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<HTML>
<HEAD><TITLE>MB01YD - SLICOT Library Routine Documentation</TITLE>
</HEAD>
<BODY>
<H2><A Name="MB01YD">MB01YD</A></H2>
<H3>
Symmetric rank k operation C := alpha op( A ) op( A )' + beta C, C symmetric
</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 perform the symmetric rank k operations
C := alpha*op( A )*op( A )' + beta*C,
where alpha and beta are scalars, C is an n-by-n symmetric matrix,
op( A ) is an n-by-k matrix, and op( A ) is one of
op( A ) = A or op( A ) = A'.
The matrix A has l nonzero codiagonals, either upper or lower.
</PRE>
<A name="Specification"><B><FONT SIZE="+1">Specification</FONT></B></A>
<PRE>
SUBROUTINE MB01YD( UPLO, TRANS, N, K, L, ALPHA, BETA, A, LDA, C,
$ LDC, INFO )
C .. Scalar Arguments ..
CHARACTER TRANS, UPLO
INTEGER INFO, LDA, LDC, K, L, N
DOUBLE PRECISION ALPHA, BETA
C .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), C( LDC, * )
</PRE>
<A name="Arguments"><B><FONT SIZE="+1">Arguments</FONT></B></A>
<P>
<B>Mode Parameters</B>
<PRE>
UPLO CHARACTER*1
Specifies which triangle of the symmetric matrix C
is given and computed, as follows:
= 'U': the upper triangular part is given/computed;
= 'L': the lower triangular part is given/computed.
UPLO also defines the pattern of the matrix A (see below).
TRANS CHARACTER*1
Specifies the form of op( A ) to be used, as follows:
= 'N': op( A ) = A;
= 'T': op( A ) = A';
= 'C': op( A ) = A'.
</PRE>
<B>Input/Output Parameters</B>
<PRE>
N (input) INTEGER
The order of the matrix C. N >= 0.
K (input) INTEGER
The number of columns of the matrix op( A ). K >= 0.
L (input) INTEGER
If UPLO = 'U', matrix A has L nonzero subdiagonals.
If UPLO = 'L', matrix A has L nonzero superdiagonals.
MAX(0,NR-1) >= L >= 0, if UPLO = 'U',
MAX(0,NC-1) >= L >= 0, if UPLO = 'L',
where NR and NC are the numbers of rows and columns of the
matrix A, respectively.
ALPHA (input) DOUBLE PRECISION
The scalar alpha. When alpha is zero then the array A is
not referenced.
BETA (input) DOUBLE PRECISION
The scalar beta. When beta is zero then the array C need
not be set before entry.
A (input) DOUBLE PRECISION array, dimension (LDA,NC), where
NC is K when TRANS = 'N', and is N otherwise.
If TRANS = 'N', the leading N-by-K part of this array must
contain the matrix A, otherwise the leading K-by-N part of
this array must contain the matrix A.
If UPLO = 'U', only the upper triangular part and the
first L subdiagonals are referenced, and the remaining
subdiagonals are assumed to be zero.
If UPLO = 'L', only the lower triangular part and the
first L superdiagonals are referenced, and the remaining
superdiagonals are assumed to be zero.
LDA INTEGER
The leading dimension of array A. LDA >= max(1,NR),
where NR = N, if TRANS = 'N', and NR = K, otherwise.
C (input/output) DOUBLE PRECISION array, dimension (LDC,N)
On entry with UPLO = 'U', the leading N-by-N upper
triangular part of this array must contain the upper
triangular part of the symmetric matrix C.
On entry with UPLO = 'L', the leading N-by-N lower
triangular part of this array must contain the lower
triangular part of the symmetric matrix C.
On exit, the leading N-by-N upper triangular part (if
UPLO = 'U'), or lower triangular part (if UPLO = 'L'), of
this array contains the corresponding triangular part of
the updated matrix C.
LDC INTEGER
The leading dimension of array C. LDC >= MAX(1,N).
</PRE>
<B>Error Indicator</B>
<PRE>
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
</PRE>
<A name="Method"><B><FONT SIZE="+1">Method</FONT></B></A>
<PRE>
The calculations are efficiently performed taking the symmetry
and structure into account.
</PRE>
<A name="Comments"><B><FONT SIZE="+1">Further Comments</FONT></B></A>
<PRE>
The matrix A may have the following patterns, when n = 7, k = 5,
and l = 2 are used for illustration:
UPLO = 'U', TRANS = 'N' UPLO = 'L', TRANS = 'N'
[ x x x x x ] [ x x x 0 0 ]
[ x x x x x ] [ x x x x 0 ]
[ x x x x x ] [ x x x x x ]
A = [ 0 x x x x ], A = [ x x x x x ],
[ 0 0 x x x ] [ x x x x x ]
[ 0 0 0 x x ] [ x x x x x ]
[ 0 0 0 0 x ] [ x x x x x ]
UPLO = 'U', TRANS = 'T' UPLO = 'L', TRANS = 'T'
[ x x x x x x x ] [ x x x 0 0 0 0 ]
[ x x x x x x x ] [ x x x x 0 0 0 ]
A = [ x x x x x x x ], A = [ x x x x x 0 0 ].
[ 0 x x x x x x ] [ x x x x x x 0 ]
[ 0 0 x x x x x ] [ x x x x x x x ]
If N = K, the matrix A is upper or lower triangular, for L = 0,
and upper or lower Hessenberg, for L = 1.
This routine is a specialization of the BLAS 3 routine DSYRK.
BLAS 1 calls are used when appropriate, instead of in-line code,
in order to increase the efficiency. If the matrix A is full, or
its zero triangle has small order, an optimized DSYRK code could
be faster than MB01YD.
</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>
<HR>
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