Purpose
To solve the real Lyapunov matrix equation
op(A)'*X + X*op(A) = scale*C
where op(A) = A or A' (A**T), A is upper quasi-triangular and C is
symmetric (C = C'). (A' denotes the transpose of the matrix A.)
A is N-by-N, the right hand side C and the solution X are N-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 must be in Schur canonical form (as returned by LAPACK routines
DGEES or 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 SB03MY( TRANA, N, A, LDA, C, LDC, SCALE, INFO )
C .. Scalar Arguments ..
CHARACTER TRANA
INTEGER INFO, LDA, LDC, N
DOUBLE PRECISION SCALE
C .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), C( LDC, * )
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).
Input/Output Parameters
N (input) INTEGER
The order of the matrices A, X, and C. N >= 0.
A (input) DOUBLE PRECISION array, dimension (LDA,N)
The leading N-by-N 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,N).
C (input/output) DOUBLE PRECISION array, dimension (LDC,N)
On entry, the leading N-by-N part of this array must
contain the symmetric matrix C.
On exit, if INFO >= 0, the leading N-by-N part of this
array contains the symmetric solution matrix X.
LDC INTEGER
The leading dimension of array C. LDC >= MAX(1,N).
SCALE (output) DOUBLE PRECISION
The scale factor, scale, set less than or equal to 1 to
prevent the solution overflowing.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= 1: if A and -A have common or very close eigenvalues;
perturbed values were used to solve the equation
(but the matrix A is unchanged).
Method
Bartels-Stewart algorithm is used. 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.
Numerical Aspects
3 The algorithm requires 0(N ) operations.Further Comments
NoneExample
Program Text
NoneProgram Data
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