( info.NE.0 ) THEN
532  CALL xerbla( 'ZGGEVX', -info )
533  RETURN
534  ELSE IF( lquery ) THEN
535  RETURN
536  END IF
537 *
538 * Quick return if possible
539 *
540  IF( n.EQ.0 )
541  \$ RETURN
542 *
543 * Get machine constants
544 *
545  eps = dlamch( 'P' )
546  smlnum = dlamch( 'S' )
547  bignum = one / smlnum
548  CALL dlabad( smlnum, bignum )
549  smlnum = sqrt( smlnum ) / eps
550  bignum = one / smlnum
551 *
552 * Scale A if max element outside range [SMLNUM,BIGNUM]
553 *
554  anrm = zlange( 'M', n, n, a, lda, rwork )
555  ilascl = .false.
556  IF( anrm.GT.zero .AND. anrm.LT.smlnum ) THEN
557  anrmto = smlnum
558  ilascl = .true.
559  ELSE IF( anrm.GT.bignum ) THEN
560  anrmto = bignum
561  ilascl = .true.
562  END IF
563  IF( ilascl )
564  \$ CALL zlascl( 'G', 0, 0, anrm, anrmto, n, n, a, lda, ierr )
565 *
566 * Scale B if max element outside range [SMLNUM,BIGNUM]
567 *
568  bnrm = zlange( 'M', n, n, b, ldb, rwork )
569  ilbscl = .false.
570  IF( bnrm.GT.zero .AND. bnrm.LT.smlnum ) THEN
571  bnrmto = smlnum
572  ilbscl = .true.
573  ELSE IF( bnrm.GT.bignum ) THEN
574  bnrmto = bignum
575  ilbscl = .true.
576  END IF
577  IF( ilbscl )
578  \$ CALL zlascl( 'G', 0, 0, bnrm, bnrmto, n, n, b, ldb, ierr )
579 *
580 * Permute and/or balance the matrix pair (A,B)
581 * (Real Workspace: need 6*N if BALANC = 'S' or 'B', 1 otherwise)
582 *
583  CALL zggbal( balanc, n, a, lda, b, ldb, ilo, ihi, lscale, rscale,
584  \$ rwork, ierr )
585 *
586 * Compute ABNRM and BBNRM
587 *
588  abnrm = zlange( '1', n, n, a, lda, rwork( 1 ) )
589  IF( ilascl ) THEN
590  rwork( 1 ) = abnrm
591  CALL dlascl( 'G', 0, 0, anrmto, anrm, 1, 1, rwork( 1 ), 1,
592  \$ ierr )
593  abnrm = rwork( 1 )
594  END IF
595 *
596  bbnrm = zlange( '1', n, n, b, ldb, rwork( 1 ) )
597  IF( ilbscl ) THEN
598  rwork( 1 ) = bbnrm
599  CALL dlascl( 'G', 0, 0, bnrmto, bnrm, 1, 1, rwork( 1 ), 1,
600  \$ ierr )
601  bbnrm = rwork( 1 )
602  END IF
603 *
604 * Reduce B to triangular form (QR decomposition of B)
605 * (Complex Workspace: need N, prefer N*NB )
606 *
607  irows = ihi + 1 - ilo
608  IF( ilv .OR. .NOT.wantsn ) THEN
609  icols = n + 1 - ilo
610  ELSE
611  icols = irows
612  END IF
613  itau = 1
614  iwrk = itau + irows
615  CALL zgeqrf( irows, icols, b( ilo, ilo ), ldb, work( itau ),
616  \$ work( iwrk ), lwork+1-iwrk, ierr )
617 *
618 * Apply the unitary transformation to A
619 * (Complex Workspace: need N, prefer N*NB)
620 *
621  CALL zunmqr( 'L', 'C', irows, icols, irows, b( ilo, ilo ), ldb,
622  \$ work( itau ), a( ilo, ilo ), lda, work( iwrk ),
623  \$ lwork+1-iwrk, ierr )
624 *
625 * Initialize VL and/or VR
626 * (Workspace: need N, prefer N*NB)
627 *
628  IF( ilvl ) THEN
629  CALL zlaset( 'Full', n, n, czero, cone, vl, ldvl )
630  IF( irows.GT.1 ) THEN
631  CALL zlacpy( 'L', irows-1, irows-1, b( ilo+1, ilo ), ldb,
632  \$ vl( ilo+1, ilo ), ldvl )
633  END IF
634  CALL zungqr( irows, irows, irows, vl( ilo, ilo ), ldvl,
635  \$ work( itau ), work( iwrk ), lwork+1-iwrk, ierr )
636  END IF
637 *
638  IF( ilvr )
639  \$ CALL zlaset( 'Full', n, n, czero, cone, vr, ldvr )
640 *
641 * Reduce to generalized Hessenberg form
642 * (Workspace: none needed)
643 *
644  IF( ilv .OR. .NOT.wantsn ) THEN
645 *
646 * Eigenvectors requested -- work on whole matrix.
647 *
648  CALL zgghrd( jobvl, jobvr, n, ilo, ihi, a, lda, b, ldb, vl,
649  \$ ldvl, vr, ldvr, ierr )
650  ELSE
651  CALL zgghrd( 'N', 'N', irows, 1, irows, a( ilo, ilo ), lda,
652  \$ b( ilo, ilo ), ldb, vl, ldvl, vr, ldvr, ierr )
653  END IF
654 *
655 * Perform QZ algorithm (Compute eigenvalues, and optionally, the
656 * Schur forms and Schur vectors)
657 * (Complex Workspace: need N)
658 * (Real Workspace: need N)
659 *
660  iwrk = itau
661  IF( ilv .OR. .NOT.wantsn ) THEN
662  chtemp = 'S'
663  ELSE
664  chtemp = 'E'
665  END IF
666 *
667  CALL zhgeqz( chtemp, jobvl, jobvr, n, ilo, ihi, a, lda, b, ldb,
668  \$ alpha, beta, vl, ldvl, vr, ldvr, work( iwrk ),
669  \$ lwork+1-iwrk, rwork, ierr )
670  IF( ierr.NE.0 ) THEN
671  IF( ierr.GT.0 .AND. ierr.LE.n ) THEN
672  info = ierr
673  ELSE IF( ierr.GT.n .AND. ierr.LE.2*n ) THEN
674  info = ierr - n
675  ELSE
676  info = n + 1
677  END IF
678  GO TO 90
679  END IF
680 *
681 * Compute Eigenvectors and estimate condition numbers if desired
682 * ZTGEVC: (Complex Workspace: need 2*N )
683 * (Real Workspace: need 2*N )
684 * ZTGSNA: (Complex Workspace: need 2*N*N if SENSE='V' or 'B')
685 * (Integer Workspace: need N+2 )
686 *
687  IF( ilv .OR. .NOT.wantsn ) THEN
688  IF( ilv ) THEN
689  IF( ilvl ) THEN
690  IF( ilvr ) THEN
691  chtemp = 'B'
692  ELSE
693  chtemp = 'L'
694  END IF
695  ELSE
696  chtemp = 'R'
697  END IF
698 *
699  CALL ztgevc( chtemp, 'B', ldumma, n, a, lda, b, ldb, vl,
700  \$ ldvl, vr, ldvr, n, in, work( iwrk ), rwork,
701  \$ ierr )
702  IF( ierr.NE.0 ) THEN
703  info = n + 2
704  GO TO 90
705  END IF
706  END IF
707 *
708  IF( .NOT.wantsn ) THEN
709 *
710 * compute eigenvectors (ZTGEVC) and estimate condition
711 * numbers (ZTGSNA). Note that the definition of the condition
712 * number is not invariant under transformation (u,v) to
713 * (Q*u, Z*v), where (u,v) are eigenvectors of the generalized
714 * Schur form (S,T), Q and Z are orthogonal matrices. In order
715 * to avoid using extra 2*N*N workspace, we have to
716 * re-calculate eigenvectors and estimate the condition numbers
717 * one at a time.
718 *
719  DO 20 i = 1, n
720 *
721  DO 10 j = 1, n
722  bwork( j ) = .false.
723  10 CONTINUE
724  bwork( i ) = .true.
725 *
726  iwrk = n + 1
727  iwrk1 = iwrk + n
728 *
729  IF( wantse .OR. wantsb ) THEN
730  CALL ztgevc( 'B', 'S', bwork, n, a, lda, b, ldb,
731  \$ work( 1 ), n, work( iwrk ), n, 1, m,
732  \$ work( iwrk1 ), rwork, ierr )
733  IF( ierr.NE.0 ) THEN
734  info = n + 2
735  GO TO 90
736  END IF
737  END IF
738 *
739  CALL ztgsna( sense, 'S', bwork, n, a, lda, b, ldb,
740  \$ work( 1 ), n, work( iwrk ), n, rconde( i ),
741  \$ rcondv( i ), 1, m, work( iwrk1 ),
742  \$ lwork-iwrk1+1, iwork, ierr )
743 *
744  20 CONTINUE
745  END IF
746  END IF
747 *
748 * Undo balancing on VL and VR and normalization
749 * (Workspace: none needed)
750 *
751  IF( ilvl ) THEN
752  CALL zggbak( balanc, 'L', n, ilo, ihi, lscale, rscale, n, vl,
753  \$ ldvl, ierr )
754 *
755  DO 50 jc = 1, n
756  temp = zero
757  DO 30 jr = 1, n
758  temp = max( temp, abs1( vl( jr, jc ) ) )
759  30 CONTINUE
760  IF( temp.LT.smlnum )
761  \$ GO TO 50
762  temp = one / temp
763  DO 40 jr = 1, n
764  vl( jr, jc ) = vl( jr, jc )*temp
765  40 CONTINUE
766  50 CONTINUE
767  END IF
768 *
769  IF( ilvr ) THEN
770  CALL zggbak( balanc, 'R', n, ilo, ihi, lscale, rscale, n, vr,
771  \$ ldvr, ierr )
772  DO 80 jc = 1, n
773  temp = zero
774  DO 60 jr = 1, n
775  temp = max( temp, abs1( vr( jr, jc ) ) )
776  60 CONTINUE
777  IF( temp.LT.smlnum )
778  \$ GO TO 80
779  temp = one / temp
780  DO 70 jr = 1, n
781  vr( jr, jc ) = vr( jr, jc )*temp
782  70 CONTINUE
783  80 CONTINUE
784  END IF
785 *
786 * Undo scaling if necessary
787 *
788  90 CONTINUE
789 *
790  IF( ilascl )
791  \$ CALL zlascl( 'G', 0, 0, anrmto, anrm, n, 1, alpha, n, ierr )
792 *
793  IF( ilbscl )
794  \$ CALL zlascl( 'G', 0, 0, bnrmto, bnrm, n, 1, beta, n, ierr )
795 *
796  work( 1 ) = maxwrk
797  RETURN
798 *
799 * End of ZGGEVX
800 *
double precision function dlamch(CMACH)
DLAMCH
Definition: dlamch.f:69
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
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 zggbal(JOB, N, A, LDA, B, LDB, ILO, IHI, LSCALE, RSCALE, WORK, INFO)
ZGGBAL
Definition: zggbal.f:177
subroutine zggbak(JOB, SIDE, N, ILO, IHI, LSCALE, RSCALE, M, V, LDV, INFO)
ZGGBAK
Definition: zggbak.f:148
double precision function zlange(NORM, M, N, A, LDA, WORK)
ZLANGE returns the value of the 1-norm, Frobenius norm, infinity-norm, or the largest absolute value ...
Definition: zlange.f:115
subroutine ztgevc(SIDE, HOWMNY, SELECT, N, S, LDS, P, LDP, VL, LDVL, VR, LDVR, MM, M, WORK, RWORK, INFO)
ZTGEVC
Definition: ztgevc.f:219
subroutine zhgeqz(JOB, COMPQ, COMPZ, N, ILO, IHI, H, LDH, T, LDT, ALPHA, BETA, Q, LDQ, Z, LDZ, WORK, LWORK, RWORK, INFO)
ZHGEQZ
Definition: zhgeqz.f:284
subroutine zlascl(TYPE, KL, KU, CFROM, CTO, M, N, A, LDA, INFO)
ZLASCL multiplies a general rectangular matrix by a real scalar defined as cto/cfrom.
Definition: zlascl.f:143
subroutine zlacpy(UPLO, M, N, A, LDA, B, LDB)
ZLACPY copies all or part of one two-dimensional array to another.
Definition: zlacpy.f:103
subroutine zlaset(UPLO, M, N, ALPHA, BETA, A, LDA)
ZLASET initializes the off-diagonal elements and the diagonal elements of a matrix to given values.
Definition: zlaset.f:106
subroutine zungqr(M, N, K, A, LDA, TAU, WORK, LWORK, INFO)
ZUNGQR
Definition: zungqr.f:128
subroutine ztgsna(JOB, HOWMNY, SELECT, N, A, LDA, B, LDB, VL, LDVL, VR, LDVR, S, DIF, MM, M, WORK, LWORK, IWORK, INFO)
ZTGSNA
Definition: ztgsna.f:311
subroutine zgghrd(COMPQ, COMPZ, N, ILO, IHI, A, LDA, B, LDB, Q, LDQ, Z, LDZ, INFO)
ZGGHRD
Definition: zgghrd.f:204
subroutine zunmqr(SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC, WORK, LWORK, INFO)
ZUNMQR
Definition: zunmqr.f:167
subroutine zgeqrf(M, N, A, LDA, TAU, WORK, LWORK, INFO)
ZGEQRF VARIANT: left-looking Level 3 BLAS of the algorithm.
Definition: zgeqrf.f:151
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