LAPACK  3.8.0
LAPACK: Linear Algebra PACKage

◆ ctgsen()

subroutine ctgsen ( integer  IJOB,
logical  WANTQ,
logical  WANTZ,
logical, dimension( * )  SELECT,
integer  N,
complex, dimension( lda, * )  A,
integer  LDA,
complex, dimension( ldb, * )  B,
integer  LDB,
complex, dimension( * )  ALPHA,
complex, dimension( * )  BETA,
complex, dimension( ldq, * )  Q,
integer  LDQ,
complex, dimension( ldz, * )  Z,
integer  LDZ,
integer  M,
real  PL,
real  PR,
real, dimension( * )  DIF,
complex, dimension( * )  WORK,
integer  LWORK,
integer, dimension( * )  IWORK,
integer  LIWORK,
integer  INFO 
)

CTGSEN

Download CTGSEN + dependencies [TGZ] [ZIP] [TXT]

Purpose:
 CTGSEN reorders the generalized Schur decomposition of a complex
 matrix pair (A, B) (in terms of an unitary equivalence trans-
 formation Q**H * (A, B) * Z), so that a selected cluster of eigenvalues
 appears in the leading diagonal blocks of the pair (A,B). The leading
 columns of Q and Z form unitary bases of the corresponding left and
 right eigenspaces (deflating subspaces). (A, B) must be in
 generalized Schur canonical form, that is, A and B are both upper
 triangular.

 CTGSEN also computes the generalized eigenvalues

          w(j)= ALPHA(j) / BETA(j)

 of the reordered matrix pair (A, B).

 Optionally, the routine computes estimates of reciprocal condition
 numbers for eigenvalues and eigenspaces. These are Difu[(A11,B11),
 (A22,B22)] and Difl[(A11,B11), (A22,B22)], i.e. the separation(s)
 between the matrix pairs (A11, B11) and (A22,B22) that correspond to
 the selected cluster and the eigenvalues outside the cluster, resp.,
 and norms of "projections" onto left and right eigenspaces w.r.t.
 the selected cluster in the (1,1)-block.
Parameters
[in]IJOB
          IJOB is INTEGER
          Specifies whether condition numbers are required for the
          cluster of eigenvalues (PL and PR) or the deflating subspaces
          (Difu and Difl):
           =0: Only reorder w.r.t. SELECT. No extras.
           =1: Reciprocal of norms of "projections" onto left and right
               eigenspaces w.r.t. the selected cluster (PL and PR).
           =2: Upper bounds on Difu and Difl. F-norm-based estimate
               (DIF(1:2)).
           =3: Estimate of Difu and Difl. 1-norm-based estimate
               (DIF(1:2)).
               About 5 times as expensive as IJOB = 2.
           =4: Compute PL, PR and DIF (i.e. 0, 1 and 2 above): Economic
               version to get it all.
           =5: Compute PL, PR and DIF (i.e. 0, 1 and 3 above)
[in]WANTQ
          WANTQ is LOGICAL
          .TRUE. : update the left transformation matrix Q;
          .FALSE.: do not update Q.
[in]WANTZ
          WANTZ is LOGICAL
          .TRUE. : update the right transformation matrix Z;
          .FALSE.: do not update Z.
[in]SELECT
          SELECT is LOGICAL array, dimension (N)
          SELECT specifies the eigenvalues in the selected cluster. To
          select an eigenvalue w(j), SELECT(j) must be set to
          .TRUE..
[in]N
          N is INTEGER
          The order of the matrices A and B. N >= 0.
[in,out]A
          A is COMPLEX array, dimension(LDA,N)
          On entry, the upper triangular matrix A, in generalized
          Schur canonical form.
          On exit, A is overwritten by the reordered matrix A.
[in]LDA
          LDA is INTEGER
          The leading dimension of the array A. LDA >= max(1,N).
[in,out]B
          B is COMPLEX array, dimension(LDB,N)
          On entry, the upper triangular matrix B, in generalized
          Schur canonical form.
          On exit, B is overwritten by the reordered matrix B.
[in]LDB
          LDB is INTEGER
          The leading dimension of the array B. LDB >= max(1,N).
[out]ALPHA
          ALPHA is COMPLEX array, dimension (N)
[out]BETA
          BETA is COMPLEX array, dimension (N)

          The diagonal elements of A and B, respectively,
          when the pair (A,B) has been reduced to generalized Schur
          form.  ALPHA(i)/BETA(i) i=1,...,N are the generalized
          eigenvalues.
[in,out]Q
          Q is COMPLEX array, dimension (LDQ,N)
          On entry, if WANTQ = .TRUE., Q is an N-by-N matrix.
          On exit, Q has been postmultiplied by the left unitary
          transformation matrix which reorder (A, B); The leading M
          columns of Q form orthonormal bases for the specified pair of
          left eigenspaces (deflating subspaces).
          If WANTQ = .FALSE., Q is not referenced.
[in]LDQ
          LDQ is INTEGER
          The leading dimension of the array Q. LDQ >= 1.
          If WANTQ = .TRUE., LDQ >= N.
[in,out]Z
          Z is COMPLEX array, dimension (LDZ,N)
          On entry, if WANTZ = .TRUE., Z is an N-by-N matrix.
          On exit, Z has been postmultiplied by the left unitary
          transformation matrix which reorder (A, B); The leading M
          columns of Z form orthonormal bases for the specified pair of
          left eigenspaces (deflating subspaces).
          If WANTZ = .FALSE., Z is not referenced.
[in]LDZ
          LDZ is INTEGER
          The leading dimension of the array Z. LDZ >= 1.
          If WANTZ = .TRUE., LDZ >= N.
[out]M
          M is INTEGER
          The dimension of the specified pair of left and right
          eigenspaces, (deflating subspaces) 0 <= M <= N.
[out]PL
          PL is REAL
[out]PR
          PR is REAL

          If IJOB = 1, 4 or 5, PL, PR are lower bounds on the
          reciprocal  of the norm of "projections" onto left and right
          eigenspace with respect to the selected cluster.
          0 < PL, PR <= 1.
          If M = 0 or M = N, PL = PR  = 1.
          If IJOB = 0, 2 or 3 PL, PR are not referenced.
[out]DIF
          DIF is REAL array, dimension (2).
          If IJOB >= 2, DIF(1:2) store the estimates of Difu and Difl.
          If IJOB = 2 or 4, DIF(1:2) are F-norm-based upper bounds on
          Difu and Difl. If IJOB = 3 or 5, DIF(1:2) are 1-norm-based
          estimates of Difu and Difl, computed using reversed
          communication with CLACN2.
          If M = 0 or N, DIF(1:2) = F-norm([A, B]).
          If IJOB = 0 or 1, DIF is not referenced.
[out]WORK
          WORK is COMPLEX 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 >=  1
          If IJOB = 1, 2 or 4, LWORK >=  2*M*(N-M)
          If IJOB = 3 or 5, LWORK >=  4*M*(N-M)

          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]IWORK
          IWORK is INTEGER array, dimension (MAX(1,LIWORK))
          On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.
[in]LIWORK
          LIWORK is INTEGER
          The dimension of the array IWORK. LIWORK >= 1.
          If IJOB = 1, 2 or 4, LIWORK >=  N+2;
          If IJOB = 3 or 5, LIWORK >= MAX(N+2, 2*M*(N-M));

          If LIWORK = -1, then a workspace query is assumed; the
          routine only calculates the optimal size of the IWORK array,
          returns this value as the first entry of the IWORK array, and
          no error message related to LIWORK is issued by XERBLA.
[out]INFO
          INFO is INTEGER
            =0: Successful exit.
            <0: If INFO = -i, the i-th argument had an illegal value.
            =1: Reordering of (A, B) failed because the transformed
                matrix pair (A, B) would be too far from generalized
                Schur form; the problem is very ill-conditioned.
                (A, B) may have been partially reordered.
                If requested, 0 is returned in DIF(*), PL and PR.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date
June 2016
Further Details:
  CTGSEN first collects the selected eigenvalues by computing unitary
  U and W that move them to the top left corner of (A, B). In other
  words, the selected eigenvalues are the eigenvalues of (A11, B11) in

              U**H*(A, B)*W = (A11 A12) (B11 B12) n1
                              ( 0  A22),( 0  B22) n2
                                n1  n2    n1  n2

  where N = n1+n2 and U**H means the conjugate transpose of U. The first
  n1 columns of U and W span the specified pair of left and right
  eigenspaces (deflating subspaces) of (A, B).

  If (A, B) has been obtained from the generalized real Schur
  decomposition of a matrix pair (C, D) = Q*(A, B)*Z', then the
  reordered generalized Schur form of (C, D) is given by

           (C, D) = (Q*U)*(U**H *(A, B)*W)*(Z*W)**H,

  and the first n1 columns of Q*U and Z*W span the corresponding
  deflating subspaces of (C, D) (Q and Z store Q*U and Z*W, resp.).

  Note that if the selected eigenvalue is sufficiently ill-conditioned,
  then its value may differ significantly from its value before
  reordering.

  The reciprocal condition numbers of the left and right eigenspaces
  spanned by the first n1 columns of U and W (or Q*U and Z*W) may
  be returned in DIF(1:2), corresponding to Difu and Difl, resp.

  The Difu and Difl are defined as:

       Difu[(A11, B11), (A22, B22)] = sigma-min( Zu )
  and
       Difl[(A11, B11), (A22, B22)] = Difu[(A22, B22), (A11, B11)],

  where sigma-min(Zu) is the smallest singular value of the
  (2*n1*n2)-by-(2*n1*n2) matrix

       Zu = [ kron(In2, A11)  -kron(A22**H, In1) ]
            [ kron(In2, B11)  -kron(B22**H, In1) ].

  Here, Inx is the identity matrix of size nx and A22**H is the
  conjuguate transpose of A22. kron(X, Y) is the Kronecker product between
  the matrices X and Y.

  When DIF(2) is small, small changes in (A, B) can cause large changes
  in the deflating subspace. An approximate (asymptotic) bound on the
  maximum angular error in the computed deflating subspaces is

       EPS * norm((A, B)) / DIF(2),

  where EPS is the machine precision.

  The reciprocal norm of the projectors on the left and right
  eigenspaces associated with (A11, B11) may be returned in PL and PR.
  They are computed as follows. First we compute L and R so that
  P*(A, B)*Q is block diagonal, where

       P = ( I -L ) n1           Q = ( I R ) n1
           ( 0  I ) n2    and        ( 0 I ) n2
             n1 n2                    n1 n2

  and (L, R) is the solution to the generalized Sylvester equation

       A11*R - L*A22 = -A12
       B11*R - L*B22 = -B12

  Then PL = (F-norm(L)**2+1)**(-1/2) and PR = (F-norm(R)**2+1)**(-1/2).
  An approximate (asymptotic) bound on the average absolute error of
  the selected eigenvalues is

       EPS * norm((A, B)) / PL.

  There are also global error bounds which valid for perturbations up
  to a certain restriction:  A lower bound (x) on the smallest
  F-norm(E,F) for which an eigenvalue of (A11, B11) may move and
  coalesce with an eigenvalue of (A22, B22) under perturbation (E,F),
  (i.e. (A + E, B + F), is

   x = min(Difu,Difl)/((1/(PL*PL)+1/(PR*PR))**(1/2)+2*max(1/PL,1/PR)).

  An approximate bound on x can be computed from DIF(1:2), PL and PR.

  If y = ( F-norm(E,F) / x) <= 1, the angles between the perturbed
  (L', R') and unperturbed (L, R) left and right deflating subspaces
  associated with the selected cluster in the (1,1)-blocks can be
  bounded as

   max-angle(L, L') <= arctan( y * PL / (1 - y * (1 - PL * PL)**(1/2))
   max-angle(R, R') <= arctan( y * PR / (1 - y * (1 - PR * PR)**(1/2))

  See LAPACK User's Guide section 4.11 or the following references
  for more information.

  Note that if the default method for computing the Frobenius-norm-
  based estimate DIF is not wanted (see CLATDF), then the parameter
  IDIFJB (see below) should be changed from 3 to 4 (routine CLATDF
  (IJOB = 2 will be used)). See CTGSYL for more details.
Contributors:
Bo Kagstrom and Peter Poromaa, Department of Computing Science, Umea University, S-901 87 Umea, Sweden.
References:
[1] B. Kagstrom; A Direct Method for Reordering Eigenvalues in the Generalized Real Schur Form of a Regular Matrix Pair (A, B), in M.S. Moonen et al (eds), Linear Algebra for Large Scale and Real-Time Applications, Kluwer Academic Publ. 1993, pp 195-218.
[2] B. Kagstrom and P. Poromaa; Computing Eigenspaces with Specified Eigenvalues of a Regular Matrix Pair (A, B) and Condition Estimation: Theory, Algorithms and Software, Report UMINF - 94.04, Department of Computing Science, Umea University, S-901 87 Umea, Sweden, 1994. Also as LAPACK Working Note 87. To appear in Numerical Algorithms, 1996.
[3] B. Kagstrom and P. Poromaa, LAPACK-Style Algorithms and Software for Solving the Generalized Sylvester Equation and Estimating the Separation between Regular Matrix Pairs, Report UMINF - 93.23, Department of Computing Science, Umea University, S-901 87 Umea, Sweden, December 1993, Revised April 1994, Also as LAPACK working Note 75. To appear in ACM Trans. on Math. Software, Vol 22, No 1, 1996.

Definition at line 435 of file ctgsen.f.

435 *
436 * -- LAPACK computational routine (version 3.7.1) --
437 * -- LAPACK is a software package provided by Univ. of Tennessee, --
438 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
439 * June 2016
440 *
441 * .. Scalar Arguments ..
442  LOGICAL wantq, wantz
443  INTEGER ijob, info, lda, ldb, ldq, ldz, liwork, lwork,
444  $ m, n
445  REAL pl, pr
446 * ..
447 * .. Array Arguments ..
448  LOGICAL select( * )
449  INTEGER iwork( * )
450  REAL dif( * )
451  COMPLEX a( lda, * ), alpha( * ), b( ldb, * ),
452  $ beta( * ), q( ldq, * ), work( * ), z( ldz, * )
453 * ..
454 *
455 * =====================================================================
456 *
457 * .. Parameters ..
458  INTEGER idifjb
459  parameter( idifjb = 3 )
460  REAL zero, one
461  parameter( zero = 0.0e+0, one = 1.0e+0 )
462 * ..
463 * .. Local Scalars ..
464  LOGICAL lquery, swap, wantd, wantd1, wantd2, wantp
465  INTEGER i, ierr, ijb, k, kase, ks, liwmin, lwmin, mn2,
466  $ n1, n2
467  REAL dscale, dsum, rdscal, safmin
468  COMPLEX temp1, temp2
469 * ..
470 * .. Local Arrays ..
471  INTEGER isave( 3 )
472 * ..
473 * .. External Subroutines ..
474  REAL slamch
475  EXTERNAL clacn2, clacpy, classq, cscal, ctgexc, ctgsyl,
476  $ slamch, xerbla
477 * ..
478 * .. Intrinsic Functions ..
479  INTRINSIC abs, cmplx, conjg, max, sqrt
480 * ..
481 * .. Executable Statements ..
482 *
483 * Decode and test the input parameters
484 *
485  info = 0
486  lquery = ( lwork.EQ.-1 .OR. liwork.EQ.-1 )
487 *
488  IF( ijob.LT.0 .OR. ijob.GT.5 ) THEN
489  info = -1
490  ELSE IF( n.LT.0 ) THEN
491  info = -5
492  ELSE IF( lda.LT.max( 1, n ) ) THEN
493  info = -7
494  ELSE IF( ldb.LT.max( 1, n ) ) THEN
495  info = -9
496  ELSE IF( ldq.LT.1 .OR. ( wantq .AND. ldq.LT.n ) ) THEN
497  info = -13
498  ELSE IF( ldz.LT.1 .OR. ( wantz .AND. ldz.LT.n ) ) THEN
499  info = -15
500  END IF
501 *
502  IF( info.NE.0 ) THEN
503  CALL xerbla( 'CTGSEN', -info )
504  RETURN
505  END IF
506 *
507  ierr = 0
508 *
509  wantp = ijob.EQ.1 .OR. ijob.GE.4
510  wantd1 = ijob.EQ.2 .OR. ijob.EQ.4
511  wantd2 = ijob.EQ.3 .OR. ijob.EQ.5
512  wantd = wantd1 .OR. wantd2
513 *
514 * Set M to the dimension of the specified pair of deflating
515 * subspaces.
516 *
517  m = 0
518  IF( .NOT.lquery .OR. ijob.NE.0 ) THEN
519  DO 10 k = 1, n
520  alpha( k ) = a( k, k )
521  beta( k ) = b( k, k )
522  IF( k.LT.n ) THEN
523  IF( SELECT( k ) )
524  $ m = m + 1
525  ELSE
526  IF( SELECT( n ) )
527  $ m = m + 1
528  END IF
529  10 CONTINUE
530  END IF
531 *
532  IF( ijob.EQ.1 .OR. ijob.EQ.2 .OR. ijob.EQ.4 ) THEN
533  lwmin = max( 1, 2*m*(n-m) )
534  liwmin = max( 1, n+2 )
535  ELSE IF( ijob.EQ.3 .OR. ijob.EQ.5 ) THEN
536  lwmin = max( 1, 4*m*(n-m) )
537  liwmin = max( 1, 2*m*(n-m), n+2 )
538  ELSE
539  lwmin = 1
540  liwmin = 1
541  END IF
542 *
543  work( 1 ) = lwmin
544  iwork( 1 ) = liwmin
545 *
546  IF( lwork.LT.lwmin .AND. .NOT.lquery ) THEN
547  info = -21
548  ELSE IF( liwork.LT.liwmin .AND. .NOT.lquery ) THEN
549  info = -23
550  END IF
551 *
552  IF( info.NE.0 ) THEN
553  CALL xerbla( 'CTGSEN', -info )
554  RETURN
555  ELSE IF( lquery ) THEN
556  RETURN
557  END IF
558 *
559 * Quick return if possible.
560 *
561  IF( m.EQ.n .OR. m.EQ.0 ) THEN
562  IF( wantp ) THEN
563  pl = one
564  pr = one
565  END IF
566  IF( wantd ) THEN
567  dscale = zero
568  dsum = one
569  DO 20 i = 1, n
570  CALL classq( n, a( 1, i ), 1, dscale, dsum )
571  CALL classq( n, b( 1, i ), 1, dscale, dsum )
572  20 CONTINUE
573  dif( 1 ) = dscale*sqrt( dsum )
574  dif( 2 ) = dif( 1 )
575  END IF
576  GO TO 70
577  END IF
578 *
579 * Get machine constant
580 *
581  safmin = slamch( 'S' )
582 *
583 * Collect the selected blocks at the top-left corner of (A, B).
584 *
585  ks = 0
586  DO 30 k = 1, n
587  swap = SELECT( k )
588  IF( swap ) THEN
589  ks = ks + 1
590 *
591 * Swap the K-th block to position KS. Compute unitary Q
592 * and Z that will swap adjacent diagonal blocks in (A, B).
593 *
594  IF( k.NE.ks )
595  $ CALL ctgexc( wantq, wantz, n, a, lda, b, ldb, q, ldq, z,
596  $ ldz, k, ks, ierr )
597 *
598  IF( ierr.GT.0 ) THEN
599 *
600 * Swap is rejected: exit.
601 *
602  info = 1
603  IF( wantp ) THEN
604  pl = zero
605  pr = zero
606  END IF
607  IF( wantd ) THEN
608  dif( 1 ) = zero
609  dif( 2 ) = zero
610  END IF
611  GO TO 70
612  END IF
613  END IF
614  30 CONTINUE
615  IF( wantp ) THEN
616 *
617 * Solve generalized Sylvester equation for R and L:
618 * A11 * R - L * A22 = A12
619 * B11 * R - L * B22 = B12
620 *
621  n1 = m
622  n2 = n - m
623  i = n1 + 1
624  CALL clacpy( 'Full', n1, n2, a( 1, i ), lda, work, n1 )
625  CALL clacpy( 'Full', n1, n2, b( 1, i ), ldb, work( n1*n2+1 ),
626  $ n1 )
627  ijb = 0
628  CALL ctgsyl( 'N', ijb, n1, n2, a, lda, a( i, i ), lda, work,
629  $ n1, b, ldb, b( i, i ), ldb, work( n1*n2+1 ), n1,
630  $ dscale, dif( 1 ), work( n1*n2*2+1 ),
631  $ lwork-2*n1*n2, iwork, ierr )
632 *
633 * Estimate the reciprocal of norms of "projections" onto
634 * left and right eigenspaces
635 *
636  rdscal = zero
637  dsum = one
638  CALL classq( n1*n2, work, 1, rdscal, dsum )
639  pl = rdscal*sqrt( dsum )
640  IF( pl.EQ.zero ) THEN
641  pl = one
642  ELSE
643  pl = dscale / ( sqrt( dscale*dscale / pl+pl )*sqrt( pl ) )
644  END IF
645  rdscal = zero
646  dsum = one
647  CALL classq( n1*n2, work( n1*n2+1 ), 1, rdscal, dsum )
648  pr = rdscal*sqrt( dsum )
649  IF( pr.EQ.zero ) THEN
650  pr = one
651  ELSE
652  pr = dscale / ( sqrt( dscale*dscale / pr+pr )*sqrt( pr ) )
653  END IF
654  END IF
655  IF( wantd ) THEN
656 *
657 * Compute estimates Difu and Difl.
658 *
659  IF( wantd1 ) THEN
660  n1 = m
661  n2 = n - m
662  i = n1 + 1
663  ijb = idifjb
664 *
665 * Frobenius norm-based Difu estimate.
666 *
667  CALL ctgsyl( 'N', ijb, n1, n2, a, lda, a( i, i ), lda, work,
668  $ n1, b, ldb, b( i, i ), ldb, work( n1*n2+1 ),
669  $ n1, dscale, dif( 1 ), work( n1*n2*2+1 ),
670  $ lwork-2*n1*n2, iwork, ierr )
671 *
672 * Frobenius norm-based Difl estimate.
673 *
674  CALL ctgsyl( 'N', ijb, n2, n1, a( i, i ), lda, a, lda, work,
675  $ n2, b( i, i ), ldb, b, ldb, work( n1*n2+1 ),
676  $ n2, dscale, dif( 2 ), work( n1*n2*2+1 ),
677  $ lwork-2*n1*n2, iwork, ierr )
678  ELSE
679 *
680 * Compute 1-norm-based estimates of Difu and Difl using
681 * reversed communication with CLACN2. In each step a
682 * generalized Sylvester equation or a transposed variant
683 * is solved.
684 *
685  kase = 0
686  n1 = m
687  n2 = n - m
688  i = n1 + 1
689  ijb = 0
690  mn2 = 2*n1*n2
691 *
692 * 1-norm-based estimate of Difu.
693 *
694  40 CONTINUE
695  CALL clacn2( mn2, work( mn2+1 ), work, dif( 1 ), kase,
696  $ isave )
697  IF( kase.NE.0 ) THEN
698  IF( kase.EQ.1 ) THEN
699 *
700 * Solve generalized Sylvester equation
701 *
702  CALL ctgsyl( 'N', ijb, n1, n2, a, lda, a( i, i ), lda,
703  $ work, n1, b, ldb, b( i, i ), ldb,
704  $ work( n1*n2+1 ), n1, dscale, dif( 1 ),
705  $ work( n1*n2*2+1 ), lwork-2*n1*n2, iwork,
706  $ ierr )
707  ELSE
708 *
709 * Solve the transposed variant.
710 *
711  CALL ctgsyl( 'C', ijb, n1, n2, a, lda, a( i, i ), lda,
712  $ work, n1, b, ldb, b( i, i ), ldb,
713  $ work( n1*n2+1 ), n1, dscale, dif( 1 ),
714  $ work( n1*n2*2+1 ), lwork-2*n1*n2, iwork,
715  $ ierr )
716  END IF
717  GO TO 40
718  END IF
719  dif( 1 ) = dscale / dif( 1 )
720 *
721 * 1-norm-based estimate of Difl.
722 *
723  50 CONTINUE
724  CALL clacn2( mn2, work( mn2+1 ), work, dif( 2 ), kase,
725  $ isave )
726  IF( kase.NE.0 ) THEN
727  IF( kase.EQ.1 ) THEN
728 *
729 * Solve generalized Sylvester equation
730 *
731  CALL ctgsyl( 'N', ijb, n2, n1, a( i, i ), lda, a, lda,
732  $ work, n2, b( i, i ), ldb, b, ldb,
733  $ work( n1*n2+1 ), n2, dscale, dif( 2 ),
734  $ work( n1*n2*2+1 ), lwork-2*n1*n2, iwork,
735  $ ierr )
736  ELSE
737 *
738 * Solve the transposed variant.
739 *
740  CALL ctgsyl( 'C', ijb, n2, n1, a( i, i ), lda, a, lda,
741  $ work, n2, b, ldb, b( i, i ), ldb,
742  $ work( n1*n2+1 ), n2, dscale, dif( 2 ),
743  $ work( n1*n2*2+1 ), lwork-2*n1*n2, iwork,
744  $ ierr )
745  END IF
746  GO TO 50
747  END IF
748  dif( 2 ) = dscale / dif( 2 )
749  END IF
750  END IF
751 *
752 * If B(K,K) is complex, make it real and positive (normalization
753 * of the generalized Schur form) and Store the generalized
754 * eigenvalues of reordered pair (A, B)
755 *
756  DO 60 k = 1, n
757  dscale = abs( b( k, k ) )
758  IF( dscale.GT.safmin ) THEN
759  temp1 = conjg( b( k, k ) / dscale )
760  temp2 = b( k, k ) / dscale
761  b( k, k ) = dscale
762  CALL cscal( n-k, temp1, b( k, k+1 ), ldb )
763  CALL cscal( n-k+1, temp1, a( k, k ), lda )
764  IF( wantq )
765  $ CALL cscal( n, temp2, q( 1, k ), 1 )
766  ELSE
767  b( k, k ) = cmplx( zero, zero )
768  END IF
769 *
770  alpha( k ) = a( k, k )
771  beta( k ) = b( k, k )
772 *
773  60 CONTINUE
774 *
775  70 CONTINUE
776 *
777  work( 1 ) = lwmin
778  iwork( 1 ) = liwmin
779 *
780  RETURN
781 *
782 * End of CTGSEN
783 *
subroutine classq(N, X, INCX, SCALE, SUMSQ)
CLASSQ updates a sum of squares represented in scaled form.
Definition: classq.f:108
subroutine cscal(N, CA, CX, INCX)
CSCAL
Definition: cscal.f:80
subroutine ctgsyl(TRANS, IJOB, M, N, A, LDA, B, LDB, C, LDC, D, LDD, E, LDE, F, LDF, SCALE, DIF, WORK, LWORK, IWORK, INFO)
CTGSYL
Definition: ctgsyl.f:297
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:62
real function slamch(CMACH)
SLAMCH
Definition: slamch.f:69
subroutine clacpy(UPLO, M, N, A, LDA, B, LDB)
CLACPY copies all or part of one two-dimensional array to another.
Definition: clacpy.f:105
subroutine clacn2(N, V, X, EST, KASE, ISAVE)
CLACN2 estimates the 1-norm of a square matrix, using reverse communication for evaluating matrix-vec...
Definition: clacn2.f:135
subroutine ctgexc(WANTQ, WANTZ, N, A, LDA, B, LDB, Q, LDQ, Z, LDZ, IFST, ILST, INFO)
CTGEXC
Definition: ctgexc.f:202
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