MB01RD

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

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.

      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,*)

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'.

  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).

  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.

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

  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 ) ).

  None.

  The algorithm requires approximately

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

  operations.

  None

Program Text

  None

  None

  None

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