LAPACK  3.10.1 LAPACK: Linear Algebra PACKage

## ◆ stgsen()

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

STGSEN

Purpose:
``` STGSEN reorders the generalized real Schur decomposition of a real
matrix pair (A, B) (in terms of an orthonormal equivalence trans-
formation Q**T * (A, B) * Z), so that a selected cluster of eigenvalues
appears in the leading diagonal blocks of the upper quasi-triangular
matrix A and the upper triangular B. The leading columns of Q and
Z form orthonormal bases of the corresponding left and right eigen-
spaces (deflating subspaces). (A, B) must be in generalized real
Schur canonical form (as returned by SGGES), i.e. A is block upper
triangular with 1-by-1 and 2-by-2 diagonal blocks. B is upper
triangular.

STGSEN also computes the generalized eigenvalues

w(j) = (ALPHAR(j) + i*ALPHAI(j))/BETA(j)

of the reordered matrix pair (A, B).

Optionally, STGSEN computes the 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 a real eigenvalue w(j), SELECT(j) must be set to .TRUE.. To select a complex conjugate pair of eigenvalues w(j) and w(j+1), corresponding to a 2-by-2 diagonal block, either SELECT(j) or SELECT(j+1) or both must be set to .TRUE.; a complex conjugate pair of eigenvalues must be either both included in the cluster or both excluded.``` [in] N ``` N is INTEGER The order of the matrices A and B. N >= 0.``` [in,out] A ``` A is REAL array, dimension(LDA,N) On entry, the upper quasi-triangular matrix A, with (A, B) in generalized real 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 REAL array, dimension(LDB,N) On entry, the upper triangular matrix B, with (A, B) in generalized real 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] ALPHAR ` ALPHAR is REAL array, dimension (N)` [out] ALPHAI ` ALPHAI is REAL array, dimension (N)` [out] BETA ``` BETA is REAL 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 generalized Schur form of (A,B) were further reduced to triangular form using 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.``` [in,out] Q ``` Q is REAL 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 orthogonal 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; and if WANTQ = .TRUE., LDQ >= N.``` [in,out] Z ``` Z is REAL 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 orthogonal 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 eigen- spaces (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 eigenspaces 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 and 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. 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 REAL 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 >= 4*N+16. If IJOB = 1, 2 or 4, LWORK >= MAX(4*N+16, 2*M*(N-M)). If IJOB = 3 or 5, LWORK >= MAX(4*N+16, 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+6. If IJOB = 3 or 5, LIWORK >= MAX(2*M*(N-M), N+6). 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:
```  STGSEN first collects the selected eigenvalues by computing
orthogonal 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**T*(A, B)*W = (A11 A12) (B11 B12) n1
( 0  A22),( 0  B22) n2
n1  n2    n1  n2

where N = n1+n2 and U**T means the 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**T, then the
reordered generalized real Schur form of (C, D) is given by

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

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**T, In1) ]
[ kron(In2, B11)  -kron(B22**T, In1) ].

Here, Inx is the identity matrix of size nx and A22**T is the
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 SLATDF), then the parameter
IDIFJB (see below) should be changed from 3 to 4 (routine SLATDF
(IJOB = 2 will be used)). See STGSYL 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 448 of file stgsen.f.

451 *
452 * -- LAPACK computational routine --
453 * -- LAPACK is a software package provided by Univ. of Tennessee, --
454 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
455 *
456 * .. Scalar Arguments ..
457  LOGICAL WANTQ, WANTZ
458  INTEGER IJOB, INFO, LDA, LDB, LDQ, LDZ, LIWORK, LWORK,
459  \$ M, N
460  REAL PL, PR
461 * ..
462 * .. Array Arguments ..
463  LOGICAL SELECT( * )
464  INTEGER IWORK( * )
465  REAL A( LDA, * ), ALPHAI( * ), ALPHAR( * ),
466  \$ B( LDB, * ), BETA( * ), DIF( * ), Q( LDQ, * ),
467  \$ WORK( * ), Z( LDZ, * )
468 * ..
469 *
470 * =====================================================================
471 *
472 * .. Parameters ..
473  INTEGER IDIFJB
474  parameter( idifjb = 3 )
475  REAL ZERO, ONE
476  parameter( zero = 0.0e+0, one = 1.0e+0 )
477 * ..
478 * .. Local Scalars ..
479  LOGICAL LQUERY, PAIR, SWAP, WANTD, WANTD1, WANTD2,
480  \$ WANTP
481  INTEGER I, IERR, IJB, K, KASE, KK, KS, LIWMIN, LWMIN,
482  \$ MN2, N1, N2
483  REAL DSCALE, DSUM, EPS, RDSCAL, SMLNUM
484 * ..
485 * .. Local Arrays ..
486  INTEGER ISAVE( 3 )
487 * ..
488 * .. External Subroutines ..
489  EXTERNAL slacn2, slacpy, slag2, slassq, stgexc, stgsyl,
490  \$ xerbla
491 * ..
492 * .. External Functions ..
493  REAL SLAMCH
494  EXTERNAL slamch
495 * ..
496 * .. Intrinsic Functions ..
497  INTRINSIC max, sign, sqrt
498 * ..
499 * .. Executable Statements ..
500 *
501 * Decode and test the input parameters
502 *
503  info = 0
504  lquery = ( lwork.EQ.-1 .OR. liwork.EQ.-1 )
505 *
506  IF( ijob.LT.0 .OR. ijob.GT.5 ) THEN
507  info = -1
508  ELSE IF( n.LT.0 ) THEN
509  info = -5
510  ELSE IF( lda.LT.max( 1, n ) ) THEN
511  info = -7
512  ELSE IF( ldb.LT.max( 1, n ) ) THEN
513  info = -9
514  ELSE IF( ldq.LT.1 .OR. ( wantq .AND. ldq.LT.n ) ) THEN
515  info = -14
516  ELSE IF( ldz.LT.1 .OR. ( wantz .AND. ldz.LT.n ) ) THEN
517  info = -16
518  END IF
519 *
520  IF( info.NE.0 ) THEN
521  CALL xerbla( 'STGSEN', -info )
522  RETURN
523  END IF
524 *
525 * Get machine constants
526 *
527  eps = slamch( 'P' )
528  smlnum = slamch( 'S' ) / eps
529  ierr = 0
530 *
531  wantp = ijob.EQ.1 .OR. ijob.GE.4
532  wantd1 = ijob.EQ.2 .OR. ijob.EQ.4
533  wantd2 = ijob.EQ.3 .OR. ijob.EQ.5
534  wantd = wantd1 .OR. wantd2
535 *
536 * Set M to the dimension of the specified pair of deflating
537 * subspaces.
538 *
539  m = 0
540  pair = .false.
541  IF( .NOT.lquery .OR. ijob.NE.0 ) THEN
542  DO 10 k = 1, n
543  IF( pair ) THEN
544  pair = .false.
545  ELSE
546  IF( k.LT.n ) THEN
547  IF( a( k+1, k ).EQ.zero ) THEN
548  IF( SELECT( k ) )
549  \$ m = m + 1
550  ELSE
551  pair = .true.
552  IF( SELECT( k ) .OR. SELECT( k+1 ) )
553  \$ m = m + 2
554  END IF
555  ELSE
556  IF( SELECT( n ) )
557  \$ m = m + 1
558  END IF
559  END IF
560  10 CONTINUE
561  END IF
562 *
563  IF( ijob.EQ.1 .OR. ijob.EQ.2 .OR. ijob.EQ.4 ) THEN
564  lwmin = max( 1, 4*n+16, 2*m*(n-m) )
565  liwmin = max( 1, n+6 )
566  ELSE IF( ijob.EQ.3 .OR. ijob.EQ.5 ) THEN
567  lwmin = max( 1, 4*n+16, 4*m*(n-m) )
568  liwmin = max( 1, 2*m*(n-m), n+6 )
569  ELSE
570  lwmin = max( 1, 4*n+16 )
571  liwmin = 1
572  END IF
573 *
574  work( 1 ) = lwmin
575  iwork( 1 ) = liwmin
576 *
577  IF( lwork.LT.lwmin .AND. .NOT.lquery ) THEN
578  info = -22
579  ELSE IF( liwork.LT.liwmin .AND. .NOT.lquery ) THEN
580  info = -24
581  END IF
582 *
583  IF( info.NE.0 ) THEN
584  CALL xerbla( 'STGSEN', -info )
585  RETURN
586  ELSE IF( lquery ) THEN
587  RETURN
588  END IF
589 *
590 * Quick return if possible.
591 *
592  IF( m.EQ.n .OR. m.EQ.0 ) THEN
593  IF( wantp ) THEN
594  pl = one
595  pr = one
596  END IF
597  IF( wantd ) THEN
598  dscale = zero
599  dsum = one
600  DO 20 i = 1, n
601  CALL slassq( n, a( 1, i ), 1, dscale, dsum )
602  CALL slassq( n, b( 1, i ), 1, dscale, dsum )
603  20 CONTINUE
604  dif( 1 ) = dscale*sqrt( dsum )
605  dif( 2 ) = dif( 1 )
606  END IF
607  GO TO 60
608  END IF
609 *
610 * Collect the selected blocks at the top-left corner of (A, B).
611 *
612  ks = 0
613  pair = .false.
614  DO 30 k = 1, n
615  IF( pair ) THEN
616  pair = .false.
617  ELSE
618 *
619  swap = SELECT( k )
620  IF( k.LT.n ) THEN
621  IF( a( k+1, k ).NE.zero ) THEN
622  pair = .true.
623  swap = swap .OR. SELECT( k+1 )
624  END IF
625  END IF
626 *
627  IF( swap ) THEN
628  ks = ks + 1
629 *
630 * Swap the K-th block to position KS.
631 * Perform the reordering of diagonal blocks in (A, B)
632 * by orthogonal transformation matrices and update
633 * Q and Z accordingly (if requested):
634 *
635  kk = k
636  IF( k.NE.ks )
637  \$ CALL stgexc( wantq, wantz, n, a, lda, b, ldb, q, ldq,
638  \$ z, ldz, kk, ks, work, lwork, ierr )
639 *
640  IF( ierr.GT.0 ) THEN
641 *
642 * Swap is rejected: exit.
643 *
644  info = 1
645  IF( wantp ) THEN
646  pl = zero
647  pr = zero
648  END IF
649  IF( wantd ) THEN
650  dif( 1 ) = zero
651  dif( 2 ) = zero
652  END IF
653  GO TO 60
654  END IF
655 *
656  IF( pair )
657  \$ ks = ks + 1
658  END IF
659  END IF
660  30 CONTINUE
661  IF( wantp ) THEN
662 *
663 * Solve generalized Sylvester equation for R and L
664 * and compute PL and PR.
665 *
666  n1 = m
667  n2 = n - m
668  i = n1 + 1
669  ijb = 0
670  CALL slacpy( 'Full', n1, n2, a( 1, i ), lda, work, n1 )
671  CALL slacpy( 'Full', n1, n2, b( 1, i ), ldb, work( n1*n2+1 ),
672  \$ n1 )
673  CALL stgsyl( 'N', ijb, n1, n2, a, lda, a( i, i ), lda, work,
674  \$ n1, b, ldb, b( i, i ), ldb, work( n1*n2+1 ), n1,
675  \$ dscale, dif( 1 ), work( n1*n2*2+1 ),
676  \$ lwork-2*n1*n2, iwork, ierr )
677 *
678 * Estimate the reciprocal of norms of "projections" onto left
679 * and right eigenspaces.
680 *
681  rdscal = zero
682  dsum = one
683  CALL slassq( n1*n2, work, 1, rdscal, dsum )
684  pl = rdscal*sqrt( dsum )
685  IF( pl.EQ.zero ) THEN
686  pl = one
687  ELSE
688  pl = dscale / ( sqrt( dscale*dscale / pl+pl )*sqrt( pl ) )
689  END IF
690  rdscal = zero
691  dsum = one
692  CALL slassq( n1*n2, work( n1*n2+1 ), 1, rdscal, dsum )
693  pr = rdscal*sqrt( dsum )
694  IF( pr.EQ.zero ) THEN
695  pr = one
696  ELSE
697  pr = dscale / ( sqrt( dscale*dscale / pr+pr )*sqrt( pr ) )
698  END IF
699  END IF
700 *
701  IF( wantd ) THEN
702 *
703 * Compute estimates of Difu and Difl.
704 *
705  IF( wantd1 ) THEN
706  n1 = m
707  n2 = n - m
708  i = n1 + 1
709  ijb = idifjb
710 *
711 * Frobenius norm-based Difu-estimate.
712 *
713  CALL stgsyl( 'N', ijb, n1, n2, a, lda, a( i, i ), lda, work,
714  \$ n1, b, ldb, b( i, i ), ldb, work( n1*n2+1 ),
715  \$ n1, dscale, dif( 1 ), work( 2*n1*n2+1 ),
716  \$ lwork-2*n1*n2, iwork, ierr )
717 *
718 * Frobenius norm-based Difl-estimate.
719 *
720  CALL stgsyl( 'N', ijb, n2, n1, a( i, i ), lda, a, lda, work,
721  \$ n2, b( i, i ), ldb, b, ldb, work( n1*n2+1 ),
722  \$ n2, dscale, dif( 2 ), work( 2*n1*n2+1 ),
723  \$ lwork-2*n1*n2, iwork, ierr )
724  ELSE
725 *
726 *
727 * Compute 1-norm-based estimates of Difu and Difl using
728 * reversed communication with SLACN2. In each step a
729 * generalized Sylvester equation or a transposed variant
730 * is solved.
731 *
732  kase = 0
733  n1 = m
734  n2 = n - m
735  i = n1 + 1
736  ijb = 0
737  mn2 = 2*n1*n2
738 *
739 * 1-norm-based estimate of Difu.
740 *
741  40 CONTINUE
742  CALL slacn2( mn2, work( mn2+1 ), work, iwork, dif( 1 ),
743  \$ kase, isave )
744  IF( kase.NE.0 ) THEN
745  IF( kase.EQ.1 ) THEN
746 *
747 * Solve generalized Sylvester equation.
748 *
749  CALL stgsyl( 'N', ijb, n1, n2, a, lda, a( i, i ), lda,
750  \$ work, n1, b, ldb, b( i, i ), ldb,
751  \$ work( n1*n2+1 ), n1, dscale, dif( 1 ),
752  \$ work( 2*n1*n2+1 ), lwork-2*n1*n2, iwork,
753  \$ ierr )
754  ELSE
755 *
756 * Solve the transposed variant.
757 *
758  CALL stgsyl( 'T', ijb, n1, n2, a, lda, a( i, i ), lda,
759  \$ work, n1, b, ldb, b( i, i ), ldb,
760  \$ work( n1*n2+1 ), n1, dscale, dif( 1 ),
761  \$ work( 2*n1*n2+1 ), lwork-2*n1*n2, iwork,
762  \$ ierr )
763  END IF
764  GO TO 40
765  END IF
766  dif( 1 ) = dscale / dif( 1 )
767 *
768 * 1-norm-based estimate of Difl.
769 *
770  50 CONTINUE
771  CALL slacn2( mn2, work( mn2+1 ), work, iwork, dif( 2 ),
772  \$ kase, isave )
773  IF( kase.NE.0 ) THEN
774  IF( kase.EQ.1 ) THEN
775 *
776 * Solve generalized Sylvester equation.
777 *
778  CALL stgsyl( 'N', ijb, n2, n1, a( i, i ), lda, a, lda,
779  \$ work, n2, b( i, i ), ldb, b, ldb,
780  \$ work( n1*n2+1 ), n2, dscale, dif( 2 ),
781  \$ work( 2*n1*n2+1 ), lwork-2*n1*n2, iwork,
782  \$ ierr )
783  ELSE
784 *
785 * Solve the transposed variant.
786 *
787  CALL stgsyl( 'T', ijb, n2, n1, a( i, i ), lda, a, lda,
788  \$ work, n2, b( i, i ), ldb, b, ldb,
789  \$ work( n1*n2+1 ), n2, dscale, dif( 2 ),
790  \$ work( 2*n1*n2+1 ), lwork-2*n1*n2, iwork,
791  \$ ierr )
792  END IF
793  GO TO 50
794  END IF
795  dif( 2 ) = dscale / dif( 2 )
796 *
797  END IF
798  END IF
799 *
800  60 CONTINUE
801 *
802 * Compute generalized eigenvalues of reordered pair (A, B) and
803 * normalize the generalized Schur form.
804 *
805  pair = .false.
806  DO 70 k = 1, n
807  IF( pair ) THEN
808  pair = .false.
809  ELSE
810 *
811  IF( k.LT.n ) THEN
812  IF( a( k+1, k ).NE.zero ) THEN
813  pair = .true.
814  END IF
815  END IF
816 *
817  IF( pair ) THEN
818 *
819 * Compute the eigenvalue(s) at position K.
820 *
821  work( 1 ) = a( k, k )
822  work( 2 ) = a( k+1, k )
823  work( 3 ) = a( k, k+1 )
824  work( 4 ) = a( k+1, k+1 )
825  work( 5 ) = b( k, k )
826  work( 6 ) = b( k+1, k )
827  work( 7 ) = b( k, k+1 )
828  work( 8 ) = b( k+1, k+1 )
829  CALL slag2( work, 2, work( 5 ), 2, smlnum*eps, beta( k ),
830  \$ beta( k+1 ), alphar( k ), alphar( k+1 ),
831  \$ alphai( k ) )
832  alphai( k+1 ) = -alphai( k )
833 *
834  ELSE
835 *
836  IF( sign( one, b( k, k ) ).LT.zero ) THEN
837 *
838 * If B(K,K) is negative, make it positive
839 *
840  DO 80 i = 1, n
841  a( k, i ) = -a( k, i )
842  b( k, i ) = -b( k, i )
843  IF( wantq ) q( i, k ) = -q( i, k )
844  80 CONTINUE
845  END IF
846 *
847  alphar( k ) = a( k, k )
848  alphai( k ) = zero
849  beta( k ) = b( k, k )
850 *
851  END IF
852  END IF
853  70 CONTINUE
854 *
855  work( 1 ) = lwmin
856  iwork( 1 ) = liwmin
857 *
858  RETURN
859 *
860 * End of STGSEN
861 *
subroutine slassq(n, x, incx, scl, sumsq)
SLASSQ updates a sum of squares represented in scaled form.
Definition: slassq.f90:137
subroutine slacpy(UPLO, M, N, A, LDA, B, LDB)
SLACPY copies all or part of one two-dimensional array to another.
Definition: slacpy.f:103
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
subroutine stgexc(WANTQ, WANTZ, N, A, LDA, B, LDB, Q, LDQ, Z, LDZ, IFST, ILST, WORK, LWORK, INFO)
STGEXC
Definition: stgexc.f:220
subroutine slacn2(N, V, X, ISGN, EST, KASE, ISAVE)
SLACN2 estimates the 1-norm of a square matrix, using reverse communication for evaluating matrix-vec...
Definition: slacn2.f:136
subroutine slag2(A, LDA, B, LDB, SAFMIN, SCALE1, SCALE2, WR1, WR2, WI)
SLAG2 computes the eigenvalues of a 2-by-2 generalized eigenvalue problem, with scaling as necessary ...
Definition: slag2.f:156
subroutine stgsyl(TRANS, IJOB, M, N, A, LDA, B, LDB, C, LDC, D, LDD, E, LDE, F, LDF, SCALE, DIF, WORK, LWORK, IWORK, INFO)
STGSYL
Definition: stgsyl.f:299
real function slamch(CMACH)
SLAMCH
Definition: slamch.f:68
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