## MB01RD

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

[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 MB01RD( 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, 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 (which is possible only in this case).

      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; the strictly
              lower triangular part of the array is used as workspace.
              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; the strictly
              upper triangular part of the array is used as workspace.
              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. If beta <> 0, the remaining
              strictly triangular part of this array contains the
              corresponding part of the matrix expression
              beta*op( A )*T*op( A )', where T is the triangular matrix
              defined in the Method section.

      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,l),
              where l is M when TRANS = 'N' and is N when TRANS = 'T' or
              TRANS = 'C'.

      X       (input/output) 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.
              On exit, each diagonal element of this array has half its
              input value, but the other elements are not modified.

      LDX     INTEGER
              The leading dimension of array X.  LDX >= MAX(1,N).

Workspace

      DWORK   DOUBLE PRECISION array, dimension (LDWORK)
              On exit, if INFO = 0, the leading M-by-N part of this
              array (with the leading dimension MAX(1,M)) returns the
              matrix product beta*op( A )*T, where T is the triangular
              matrix defined in the Method section.
              This array is not referenced when beta = 0.

      LDWORK  The length of the array DWORK.
              LDWORK >= MAX(1,M*N), if  beta <> 0;
              LDWORK >= 1,          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,

         op( A )*X*op( A )' = B + B',

      where B := op( A )*T*op( A )'. Matrix B is not symmetric, but it
      can be written as tri( B ) + stri( B ), where tri denotes the
      triangular part specified by UPLO, and stri denotes the remaining
      strictly triangular part. Let R = V + V', with V defined as T
      above. Then, the required triangular part of the result can be
      written as

         alpha*V + beta*tri( B )  + beta*(stri( B ))' +
                  alpha*diag( V ) + beta*diag( tri( B ) ).

References

      None.

Numerical Aspects

      The algorithm requires approximately

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

      operations.

Further Comments

      None

Example

Program Text

      None

Program Data

      None

Program Results

      None

Return to index