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<HTML>
<HEAD><TITLE>MB03RW - SLICOT Library Routine Documentation</TITLE>
</HEAD>
<BODY>
<H2><A Name="MB03RW">MB03RW</A></H2>
<H3>
Solution of a Sylvester equation -AX + XB = C, with A and B in complex Schur form, aborting the computations when the norm of X is too large
</H3>
<A HREF ="#Specification"><B>[Specification]</B></A>
<A HREF ="#Arguments"><B>[Arguments]</B></A>
<A HREF ="#Method"><B>[Method]</B></A>
<A HREF ="#References"><B>[References]</B></A>
<A HREF ="#Comments"><B>[Comments]</B></A>
<A HREF ="#Example"><B>[Example]</B></A>
<P>
<B><FONT SIZE="+1">Purpose</FONT></B>
<PRE>
To solve the Sylvester equation -AX + XB = C, where A and B are
complex M-by-M and N-by-N matrices, respectively, in Schur form.
This routine is intended to be called only by SLICOT Library
routine MB03RZ. For efficiency purposes, the computations are
aborted when the absolute value of an element of X is greater than
a given value PMAX.
</PRE>
<A name="Specification"><B><FONT SIZE="+1">Specification</FONT></B></A>
<PRE>
SUBROUTINE MB03RW( 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 ..
COMPLEX*16 A(LDA,*), B(LDB,*), C(LDC,*)
</PRE>
<A name="Arguments"><B><FONT SIZE="+1">Arguments</FONT></B></A>
<P>
</PRE>
<B>Input/Output Parameters</B>
<PRE>
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 absolute value of the elements of X
(see METHOD).
A (input) COMPLEX*16 array, dimension (LDA,M)
The leading M-by-M upper triangular part of this array
must contain the matrix A of the Sylvester equation.
The elements below the diagonal are not referenced.
LDA INTEGER
The leading dimension of array A. LDA >= MAX(1,M).
B (input) COMPLEX*16 array, dimension (LDB,N)
The leading N-by-N upper triangular part of this array
must contain the matrix B of the Sylvester equation.
The elements below the diagonal are not referenced.
LDB INTEGER
The leading dimension of array B. LDB >= MAX(1,N).
C (input/output) COMPLEX*16 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 element of X (see METHOD) has the
absolute value less than or equal to PMAX.
On exit, if INFO = 1, the solution matrix X has not been
computed completely, because an element of X had the
absolute value 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).
</PRE>
<B>Error Indicator</B>
<PRE>
INFO INTEGER
= 0: successful exit;
= 1: an element of X had the absolute value greater than
the given value PMAX.
= 2: A and B have common or very close eigenvalues;
perturbed values were used to solve the equation
(but the matrices A and B are unchanged). This is a
warning.
</PRE>
<A name="Method"><B><FONT SIZE="+1">Method</FONT></B></A>
<PRE>
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
m 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:n, and k = m:-1:1, where A , B , C , and X , are the
kk ll kl kl
elements defined by the partitioning induced by the Schur form
of A and B. So, the elements of X are found column by column,
starting from the bottom. If any such element has the absolute
value greater than the given value PMAX, the calculations are
ended.
</PRE>
<A name="References"><B><FONT SIZE="+1">References</FONT></B></A>
<PRE>
[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.
</PRE>
<A name="Numerical Aspects"><B><FONT SIZE="+1">Numerical Aspects</FONT></B></A>
<PRE> 2 2
The algorithm requires 0(M N + MN ) operations.
</PRE>
<A name="Comments"><B><FONT SIZE="+1">Further Comments</FONT></B></A>
<PRE>
Let
( A C ) ( I X )
M = ( ), Y = ( ).
( 0 B ) ( 0 I )
Then
-1 ( A 0 )
Y M Y = ( ),
( 0 B )
hence Y is a non-unitary 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.
</PRE>
<A name="Example"><B><FONT SIZE="+1">Example</FONT></B></A>
<P>
<B>Program Text</B>
<PRE>
None
</PRE>
<B>Program Data</B>
<PRE>
None
</PRE>
<B>Program Results</B>
<PRE>
None
</PRE>
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