 CTGEX2(3) swaps adjacent diagonal 1 by 1 blocks (A11,B11) and (A22,B22)

## SYNOPSIS

SUBROUTINE CTGEX2(
WANTQ, WANTZ, N, A, LDA, B, LDB, Q, LDQ, Z, LDZ, J1, INFO )

LOGICAL WANTQ, WANTZ

INTEGER INFO, J1, LDA, LDB, LDQ, LDZ, N

COMPLEX A( LDA, * ), B( LDB, * ), Q( LDQ, * ), Z( LDZ, * )

## PURPOSE

CTGEX2 swaps adjacent diagonal 1 by 1 blocks (A11,B11) and (A22,B22) in an upper triangular matrix pair (A, B) by an unitary equivalence transformation.
(A, B) must be in generalized Schur canonical form, that is, A and B are both upper triangular.
Optionally, the matrices Q and Z of generalized Schur vectors are updated.

Q(in) * A(in) * Z(in)' = Q(out) * A(out) * Z(out)'
Q(in) * B(in) * Z(in)' = Q(out) * B(out) * Z(out)'

## ARGUMENTS

WANTQ (input) LOGICAL .TRUE. : update the left transformation matrix Q;

WANTZ (input) LOGICAL

N (input) INTEGER
The order of the matrices A and B. N >= 0.
A (input/output) COMPLEX arrays, dimensions (LDA,N)
On entry, the matrix A in the pair (A, B). On exit, the updated matrix A.
LDA (input) INTEGER
The leading dimension of the array A. LDA >= max(1,N).
B (input/output) COMPLEX arrays, dimensions (LDB,N)
On entry, the matrix B in the pair (A, B). On exit, the updated matrix B.
LDB (input) INTEGER
The leading dimension of the array B. LDB >= max(1,N).
Q (input/output) COMPLEX array, dimension (LDZ,N)
If WANTQ = .TRUE, on entry, the unitary matrix Q. On exit, the updated matrix Q. Not referenced if WANTQ = .FALSE..
LDQ (input) INTEGER
The leading dimension of the array Q. LDQ >= 1; If WANTQ = .TRUE., LDQ >= N.
Z (input/output) COMPLEX array, dimension (LDZ,N)
If WANTZ = .TRUE, on entry, the unitary matrix Z. On exit, the updated matrix Z. Not referenced if WANTZ = .FALSE..
LDZ (input) INTEGER
The leading dimension of the array Z. LDZ >= 1; If WANTZ = .TRUE., LDZ >= N.
J1 (input) INTEGER
The index to the first block (A11, B11).
INFO (output) INTEGER
=0: Successful exit.
=1: The transformed matrix pair (A, B) would be too far from generalized Schur form; the problem is ill- conditioned.

## FURTHER DETAILS

Based on contributions by

Bo Kagstrom and Peter Poromaa, Department of Computing Science,
Umea University, S-901 87 Umea, Sweden.
In the current code both weak and strong stability tests are performed. The user can omit the strong stability test by changing the internal logical parameter WANDS to .FALSE.. See ref.  for details.
 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.