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<HTML>
<HEAD><TITLE>SB03OR - SLICOT Library Routine Documentation</TITLE>
</HEAD>
<BODY>
<H2><A Name="SB03OR">SB03OR</A></H2>
<H3>
Solving continuous- or discrete-time Sylvester equations, with matrix S quasi-triangular, for an n-by-m matrix X, 1 <= m <= 2
</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 compute the solution of the Sylvester equations
op(S)'*X + X*op(A) = scale*C, if DISCR = .FALSE. or
op(S)'*X*op(A) - X = scale*C, if DISCR = .TRUE.
where op(K) = K or K' (i.e., the transpose of the matrix K), S is
an N-by-N block upper triangular matrix with one-by-one and
two-by-two blocks on the diagonal, A is an M-by-M matrix (M = 1 or
M = 2), X and C are each N-by-M matrices, and scale is an output
scale factor, set less than or equal to 1 to avoid overflow in X.
The solution X is overwritten on C.
SB03OR is a service routine for the Lyapunov solver SB03OT.
</PRE>
<A name="Specification"><B><FONT SIZE="+1">Specification</FONT></B></A>
<PRE>
SUBROUTINE SB03OR( DISCR, LTRANS, N, M, S, LDS, A, LDA, C, LDC,
$ SCALE, INFO )
C .. Scalar Arguments ..
LOGICAL DISCR, LTRANS
INTEGER INFO, LDA, LDS, LDC, M, N
DOUBLE PRECISION SCALE
C .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), C( LDC, * ), S( LDS, * )
</PRE>
<A name="Arguments"><B><FONT SIZE="+1">Arguments</FONT></B></A>
<P>
<B>Mode Parameters</B>
<PRE>
DISCR LOGICAL
Specifies the equation to be solved:
= .FALSE.: op(S)'*X + X*op(A) = scale*C;
= .TRUE. : op(S)'*X*op(A) - X = scale*C.
LTRANS LOGICAL
Specifies the form of op(K) to be used, as follows:
= .FALSE.: op(K) = K (No transpose);
= .TRUE. : op(K) = K**T (Transpose).
</PRE>
<B>Input/Output Parameters</B>
<PRE>
N (input) INTEGER
The order of the matrix S and also the number of rows of
matrices X and C. N >= 0.
M (input) INTEGER
The order of the matrix A and also the number of columns
of matrices X and C. M = 1 or M = 2.
S (input) DOUBLE PRECISION array, dimension (LDS,N)
The leading N-by-N upper Hessenberg part of the array S
must contain the block upper triangular matrix. The
elements below the upper Hessenberg part of the array S
are not referenced. The array S must not contain
diagonal blocks larger than two-by-two and the two-by-two
blocks must only correspond to complex conjugate pairs of
eigenvalues, not to real eigenvalues.
LDS INTEGER
The leading dimension of array S. LDS >= MAX(1,N).
A (input) DOUBLE PRECISION array, dimension (LDS,M)
The leading M-by-M part of this array must contain a
given matrix, where M = 1 or M = 2.
LDA INTEGER
The leading dimension of array A. LDA >= M.
C (input/output) DOUBLE PRECISION array, dimension (LDC,M)
On entry, C must contain an N-by-M matrix, where M = 1 or
M = 2.
On exit, C contains the N-by-M matrix X, the solution of
the Sylvester equation.
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.
</PRE>
<B>Error Indicator</B>
<PRE>
INFO INTEGER
= 0: successful exit;
= 1: if DISCR = .FALSE., and S and -A have common
eigenvalues, or if DISCR = .TRUE., and S and A have
eigenvalues whose product is equal to unity;
a solution has been computed using slightly
perturbed values.
</PRE>
<A name="Method"><B><FONT SIZE="+1">Method</FONT></B></A>
<PRE>
The LAPACK scheme for solving Sylvester equations is adapted.
</PRE>
<A name="References"><B><FONT SIZE="+1">References</FONT></B></A>
<PRE>
[1] Hammarling, S.J.
Numerical solution of the stable, non-negative definite
Lyapunov equation.
IMA J. Num. Anal., 2, pp. 303-325, 1982.
</PRE>
<A name="Numerical Aspects"><B><FONT SIZE="+1">Numerical Aspects</FONT></B></A>
<PRE> 2
The algorithm requires 0(N M) operations and is backward stable.
</PRE>
<A name="Comments"><B><FONT SIZE="+1">Further Comments</FONT></B></A>
<PRE>
None
</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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