 LAPACK 3.11.0 LAPACK: Linear Algebra PACKage
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## ◆ 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

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.```
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

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:
 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 430 of file ctgsen.f.

433*
434* -- LAPACK computational routine --
435* -- LAPACK is a software package provided by Univ. of Tennessee, --
436* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
437*
438* .. Scalar Arguments ..
439 LOGICAL WANTQ, WANTZ
440 INTEGER IJOB, INFO, LDA, LDB, LDQ, LDZ, LIWORK, LWORK,
441 \$ M, N
442 REAL PL, PR
443* ..
444* .. Array Arguments ..
445 LOGICAL SELECT( * )
446 INTEGER IWORK( * )
447 REAL DIF( * )
448 COMPLEX A( LDA, * ), ALPHA( * ), B( LDB, * ),
449 \$ BETA( * ), Q( LDQ, * ), WORK( * ), Z( LDZ, * )
450* ..
451*
452* =====================================================================
453*
454* .. Parameters ..
455 INTEGER IDIFJB
456 parameter( idifjb = 3 )
457 REAL ZERO, ONE
458 parameter( zero = 0.0e+0, one = 1.0e+0 )
459* ..
460* .. Local Scalars ..
461 LOGICAL LQUERY, SWAP, WANTD, WANTD1, WANTD2, WANTP
462 INTEGER I, IERR, IJB, K, KASE, KS, LIWMIN, LWMIN, MN2,
463 \$ N1, N2
464 REAL DSCALE, DSUM, RDSCAL, SAFMIN
465 COMPLEX TEMP1, TEMP2
466* ..
467* .. Local Arrays ..
468 INTEGER ISAVE( 3 )
469* ..
470* .. External Subroutines ..
471 REAL SLAMCH
472 EXTERNAL clacn2, clacpy, classq, cscal, ctgexc, ctgsyl,
473 \$ slamch, xerbla
474* ..
475* .. Intrinsic Functions ..
476 INTRINSIC abs, cmplx, conjg, max, sqrt
477* ..
478* .. Executable Statements ..
479*
480* Decode and test the input parameters
481*
482 info = 0
483 lquery = ( lwork.EQ.-1 .OR. liwork.EQ.-1 )
484*
485 IF( ijob.LT.0 .OR. ijob.GT.5 ) THEN
486 info = -1
487 ELSE IF( n.LT.0 ) THEN
488 info = -5
489 ELSE IF( lda.LT.max( 1, n ) ) THEN
490 info = -7
491 ELSE IF( ldb.LT.max( 1, n ) ) THEN
492 info = -9
493 ELSE IF( ldq.LT.1 .OR. ( wantq .AND. ldq.LT.n ) ) THEN
494 info = -13
495 ELSE IF( ldz.LT.1 .OR. ( wantz .AND. ldz.LT.n ) ) THEN
496 info = -15
497 END IF
498*
499 IF( info.NE.0 ) THEN
500 CALL xerbla( 'CTGSEN', -info )
501 RETURN
502 END IF
503*
504 ierr = 0
505*
506 wantp = ijob.EQ.1 .OR. ijob.GE.4
507 wantd1 = ijob.EQ.2 .OR. ijob.EQ.4
508 wantd2 = ijob.EQ.3 .OR. ijob.EQ.5
509 wantd = wantd1 .OR. wantd2
510*
511* Set M to the dimension of the specified pair of deflating
512* subspaces.
513*
514 m = 0
515 IF( .NOT.lquery .OR. ijob.NE.0 ) THEN
516 DO 10 k = 1, n
517 alpha( k ) = a( k, k )
518 beta( k ) = b( k, k )
519 IF( k.LT.n ) THEN
520 IF( SELECT( k ) )
521 \$ m = m + 1
522 ELSE
523 IF( SELECT( n ) )
524 \$ m = m + 1
525 END IF
526 10 CONTINUE
527 END IF
528*
529 IF( ijob.EQ.1 .OR. ijob.EQ.2 .OR. ijob.EQ.4 ) THEN
530 lwmin = max( 1, 2*m*(n-m) )
531 liwmin = max( 1, n+2 )
532 ELSE IF( ijob.EQ.3 .OR. ijob.EQ.5 ) THEN
533 lwmin = max( 1, 4*m*(n-m) )
534 liwmin = max( 1, 2*m*(n-m), n+2 )
535 ELSE
536 lwmin = 1
537 liwmin = 1
538 END IF
539*
540 work( 1 ) = lwmin
541 iwork( 1 ) = liwmin
542*
543 IF( lwork.LT.lwmin .AND. .NOT.lquery ) THEN
544 info = -21
545 ELSE IF( liwork.LT.liwmin .AND. .NOT.lquery ) THEN
546 info = -23
547 END IF
548*
549 IF( info.NE.0 ) THEN
550 CALL xerbla( 'CTGSEN', -info )
551 RETURN
552 ELSE IF( lquery ) THEN
553 RETURN
554 END IF
555*
556* Quick return if possible.
557*
558 IF( m.EQ.n .OR. m.EQ.0 ) THEN
559 IF( wantp ) THEN
560 pl = one
561 pr = one
562 END IF
563 IF( wantd ) THEN
564 dscale = zero
565 dsum = one
566 DO 20 i = 1, n
567 CALL classq( n, a( 1, i ), 1, dscale, dsum )
568 CALL classq( n, b( 1, i ), 1, dscale, dsum )
569 20 CONTINUE
570 dif( 1 ) = dscale*sqrt( dsum )
571 dif( 2 ) = dif( 1 )
572 END IF
573 GO TO 70
574 END IF
575*
576* Get machine constant
577*
578 safmin = slamch( 'S' )
579*
580* Collect the selected blocks at the top-left corner of (A, B).
581*
582 ks = 0
583 DO 30 k = 1, n
584 swap = SELECT( k )
585 IF( swap ) THEN
586 ks = ks + 1
587*
588* Swap the K-th block to position KS. Compute unitary Q
589* and Z that will swap adjacent diagonal blocks in (A, B).
590*
591 IF( k.NE.ks )
592 \$ CALL ctgexc( wantq, wantz, n, a, lda, b, ldb, q, ldq, z,
593 \$ ldz, k, ks, ierr )
594*
595 IF( ierr.GT.0 ) THEN
596*
597* Swap is rejected: exit.
598*
599 info = 1
600 IF( wantp ) THEN
601 pl = zero
602 pr = zero
603 END IF
604 IF( wantd ) THEN
605 dif( 1 ) = zero
606 dif( 2 ) = zero
607 END IF
608 GO TO 70
609 END IF
610 END IF
611 30 CONTINUE
612 IF( wantp ) THEN
613*
614* Solve generalized Sylvester equation for R and L:
615* A11 * R - L * A22 = A12
616* B11 * R - L * B22 = B12
617*
618 n1 = m
619 n2 = n - m
620 i = n1 + 1
621 CALL clacpy( 'Full', n1, n2, a( 1, i ), lda, work, n1 )
622 CALL clacpy( 'Full', n1, n2, b( 1, i ), ldb, work( n1*n2+1 ),
623 \$ n1 )
624 ijb = 0
625 CALL ctgsyl( 'N', ijb, n1, n2, a, lda, a( i, i ), lda, work,
626 \$ n1, b, ldb, b( i, i ), ldb, work( n1*n2+1 ), n1,
627 \$ dscale, dif( 1 ), work( n1*n2*2+1 ),
628 \$ lwork-2*n1*n2, iwork, ierr )
629*
630* Estimate the reciprocal of norms of "projections" onto
631* left and right eigenspaces
632*
633 rdscal = zero
634 dsum = one
635 CALL classq( n1*n2, work, 1, rdscal, dsum )
636 pl = rdscal*sqrt( dsum )
637 IF( pl.EQ.zero ) THEN
638 pl = one
639 ELSE
640 pl = dscale / ( sqrt( dscale*dscale / pl+pl )*sqrt( pl ) )
641 END IF
642 rdscal = zero
643 dsum = one
644 CALL classq( n1*n2, work( n1*n2+1 ), 1, rdscal, dsum )
645 pr = rdscal*sqrt( dsum )
646 IF( pr.EQ.zero ) THEN
647 pr = one
648 ELSE
649 pr = dscale / ( sqrt( dscale*dscale / pr+pr )*sqrt( pr ) )
650 END IF
651 END IF
652 IF( wantd ) THEN
653*
654* Compute estimates Difu and Difl.
655*
656 IF( wantd1 ) THEN
657 n1 = m
658 n2 = n - m
659 i = n1 + 1
660 ijb = idifjb
661*
662* Frobenius norm-based Difu estimate.
663*
664 CALL ctgsyl( 'N', ijb, n1, n2, a, lda, a( i, i ), lda, work,
665 \$ n1, b, ldb, b( i, i ), ldb, work( n1*n2+1 ),
666 \$ n1, dscale, dif( 1 ), work( n1*n2*2+1 ),
667 \$ lwork-2*n1*n2, iwork, ierr )
668*
669* Frobenius norm-based Difl estimate.
670*
671 CALL ctgsyl( 'N', ijb, n2, n1, a( i, i ), lda, a, lda, work,
672 \$ n2, b( i, i ), ldb, b, ldb, work( n1*n2+1 ),
673 \$ n2, dscale, dif( 2 ), work( n1*n2*2+1 ),
674 \$ lwork-2*n1*n2, iwork, ierr )
675 ELSE
676*
677* Compute 1-norm-based estimates of Difu and Difl using
678* reversed communication with CLACN2. In each step a
679* generalized Sylvester equation or a transposed variant
680* is solved.
681*
682 kase = 0
683 n1 = m
684 n2 = n - m
685 i = n1 + 1
686 ijb = 0
687 mn2 = 2*n1*n2
688*
689* 1-norm-based estimate of Difu.
690*
691 40 CONTINUE
692 CALL clacn2( mn2, work( mn2+1 ), work, dif( 1 ), kase,
693 \$ isave )
694 IF( kase.NE.0 ) THEN
695 IF( kase.EQ.1 ) THEN
696*
697* Solve generalized Sylvester equation
698*
699 CALL ctgsyl( 'N', ijb, n1, n2, a, lda, a( i, i ), lda,
700 \$ work, n1, b, ldb, b( i, i ), ldb,
701 \$ work( n1*n2+1 ), n1, dscale, dif( 1 ),
702 \$ work( n1*n2*2+1 ), lwork-2*n1*n2, iwork,
703 \$ ierr )
704 ELSE
705*
706* Solve the transposed variant.
707*
708 CALL ctgsyl( 'C', ijb, n1, n2, a, lda, a( i, i ), lda,
709 \$ work, n1, b, ldb, b( i, i ), ldb,
710 \$ work( n1*n2+1 ), n1, dscale, dif( 1 ),
711 \$ work( n1*n2*2+1 ), lwork-2*n1*n2, iwork,
712 \$ ierr )
713 END IF
714 GO TO 40
715 END IF
716 dif( 1 ) = dscale / dif( 1 )
717*
718* 1-norm-based estimate of Difl.
719*
720 50 CONTINUE
721 CALL clacn2( mn2, work( mn2+1 ), work, dif( 2 ), kase,
722 \$ isave )
723 IF( kase.NE.0 ) THEN
724 IF( kase.EQ.1 ) THEN
725*
726* Solve generalized Sylvester equation
727*
728 CALL ctgsyl( 'N', ijb, n2, n1, a( i, i ), lda, a, lda,
729 \$ work, n2, b( i, i ), ldb, b, ldb,
730 \$ work( n1*n2+1 ), n2, dscale, dif( 2 ),
731 \$ work( n1*n2*2+1 ), lwork-2*n1*n2, iwork,
732 \$ ierr )
733 ELSE
734*
735* Solve the transposed variant.
736*
737 CALL ctgsyl( 'C', ijb, n2, n1, a( i, i ), lda, a, lda,
738 \$ work, n2, b, ldb, b( i, i ), ldb,
739 \$ work( n1*n2+1 ), n2, dscale, dif( 2 ),
740 \$ work( n1*n2*2+1 ), lwork-2*n1*n2, iwork,
741 \$ ierr )
742 END IF
743 GO TO 50
744 END IF
745 dif( 2 ) = dscale / dif( 2 )
746 END IF
747 END IF
748*
749* If B(K,K) is complex, make it real and positive (normalization
750* of the generalized Schur form) and Store the generalized
751* eigenvalues of reordered pair (A, B)
752*
753 DO 60 k = 1, n
754 dscale = abs( b( k, k ) )
755 IF( dscale.GT.safmin ) THEN
756 temp1 = conjg( b( k, k ) / dscale )
757 temp2 = b( k, k ) / dscale
758 b( k, k ) = dscale
759 CALL cscal( n-k, temp1, b( k, k+1 ), ldb )
760 CALL cscal( n-k+1, temp1, a( k, k ), lda )
761 IF( wantq )
762 \$ CALL cscal( n, temp2, q( 1, k ), 1 )
763 ELSE
764 b( k, k ) = cmplx( zero, zero )
765 END IF
766*
767 alpha( k ) = a( k, k )
768 beta( k ) = b( k, k )
769*
770 60 CONTINUE
771*
772 70 CONTINUE
773*
774 work( 1 ) = lwmin
775 iwork( 1 ) = liwmin
776*
777 RETURN
778*
779* End of CTGSEN
780*
subroutine classq(n, x, incx, scl, sumsq)
CLASSQ updates a sum of squares represented in scaled form.
Definition: classq.f90:137
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
subroutine cscal(N, CA, CX, INCX)
CSCAL
Definition: cscal.f:78
subroutine ctgexc(WANTQ, WANTZ, N, A, LDA, B, LDB, Q, LDQ, Z, LDZ, IFST, ILST, INFO)
CTGEXC
Definition: ctgexc.f:200
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:133
subroutine clacpy(UPLO, M, N, A, LDA, B, LDB)
CLACPY copies all or part of one two-dimensional array to another.
Definition: clacpy.f:103
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:295
real function slamch(CMACH)
SLAMCH
Definition: slamch.f:68
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