 LAPACK  3.8.0 LAPACK: Linear Algebra PACKage

## ◆ ztgsen()

 subroutine ztgsen ( integer IJOB, logical WANTQ, logical WANTZ, logical, dimension( * ) SELECT, 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( ldq, * ) Q, integer LDQ, complex*16, dimension( ldz, * ) Z, integer LDZ, integer M, double precision PL, double precision PR, double precision, dimension( * ) DIF, complex*16, dimension( * ) WORK, integer LWORK, integer, dimension( * ) IWORK, integer LIWORK, integer INFO )

ZTGSEN

Purpose:
``` ZTGSEN 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.

ZTGSEN 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*16 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*16 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*16 array, dimension (N)` [out] BETA ``` BETA is COMPLEX*16 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*16 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*16 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 DOUBLE PRECISION` [out] PR ``` PR is DOUBLE PRECISION 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 DOUBLE PRECISION 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 ZLACN2. 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*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 >= 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.```
Date
June 2016
Further Details:
```  ZTGSEN 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**H, 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
conjugate 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

Note that if the default method for computing the Frobenius-norm-
based estimate DIF is not wanted (see ZLATDF), then the parameter
IDIFJB (see below) should be changed from 3 to 4 (routine ZLATDF
(IJOB = 2 will be used)). See ZTGSYL for more details.```
Contributors:
Bo Kagstrom and Peter Poromaa, Department of Computing Science, Umea University, S-901 87 Umea, Sweden.
References:
 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.
 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.
 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 ztgsen.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  DOUBLE PRECISION pl, pr
446 * ..
447 * .. Array Arguments ..
448  LOGICAL select( * )
449  INTEGER iwork( * )
450  DOUBLE PRECISION dif( * )
451  COMPLEX*16 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  DOUBLE PRECISION zero, one
461  parameter( zero = 0.0d+0, one = 1.0d+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  DOUBLE PRECISION dscale, dsum, rdscal, safmin
468  COMPLEX*16 temp1, temp2
469 * ..
470 * .. Local Arrays ..
471  INTEGER isave( 3 )
472 * ..
473 * .. External Subroutines ..
474  EXTERNAL xerbla, zlacn2, zlacpy, zlassq, zscal, ztgexc,
475  \$ ztgsyl
476 * ..
477 * .. Intrinsic Functions ..
478  INTRINSIC abs, dcmplx, dconjg, max, sqrt
479 * ..
480 * .. External Functions ..
481  DOUBLE PRECISION dlamch
482  EXTERNAL dlamch
483 * ..
484 * .. Executable Statements ..
485 *
486 * Decode and test the input parameters
487 *
488  info = 0
489  lquery = ( lwork.EQ.-1 .OR. liwork.EQ.-1 )
490 *
491  IF( ijob.LT.0 .OR. ijob.GT.5 ) THEN
492  info = -1
493  ELSE IF( n.LT.0 ) THEN
494  info = -5
495  ELSE IF( lda.LT.max( 1, n ) ) THEN
496  info = -7
497  ELSE IF( ldb.LT.max( 1, n ) ) THEN
498  info = -9
499  ELSE IF( ldq.LT.1 .OR. ( wantq .AND. ldq.LT.n ) ) THEN
500  info = -13
501  ELSE IF( ldz.LT.1 .OR. ( wantz .AND. ldz.LT.n ) ) THEN
502  info = -15
503  END IF
504 *
505  IF( info.NE.0 ) THEN
506  CALL xerbla( 'ZTGSEN', -info )
507  RETURN
508  END IF
509 *
510  ierr = 0
511 *
512  wantp = ijob.EQ.1 .OR. ijob.GE.4
513  wantd1 = ijob.EQ.2 .OR. ijob.EQ.4
514  wantd2 = ijob.EQ.3 .OR. ijob.EQ.5
515  wantd = wantd1 .OR. wantd2
516 *
517 * Set M to the dimension of the specified pair of deflating
518 * subspaces.
519 *
520  m = 0
521  IF( .NOT.lquery .OR. ijob.NE.0 ) THEN
522  DO 10 k = 1, n
523  alpha( k ) = a( k, k )
524  beta( k ) = b( k, k )
525  IF( k.LT.n ) THEN
526  IF( SELECT( k ) )
527  \$ m = m + 1
528  ELSE
529  IF( SELECT( n ) )
530  \$ m = m + 1
531  END IF
532  10 CONTINUE
533  END IF
534 *
535  IF( ijob.EQ.1 .OR. ijob.EQ.2 .OR. ijob.EQ.4 ) THEN
536  lwmin = max( 1, 2*m*( n-m ) )
537  liwmin = max( 1, n+2 )
538  ELSE IF( ijob.EQ.3 .OR. ijob.EQ.5 ) THEN
539  lwmin = max( 1, 4*m*( n-m ) )
540  liwmin = max( 1, 2*m*( n-m ), n+2 )
541  ELSE
542  lwmin = 1
543  liwmin = 1
544  END IF
545 *
546  work( 1 ) = lwmin
547  iwork( 1 ) = liwmin
548 *
549  IF( lwork.LT.lwmin .AND. .NOT.lquery ) THEN
550  info = -21
551  ELSE IF( liwork.LT.liwmin .AND. .NOT.lquery ) THEN
552  info = -23
553  END IF
554 *
555  IF( info.NE.0 ) THEN
556  CALL xerbla( 'ZTGSEN', -info )
557  RETURN
558  ELSE IF( lquery ) THEN
559  RETURN
560  END IF
561 *
562 * Quick return if possible.
563 *
564  IF( m.EQ.n .OR. m.EQ.0 ) THEN
565  IF( wantp ) THEN
566  pl = one
567  pr = one
568  END IF
569  IF( wantd ) THEN
570  dscale = zero
571  dsum = one
572  DO 20 i = 1, n
573  CALL zlassq( n, a( 1, i ), 1, dscale, dsum )
574  CALL zlassq( n, b( 1, i ), 1, dscale, dsum )
575  20 CONTINUE
576  dif( 1 ) = dscale*sqrt( dsum )
577  dif( 2 ) = dif( 1 )
578  END IF
579  GO TO 70
580  END IF
581 *
582 * Get machine constant
583 *
584  safmin = dlamch( 'S' )
585 *
586 * Collect the selected blocks at the top-left corner of (A, B).
587 *
588  ks = 0
589  DO 30 k = 1, n
590  swap = SELECT( k )
591  IF( swap ) THEN
592  ks = ks + 1
593 *
594 * Swap the K-th block to position KS. Compute unitary Q
595 * and Z that will swap adjacent diagonal blocks in (A, B).
596 *
597  IF( k.NE.ks )
598  \$ CALL ztgexc( wantq, wantz, n, a, lda, b, ldb, q, ldq, z,
599  \$ ldz, k, ks, ierr )
600 *
601  IF( ierr.GT.0 ) THEN
602 *
603 * Swap is rejected: exit.
604 *
605  info = 1
606  IF( wantp ) THEN
607  pl = zero
608  pr = zero
609  END IF
610  IF( wantd ) THEN
611  dif( 1 ) = zero
612  dif( 2 ) = zero
613  END IF
614  GO TO 70
615  END IF
616  END IF
617  30 CONTINUE
618  IF( wantp ) THEN
619 *
620 * Solve generalized Sylvester equation for R and L:
621 * A11 * R - L * A22 = A12
622 * B11 * R - L * B22 = B12
623 *
624  n1 = m
625  n2 = n - m
626  i = n1 + 1
627  CALL zlacpy( 'Full', n1, n2, a( 1, i ), lda, work, n1 )
628  CALL zlacpy( 'Full', n1, n2, b( 1, i ), ldb, work( n1*n2+1 ),
629  \$ n1 )
630  ijb = 0
631  CALL ztgsyl( 'N', ijb, n1, n2, a, lda, a( i, i ), lda, work,
632  \$ n1, b, ldb, b( i, i ), ldb, work( n1*n2+1 ), n1,
633  \$ dscale, dif( 1 ), work( n1*n2*2+1 ),
634  \$ lwork-2*n1*n2, iwork, ierr )
635 *
636 * Estimate the reciprocal of norms of "projections" onto
637 * left and right eigenspaces
638 *
639  rdscal = zero
640  dsum = one
641  CALL zlassq( n1*n2, work, 1, rdscal, dsum )
642  pl = rdscal*sqrt( dsum )
643  IF( pl.EQ.zero ) THEN
644  pl = one
645  ELSE
646  pl = dscale / ( sqrt( dscale*dscale / pl+pl )*sqrt( pl ) )
647  END IF
648  rdscal = zero
649  dsum = one
650  CALL zlassq( n1*n2, work( n1*n2+1 ), 1, rdscal, dsum )
651  pr = rdscal*sqrt( dsum )
652  IF( pr.EQ.zero ) THEN
653  pr = one
654  ELSE
655  pr = dscale / ( sqrt( dscale*dscale / pr+pr )*sqrt( pr ) )
656  END IF
657  END IF
658  IF( wantd ) THEN
659 *
660 * Compute estimates Difu and Difl.
661 *
662  IF( wantd1 ) THEN
663  n1 = m
664  n2 = n - m
665  i = n1 + 1
666  ijb = idifjb
667 *
668 * Frobenius norm-based Difu estimate.
669 *
670  CALL ztgsyl( 'N', ijb, n1, n2, a, lda, a( i, i ), lda, work,
671  \$ n1, b, ldb, b( i, i ), ldb, work( n1*n2+1 ),
672  \$ n1, dscale, dif( 1 ), work( n1*n2*2+1 ),
673  \$ lwork-2*n1*n2, iwork, ierr )
674 *
675 * Frobenius norm-based Difl estimate.
676 *
677  CALL ztgsyl( 'N', ijb, n2, n1, a( i, i ), lda, a, lda, work,
678  \$ n2, b( i, i ), ldb, b, ldb, work( n1*n2+1 ),
679  \$ n2, dscale, dif( 2 ), work( n1*n2*2+1 ),
680  \$ lwork-2*n1*n2, iwork, ierr )
681  ELSE
682 *
683 * Compute 1-norm-based estimates of Difu and Difl using
684 * reversed communication with ZLACN2. In each step a
685 * generalized Sylvester equation or a transposed variant
686 * is solved.
687 *
688  kase = 0
689  n1 = m
690  n2 = n - m
691  i = n1 + 1
692  ijb = 0
693  mn2 = 2*n1*n2
694 *
695 * 1-norm-based estimate of Difu.
696 *
697  40 CONTINUE
698  CALL zlacn2( mn2, work( mn2+1 ), work, dif( 1 ), kase,
699  \$ isave )
700  IF( kase.NE.0 ) THEN
701  IF( kase.EQ.1 ) THEN
702 *
703 * Solve generalized Sylvester equation
704 *
705  CALL ztgsyl( 'N', ijb, n1, n2, a, lda, a( i, i ), lda,
706  \$ work, n1, b, ldb, b( i, i ), ldb,
707  \$ work( n1*n2+1 ), n1, dscale, dif( 1 ),
708  \$ work( n1*n2*2+1 ), lwork-2*n1*n2, iwork,
709  \$ ierr )
710  ELSE
711 *
712 * Solve the transposed variant.
713 *
714  CALL ztgsyl( 'C', ijb, n1, n2, a, lda, a( i, i ), lda,
715  \$ work, n1, b, ldb, b( i, i ), ldb,
716  \$ work( n1*n2+1 ), n1, dscale, dif( 1 ),
717  \$ work( n1*n2*2+1 ), lwork-2*n1*n2, iwork,
718  \$ ierr )
719  END IF
720  GO TO 40
721  END IF
722  dif( 1 ) = dscale / dif( 1 )
723 *
724 * 1-norm-based estimate of Difl.
725 *
726  50 CONTINUE
727  CALL zlacn2( mn2, work( mn2+1 ), work, dif( 2 ), kase,
728  \$ isave )
729  IF( kase.NE.0 ) THEN
730  IF( kase.EQ.1 ) THEN
731 *
732 * Solve generalized Sylvester equation
733 *
734  CALL ztgsyl( 'N', ijb, n2, n1, a( i, i ), lda, a, lda,
735  \$ work, n2, b( i, i ), ldb, b, ldb,
736  \$ work( n1*n2+1 ), n2, dscale, dif( 2 ),
737  \$ work( n1*n2*2+1 ), lwork-2*n1*n2, iwork,
738  \$ ierr )
739  ELSE
740 *
741 * Solve the transposed variant.
742 *
743  CALL ztgsyl( 'C', ijb, n2, n1, a( i, i ), lda, a, lda,
744  \$ work, n2, b, ldb, b( i, i ), ldb,
745  \$ work( n1*n2+1 ), n2, dscale, dif( 2 ),
746  \$ work( n1*n2*2+1 ), lwork-2*n1*n2, iwork,
747  \$ ierr )
748  END IF
749  GO TO 50
750  END IF
751  dif( 2 ) = dscale / dif( 2 )
752  END IF
753  END IF
754 *
755 * If B(K,K) is complex, make it real and positive (normalization
756 * of the generalized Schur form) and Store the generalized
757 * eigenvalues of reordered pair (A, B)
758 *
759  DO 60 k = 1, n
760  dscale = abs( b( k, k ) )
761  IF( dscale.GT.safmin ) THEN
762  temp1 = dconjg( b( k, k ) / dscale )
763  temp2 = b( k, k ) / dscale
764  b( k, k ) = dscale
765  CALL zscal( n-k, temp1, b( k, k+1 ), ldb )
766  CALL zscal( n-k+1, temp1, a( k, k ), lda )
767  IF( wantq )
768  \$ CALL zscal( n, temp2, q( 1, k ), 1 )
769  ELSE
770  b( k, k ) = dcmplx( zero, zero )
771  END IF
772 *
773  alpha( k ) = a( k, k )
774  beta( k ) = b( k, k )
775 *
776  60 CONTINUE
777 *
778  70 CONTINUE
779 *
780  work( 1 ) = lwmin
781  iwork( 1 ) = liwmin
782 *
783  RETURN
784 *
785 * End of ZTGSEN
786 *
subroutine ztgexc(WANTQ, WANTZ, N, A, LDA, B, LDB, Q, LDQ, Z, LDZ, IFST, ILST, INFO)
ZTGEXC
Definition: ztgexc.f:202
subroutine zlacn2(N, V, X, EST, KASE, ISAVE)
ZLACN2 estimates the 1-norm of a square matrix, using reverse communication for evaluating matrix-vec...
Definition: zlacn2.f:135
subroutine ztgsyl(TRANS, IJOB, M, N, A, LDA, B, LDB, C, LDC, D, LDD, E, LDE, F, LDF, SCALE, DIF, WORK, LWORK, IWORK, INFO)
ZTGSYL
Definition: ztgsyl.f:297
subroutine zlassq(N, X, INCX, SCALE, SUMSQ)
ZLASSQ updates a sum of squares represented in scaled form.
Definition: zlassq.f:108
subroutine zscal(N, ZA, ZX, INCX)
ZSCAL
Definition: zscal.f:80
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:62
double precision function dlamch(CMACH)
DLAMCH
Definition: dlamch.f:65
subroutine zlacpy(UPLO, M, N, A, LDA, B, LDB)
ZLACPY copies all or part of one two-dimensional array to another.
Definition: zlacpy.f:105
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