Purpose
To solve for X the discrete-time Sylvester equation
op(A)*X*op(B) + ISGN*X = scale*C,
where op(A) = A or A**T, A and B are both upper quasi-triangular,
and ISGN = 1 or -1. A is M-by-M and B is N-by-N; the right hand
side C and the solution X are M-by-N; and scale is an output scale
factor, set less than or equal to 1 to avoid overflow in X. The
solution matrix X is overwritten onto C.
A and B must be in Schur canonical form (as returned by LAPACK
Library routine DHSEQR), that is, block upper triangular with
1-by-1 and 2-by-2 diagonal blocks; each 2-by-2 diagonal block has
its diagonal elements equal and its off-diagonal elements of
opposite sign.
Specification
SUBROUTINE SB04PY( TRANA, TRANB, ISGN, M, N, A, LDA, B, LDB, C,
$ LDC, SCALE, DWORK, INFO )
C .. Scalar Arguments ..
CHARACTER TRANA, TRANB
INTEGER INFO, ISGN, LDA, LDB, LDC, M, N
DOUBLE PRECISION SCALE
C .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), B( LDB, * ), C( LDC, * ),
$ DWORK( * )
Arguments
Mode Parameters
TRANA CHARACTER*1
Specifies the form of op(A) to be used, as follows:
= 'N': op(A) = A (No transpose);
= 'T': op(A) = A**T (Transpose);
= 'C': op(A) = A**T (Conjugate transpose = Transpose).
TRANB CHARACTER*1
Specifies the form of op(B) to be used, as follows:
= 'N': op(B) = B (No transpose);
= 'T': op(B) = B**T (Transpose);
= 'C': op(B) = B**T (Conjugate transpose = Transpose).
ISGN INTEGER
Specifies the sign of the equation as described before.
ISGN may only be 1 or -1.
Input/Output Parameters
M (input) INTEGER
The order of the matrix A, and the number of rows in the
matrices X and C. M >= 0.
N (input) INTEGER
The order of the matrix B, and the number of columns in
the matrices X and C. N >= 0.
A (input) DOUBLE PRECISION array, dimension (LDA,M)
The leading M-by-M part of this array must contain the
upper quasi-triangular matrix A, in Schur canonical form.
The part of A below the first sub-diagonal is not
referenced.
LDA INTEGER
The leading dimension of array A. LDA >= MAX(1,M).
B (input) DOUBLE PRECISION array, dimension (LDB,N)
The leading N-by-N part of this array must contain the
upper quasi-triangular matrix B, in Schur canonical form.
The part of B below the first sub-diagonal is not
referenced.
LDB (input) INTEGER
The leading dimension of the array B. LDB >= max(1,N).
C (input/output) DOUBLE PRECISION array, dimension (LDC,N)
On entry, the leading M-by-N part of this array must
contain the right hand side matrix C.
On exit, if INFO >= 0, the leading M-by-N part of this
array contains the solution matrix X.
LDC INTEGER
The leading dimension of array C. LDC >= MAX(1,M).
SCALE (output) DOUBLE PRECISION
The scale factor, scale, set less than or equal to 1 to
prevent the solution overflowing.
Workspace
DWORK DOUBLE PRECISION array, dimension (2*M)Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= 1: A and -ISGN*B have almost reciprocal eigenvalues;
perturbed values were used to solve the equation
(but the matrices A and B are unchanged).
Method
The solution matrix X is computed column-wise via a back substitution scheme, an extension and refinement of the algorithm in [1], similar to that used in [2] for continuous-time Sylvester equations. A set of equivalent linear algebraic systems of equations of order at most four are formed and solved using Gaussian elimination with complete pivoting.References
[1] Bartels, R.H. and Stewart, G.W. T
Solution of the matrix equation A X + XB = C.
Comm. A.C.M., 15, pp. 820-826, 1972.
[2] Anderson, E., Bai, Z., Bischof, C., Demmel, J., Dongarra, J.,
Du Croz, J., Greenbaum, A., Hammarling, S., McKenney, A.,
Ostrouchov, S., and Sorensen, D.
LAPACK Users' Guide: Second Edition.
SIAM, Philadelphia, 1995.
Numerical Aspects
The algorithm is stable and reliable, since Gaussian elimination with complete pivoting is used.Further Comments
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