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<HTML>
<HEAD><TITLE>MB01RU - SLICOT Library Routine Documentation</TITLE>
</HEAD>
<BODY>
<H2><A Name="MB01RU">MB01RU</A></H2>
<H3>
Computation of matrix expression alpha R + beta A X trans(A), R, X symmetric (MB01RD variant)
</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 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 MB01RU( 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 symmetric matrices R
and X are given as follows:
= 'U': the upper triangular part is given;
= 'L': the 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
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 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 upper
triangular part of this array must contain the upper
triangular part of the symmetric matrix R.
On entry with UPLO = 'L', the leading M-by-M lower
triangular part of this array must contain the lower
triangular part of the symmetric matrix R.
On exit, the leading M-by-M upper triangular part (if
UPLO = 'U'), or lower triangular part (if UPLO = 'L'), of
this array contains the corresponding triangular part of
_
the computed matrix R. When R is identified with X in
the call, after exit, the diagonal entries of R must be
divided by 2.
LDR INTEGER
The leading dimension of 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 array A. LDA >= MAX(1,k),
where k is M when TRANS = 'N' and is N when TRANS = 'T' or
TRANS = 'C'.
X (input) DOUBLE PRECISION array, dimension (LDX,N)
On entry, if UPLO = 'U', the leading N-by-N upper
triangular part of this array must contain the upper
triangular part of the symmetric matrix X and the strictly
lower triangular part of the array is not referenced.
On entry, if UPLO = 'L', the leading N-by-N lower
triangular part of this array must contain the lower
triangular part of the symmetric matrix X and the strictly
upper triangular part of the array is not referenced.
The diagonal elements of this array are modified
internally, but are restored on exit.
LDX INTEGER
The leading dimension of array X. LDX >= MAX(1,N).
</PRE>
<B>Workspace</B>
<PRE>
DWORK DOUBLE PRECISION array, dimension (LDWORK)
This array is not referenced when beta = 0, or M*N = 0.
LDWORK The length of the array DWORK.
LDWORK >= M*N, if beta <> 0;
LDWORK >= 0, if beta = 0.
</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 symmetry
into account. Specifically, let X = T + T', with T an upper or
lower triangular matrix, defined by
T = triu( X ) - (1/2)*diag( X ), if UPLO = 'U',
T = tril( X ) - (1/2)*diag( X ), if UPLO = 'L',
where triu, tril, and diag denote the upper triangular part, lower
triangular part, and diagonal 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 DSYR2K).
</PRE>
<A name="Numerical Aspects"><B><FONT SIZE="+1">Numerical Aspects</FONT></B></A>
<PRE>
The algorithm requires approximately
2 2
M x N + 1/2 x N x M
operations.
</PRE>
<A name="Comments"><B><FONT SIZE="+1">Further Comments</FONT></B></A>
<PRE>
This is a simpler version for MB01RD.
</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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