DTGEXC - reorder the generalized real Schur decomposition of a real matrix pair (A,B) using an orthogonal equivalence transformation (A, B) = Q * (A, B) * Z',

SUBROUTINE DTGEXC(

WANTQ, WANTZ, N, A, LDA, B, LDB, Q, LDQ, Z, LDZ, IFST, ILST, WORK, LWORK, INFO )

LOGICAL WANTQ, WANTZ

INTEGER IFST, ILST, INFO, LDA, LDB, LDQ, LDZ, LWORK, N

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

DTGEXC reorders the generalized real Schur decomposition of a real matrix pair (A,B) using an orthogonal equivalence transformation (A, B) = Q * (A, B) * Z', so that the diagonal block of (A, B) with row index IFST is moved to row ILST.

(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 array, dimension (LDA,N)

On entry, the matrix A in generalized real Schur canonical form. On exit, the updated matrix A, again in generalized real Schur canonical form.

LDA (input) INTEGER

The leading dimension of the array A. LDA >= max(1,N).

B (input/output) DOUBLE PRECISION array, dimension (LDB,N)

On entry, the matrix B in generalized real Schur canonical form (A,B). On exit, the updated matrix B, again in generalized real Schur canonical form (A,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. If WANTQ = .FALSE., Q is not referenced.

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. If WANTZ = .FALSE., Z is not referenced.

LDZ (input) INTEGER

The leading dimension of the array Z. LDZ >= 1. If WANTZ = .TRUE., LDZ >= N.

IFST (input/output) INTEGER

ILST (input/output) INTEGER Specify the reordering of the diagonal blocks of (A, B). The block with row index IFST is moved to row ILST, by a sequence of swapping between adjacent blocks. On exit, if IFST pointed on entry to the second row of a 2-by-2 block, it is changed to point to the first row; ILST always points to the first row of the block in its final position (which may differ from its input value by +1 or -1). 1 <= IFST, ILST <= N.

WORK (workspace/output) DOUBLE PRECISION array, dimension (LWORK)

On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

LWORK (input) INTEGER

The dimension of the array WORK. LWORK >= 4*N + 16.

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.

INFO (output) INTEGER

=0: successful exit.

<0: if INFO = -i, the i-th argument had an illegal value.

=1: The transformed matrix pair (A, B) would be too far from generalized Schur form; the problem is ill- conditioned. (A, B) may have been partially reordered, and ILST points to the first row of the current position of the block being moved.

Based on contributions by

Bo Kagstrom and Peter Poromaa, Department of Computing Science, Umea University, S-901 87 Umea, Sweden.

[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.