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<HTML>
<HEAD><TITLE>MB01LD - SLICOT Library Routine Documentation</TITLE>
</HEAD>
<BODY>
<H2><A Name="MB01LD">MB01LD</A></H2>
<H3>
Computation of matrix expression alpha*R + beta*A*X*trans(A) with skew-symmetric matrices R and X
</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 matrix formula
_
R = alpha*R + beta*op( A )*X*op( A )',
_
where alpha and beta are scalars, R, X, and R are skew-symmetric
matrices, A is a general matrix, and op( A ) is one of
op( A ) = A or op( A ) = A'.
The result is overwritten on R.
</PRE>
<A name="Specification"><B><FONT SIZE="+1">Specification</FONT></B></A>
<PRE>
SUBROUTINE MB01LD( UPLO, TRANS, M, N, ALPHA, BETA, R, LDR, A, LDA,
$ X, LDX, DWORK, LDWORK, INFO )
C .. Scalar Arguments ..
CHARACTER TRANS, UPLO
INTEGER INFO, LDA, LDR, LDWORK, LDX, M, N
DOUBLE PRECISION ALPHA, BETA
C .. Array Arguments ..
DOUBLE PRECISION A(LDA,*), DWORK(*), R(LDR,*), X(LDX,*)
</PRE>
<A name="Arguments"><B><FONT SIZE="+1">Arguments</FONT></B></A>
<P>
<B>Mode Parameters</B>
<PRE>
UPLO CHARACTER*1
Specifies which triangles of the skew-symmetric matrices R
and X are given, as follows:
= 'U': the strictly upper triangular part is given;
= 'L': the strictly lower triangular part is given.
TRANS CHARACTER*1
Specifies the form of op( A ) to be used in the matrix
multiplication, as follows:
= 'N': op( A ) = A;
= 'T': op( A ) = A';
= 'C': op( A ) = A'.
</PRE>
<B>Input/Output Parameters</B>
<PRE>
M (input) INTEGER _
The order of the matrices R and R and the number of rows
of the matrix op( A ). M >= 0.
N (input) INTEGER
The order of the matrix X and the number of columns of the
matrix op( A ). N >= 0.
ALPHA (input) DOUBLE PRECISION
The scalar alpha. When alpha is zero then R need not be
set before entry, except when R is identified with X in
the call.
BETA (input) DOUBLE PRECISION
The scalar beta. When beta is zero or N <= 1, or M <= 1,
then A and X are not referenced.
R (input/output) DOUBLE PRECISION array, dimension (LDR,M)
On entry with UPLO = 'U', the leading M-by-M strictly
upper triangular part of this array must contain the
strictly upper triangular part of the skew-symmetric
matrix R. The lower triangle is not referenced.
On entry with UPLO = 'L', the leading M-by-M strictly
lower triangular part of this array must contain the
strictly lower triangular part of the skew-symmetric
matrix R. The upper triangle is not referenced.
On exit, the leading M-by-M strictly upper triangular part
(if UPLO = 'U'), or strictly lower triangular part
(if UPLO = 'L'), of this array contains the corresponding
_
strictly triangular part of the computed matrix R.
LDR INTEGER
The leading dimension of the array R. LDR >= MAX(1,M).
A (input) DOUBLE PRECISION array, dimension (LDA,k)
where k is N when TRANS = 'N' and is M when TRANS = 'T' or
TRANS = 'C'.
On entry with TRANS = 'N', the leading M-by-N part of this
array must contain the matrix A.
On entry with TRANS = 'T' or TRANS = 'C', the leading
N-by-M part of this array must contain the matrix A.
LDA INTEGER
The leading dimension of the array A. LDA >= MAX(1,k),
where k is M when TRANS = 'N' and is N when TRANS = 'T' or
TRANS = 'C'.
X (input or input/output) DOUBLE PRECISION array, dimension
(LDX,K), where K = N, if UPLO = 'U' or LDWORK >= M*(N-1),
or K = MAX(N,M), if UPLO = 'L' and LDWORK < M*(N-1).
On entry, if UPLO = 'U', the leading N-by-N strictly upper
triangular part of this array must contain the strictly
upper triangular part of the skew-symmetric matrix X and
the lower triangular part of the array is not referenced.
On entry, if UPLO = 'L', the leading N-by-N strictly lower
triangular part of this array must contain the strictly
lower triangular part of the skew-symmetric matrix X and
the upper triangular part of the array is not referenced.
If LDWORK < M*(N-1), this array is overwritten with the
matrix op(A)*X, if UPLO = 'U', or X*op(A)', if UPLO = 'L'.
LDX INTEGER
The leading dimension of the array X.
LDX >= MAX(1,N), if UPLO = 'L' or LDWORK >= M*(N-1);
LDX >= MAX(1,N,M), if UPLO = 'U' and LDWORK < M*(N-1).
</PRE>
<B>Workspace</B>
<PRE>
DWORK DOUBLE PRECISION array, dimension (LDWORK)
This array is not referenced when beta = 0, or M <= 1, or
N <= 1.
LDWORK The length of the array DWORK.
LDWORK >= N, if beta <> 0, and M > 0, and N > 1;
LDWORK >= 0, if beta = 0, or M = 0, or N <= 1.
For optimum performance, LDWORK >= M*(N-1), if beta <> 0,
M > 1, and N > 1.
</PRE>
<B>Error Indicator</B>
<PRE>
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -k, the k-th argument had an illegal
value.
</PRE>
<A name="Method"><B><FONT SIZE="+1">Method</FONT></B></A>
<PRE>
The matrix expression is efficiently evaluated taking the skew-
symmetry into account. If LDWORK >= M*(N-1), a BLAS 3 like
implementation is used. Specifically, let X = T - T', with T a
strictly upper or strictly lower triangular matrix, defined by
T = striu( X ), if UPLO = 'U',
T = stril( X ), if UPLO = 'L',
where striu and stril denote the strictly upper triangular part
and strictly lower triangular part of X, respectively. Then,
A*X*A' = ( A*T )*A' - A*( A*T )', for TRANS = 'N',
A'*X*A = A'*( T*A ) - ( T*A )'*A, for TRANS = 'T', or 'C',
which involve BLAS 3 operations DTRMM and the skew-symmetric
correspondent of DSYR2K (with a Fortran implementation available
in the SLICOT Library routine MB01KD).
If LDWORK < M*(N-1), a BLAS 2 implementation is used.
</PRE>
<A name="Numerical Aspects"><B><FONT SIZE="+1">Numerical Aspects</FONT></B></A>
<PRE>
The algorithm requires approximately
2 2
3/2 x M x N + 1/2 x M
operations.
</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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