## MB01RU

### Computation of matrix expression alpha R + beta A X trans(A), R, X symmetric (MB01RD variant)

[Specification] [Arguments] [Method] [References] [Comments] [Example]

Purpose

      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.

Specification

          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,*)

Arguments

Mode Parameters

      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'.

Input/Output Parameters

      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).

Workspace

      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.

Error Indicator

      INFO    INTEGER
              = 0:  successful exit;
              < 0:  if INFO = -k, the k-th argument had an illegal
                    value.

Method

      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).

Numerical Aspects

      The algorithm requires approximately

          2             2
         M x N + 1/2 x N x M

      operations.

Further Comments

      This is a simpler version for MB01RD.

Example

Program Text

      None

Program Data

      None

Program Results

      None

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