DTGEX2 - swap adjacent diagonal blocks (A11, B11) and (A22, B22) of size 1-by-1 or 2-by-2 in an upper (quasi) triangular matrix pair (A, B) by an orthogonal equivalence transformation

SUBROUTINE DTGEX2(

WANTQ, WANTZ, N, A, LDA, B, LDB, Q, LDQ, Z, LDZ, J1, N1, N2, WORK, LWORK, INFO )

LOGICAL WANTQ, WANTZ

INTEGER INFO, J1, LDA, LDB, LDQ, LDZ, LWORK, N, N1, N2

DOUBLE PRECISION A( LDA, * ), B( LDB, * ), Q( LDQ, * ), WORK( * ), Z( LDZ, * )

DTGEX2 swaps adjacent diagonal blocks (A11, B11) and (A22, B22) of size 1-by-1 or 2-by-2 in an upper (quasi) triangular matrix pair (A, B) by an orthogonal equivalence transformation. (A, B) must be in generalized real Schur canonical form (as returned by DGGES), i.e. A is block upper triangular with 1-by-1 and 2-by-2 diagonal blocks. B is 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)'

WANTQ (input) LOGICAL

WANTZ (input) LOGICAL

N (input) INTEGER

The order of the matrices A and B. N >= 0.

A (input/output) DOUBLE PRECISION 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) DOUBLE PRECISION 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) DOUBLE PRECISION array, dimension (LDZ,N)

On entry, if WANTQ = .TRUE., the orthogonal 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) DOUBLE PRECISION array, dimension (LDZ,N)

On entry, if WANTZ =.TRUE., the orthogonal 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). 1 <= J1 <= N.

N1 (input) INTEGER

The order of the first block (A11, B11). N1 = 0, 1 or 2.

N2 (input) INTEGER

The order of the second block (A22, B22). N2 = 0, 1 or 2.

WORK (workspace) DOUBLE PRECISION array, dimension (LWORK).

LWORK (input) INTEGER

The dimension of the array WORK. LWORK >= MAX( N*(N2+N1), (N2+N1)*(N2+N1)*2 )

INFO (output) INTEGER

=0: Successful exit

>0: If INFO = 1, the transformed matrix (A, B) would be too far from generalized Schur form; the blocks are not swapped and (A, B) and (Q, Z) are unchanged. The problem of swapping is too ill-conditioned. <0: If INFO = -16: LWORK is too small. Appropriate value for LWORK is returned in WORK(1).

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. [2] for details.

[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.