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<HTML>
<HEAD><TITLE>MB02OD - SLICOT Library Routine Documentation</TITLE>
</HEAD>
<BODY>
<H2><A Name="MB02OD">MB02OD</A></H2>
<H3>
Solution of matrix equations A X = alpha B, or X A = alpha B, A triangular
</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 (if well-conditioned) one of the matrix equations
op( A )*X = alpha*B, or X*op( A ) = alpha*B,
where alpha is a scalar, X and B are m-by-n matrices, A is a unit,
or non-unit, upper or lower triangular matrix and op( A ) is one
of
op( A ) = A or op( A ) = A'.
An estimate of the reciprocal of the condition number of the
triangular matrix A, in either the 1-norm or the infinity-norm, is
also computed as
RCOND = 1 / ( norm(A) * norm(inv(A)) ).
and the specified matrix equation is solved only if RCOND is
larger than a given tolerance TOL. In that case, the matrix X is
overwritten on B.
</PRE>
<A name="Specification"><B><FONT SIZE="+1">Specification</FONT></B></A>
<PRE>
SUBROUTINE MB02OD( SIDE, UPLO, TRANS, DIAG, NORM, M, N, ALPHA, A,
$ LDA, B, LDB, RCOND, TOL, IWORK, DWORK, INFO )
C .. Scalar Arguments ..
CHARACTER DIAG, NORM, SIDE, TRANS, UPLO
INTEGER INFO, LDA, LDB, M, N
DOUBLE PRECISION ALPHA, RCOND, TOL
C .. Array Arguments ..
INTEGER IWORK(*)
DOUBLE PRECISION A(LDA,*), B(LDB,*), DWORK(*)
</PRE>
<A name="Arguments"><B><FONT SIZE="+1">Arguments</FONT></B></A>
<P>
<B>Mode Parameters</B>
<PRE>
SIDE CHARACTER*1
Specifies whether op( A ) appears on the left or right
of X as follows:
= 'L': op( A )*X = alpha*B;
= 'R': X*op( A ) = alpha*B.
UPLO CHARACTER*1
Specifies whether the matrix A is an upper or lower
triangular matrix as follows:
= 'U': A is an upper triangular matrix;
= 'L': A is a lower triangular matrix.
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'.
DIAG CHARACTER*1
Specifies whether or not A is unit triangular as follows:
= 'U': A is assumed to be unit triangular;
= 'N': A is not assumed to be unit triangular.
NORM CHARACTER*1
Specifies whether the 1-norm condition number or the
infinity-norm condition number is required:
= '1' or 'O': 1-norm;
= 'I': Infinity-norm.
</PRE>
<B>Input/Output Parameters</B>
<PRE>
M (input) INTEGER
The number of rows of B. M >= 0.
N (input) INTEGER
The number of columns of B. N >= 0.
ALPHA (input) DOUBLE PRECISION
The scalar alpha. When alpha is zero then A is not
referenced and B need not be set before entry.
A (input) DOUBLE PRECISION array, dimension (LDA,k),
where k is M when SIDE = 'L' and is N when SIDE = 'R'.
On entry with UPLO = 'U', the leading k-by-k upper
triangular part of this array must contain the upper
triangular matrix and the strictly lower triangular part
of A is not referenced.
On entry with UPLO = 'L', the leading k-by-k lower
triangular part of this array must contain the lower
triangular matrix and the strictly upper triangular part
of A is not referenced.
Note that when DIAG = 'U', the diagonal elements of A are
not referenced either, but are assumed to be unity.
LDA INTEGER
The leading dimension of array A.
LDA >= max(1,M) when SIDE = 'L';
LDA >= max(1,N) when SIDE = 'R'.
B (input/output) DOUBLE PRECISION array, dimension (LDB,N)
On entry, the leading M-by-N part of this array must
contain the right-hand side matrix B.
On exit, if INFO = 0, the leading M-by-N part of this
array contains the solution matrix X.
Otherwise, this array is not modified by the routine.
LDB INTEGER
The leading dimension of array B. LDB >= max(1,M).
RCOND (output) DOUBLE PRECISION
The reciprocal of the condition number of the matrix A,
computed as RCOND = 1/(norm(A) * norm(inv(A))).
</PRE>
<B>Tolerances</B>
<PRE>
TOL DOUBLE PRECISION
The tolerance to be used to test for near singularity of
the matrix A. If the user sets TOL > 0, then the given
value of TOL is used as a lower bound for the reciprocal
condition number of that matrix; a matrix whose estimated
condition number is less than 1/TOL is considered to be
nonsingular. If the user sets TOL <= 0, then an implicitly
computed, default tolerance, defined by TOLDEF = k*k*EPS,
is used instead, where EPS is the machine precision (see
LAPACK Library routine DLAMCH).
</PRE>
<B>Workspace</B>
<PRE>
IWORK INTEGER array, dimension (k)
DWORK DOUBLE PRECISION array, dimension (3*k)
</PRE>
<B>Error Indicator</B>
<PRE>
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= 1: the matrix A is numerically singular, i.e. the
condition number estimate of A (in the specified
norm) exceeds 1/TOL.
</PRE>
<A name="Method"><B><FONT SIZE="+1">Method</FONT></B></A>
<PRE>
An estimate of the reciprocal of the condition number of the
triangular matrix A (in the specified norm) is computed, and if
this estimate is larger then the given (or default) tolerance,
the specified matrix equation is solved using Level 3 BLAS
routine DTRSM.
</PRE>
<A name="References"><B><FONT SIZE="+1">References</FONT></B></A>
<PRE>
None.
</PRE>
<A name="Numerical Aspects"><B><FONT SIZE="+1">Numerical Aspects</FONT></B></A>
<PRE> 2
The algorithm requires k N/2 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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