## MB03RY

### Solution of a Sylvester equation -AX + XB = C, with A and B in real Schur form, aborting the computations when the norm of X is too large

[Specification] [Arguments] [Method] [References] [Comments] [Example]

Purpose

To solve the Sylvester equation -AX + XB = C, where A and B are
M-by-M and N-by-N matrices, respectively, in real Schur form.

This routine is intended to be called only by SLICOT Library
routine MB03RD. For efficiency purposes, the computations are
aborted when the infinity norm of an elementary submatrix of X is
greater than a given value PMAX.

Specification
SUBROUTINE MB03RY( M, N, PMAX, A, LDA, B, LDB, C, LDC, INFO )
C     .. Scalar Arguments ..
INTEGER           INFO, LDA, LDB, LDC, M, N
DOUBLE PRECISION  PMAX
C     .. Array Arguments ..
DOUBLE PRECISION  A(LDA,*), B(LDB,*), C(LDC,*)

Arguments

Input/Output Parameters

M       (input) INTEGER
The order of the matrix A and the number of rows of the
matrices C and X.  M >= 0.

N       (input) INTEGER
The order of the matrix B and the number of columns of the
matrices C and X.  N >= 0.

PMAX    (input) DOUBLE PRECISION
An upper bound for the infinity norm of an elementary
submatrix of X (see METHOD).

A       (input) DOUBLE PRECISION array, dimension (LDA,M)
The leading M-by-M part of this array must contain the
matrix A of the Sylvester equation, in real Schur form.
The elements below the real Schur form are 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
matrix B of the Sylvester equation, in real Schur form.
The elements below the real Schur form are not referenced.

LDB     INTEGER
The leading dimension of 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 matrix C of the Sylvester equation.
On exit, if INFO = 0, the leading M-by-N part of this
array contains the solution matrix X of the Sylvester
equation, and each elementary submatrix of X (see METHOD)
has the infinity norm less than or equal to PMAX.
On exit, if INFO = 1, the solution matrix X has not been
computed completely, because an elementary submatrix of X
had the infinity norm greater than PMAX. Part of the
matrix C has possibly been overwritten with the
corresponding part of X.

LDC     INTEGER
The leading dimension of array C.  LDC >= MAX(1,M).

Error Indicator
INFO    INTEGER
= 0:  successful exit;
= 1:  an elementary submatrix of X had the infinity norm
greater than the given value PMAX.

Method
The routine uses an adaptation of the standard method for solving
Sylvester equations [1], which controls the magnitude of the
individual elements of the computed solution [2]. The equation
-AX + XB = C can be rewritten as
p            l-1
-A  X   + X  B   = C   + sum  A  X   - sum  X  B
kk kl    kl ll    kl  i=k+1  ki il   j=1   kj jl

for l = 1:q, and k = p:-1:1, where A  , B  , C  , and X  , are
kk   ll   kl       kl
block submatrices defined by the partitioning induced by the Schur
form of A and B, and p and q are the numbers of the diagonal
blocks of A and B, respectively. So, the elementary submatrices of
X are found block column by block column, starting from the
bottom. If any such elementary submatrix has the infinity norm
greater than the given value PMAX, the calculations are ended.

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] Bavely, C. and Stewart, G.W.
An Algorithm for Computing Reducing Subspaces by Block
Diagonalization.
SIAM J. Numer. Anal., 16, pp. 359-367, 1979.

Numerical Aspects
2      2
The algorithm requires 0(M N + MN ) operations.

Let

( A   C )       ( I   X )
M = (       ),  Y = (       ).
( 0   B )       ( 0   I )

Then

-1      ( A   0 )
Y  M Y = (       ),
( 0   B )

hence Y is an non-orthogonal transformation matrix which performs
the reduction of M to a block-diagonal form. Bounding a norm of
X is equivalent to setting an upper bound to the condition number
of the transformation matrix Y.

Example

Program Text

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Program Data
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Program Results
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