MB01RT

Computation of matrix expression alpha R + beta op(E) X op(E)' with R, X symmetric and E upper triangular

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

Purpose

  To compute the matrix formula

     R := alpha*R + beta*op( E )*X*op( E )',

  where alpha and beta are scalars, R and X are symmetric matrices,
  E is an upper triangular matrix, and op( E ) is one of

     op( E ) = E   or   op( E ) = E'.

  The result is overwritten on R.

Specification
      SUBROUTINE MB01RT( UPLO, TRANS, N, ALPHA, BETA, R, LDR, E, LDE,
     $                   X, LDX, DWORK, LDWORK, INFO )
C     .. Scalar Arguments ..
      CHARACTER         TRANS, UPLO
      INTEGER           INFO, LDE, LDR, LDWORK, LDX, N
      DOUBLE PRECISION  ALPHA, BETA
C     .. Array Arguments ..
      DOUBLE PRECISION  DWORK(*), E(LDE,*), 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( E ) to be used in the matrix
          multiplication as follows:
          = 'N':  op( E ) = E;
          = 'T':  op( E ) = E';
          = 'C':  op( E ) = E'.

Input/Output Parameters
  N       (input) INTEGER
          The order of the matrices R, E, and X.  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 E and X are not
          referenced.

  R       (input/output) DOUBLE PRECISION array, dimension (LDR,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 R.
          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 R.
          In both cases, the other strictly triangular part is not
          referenced.
          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 computed matrix R.

  LDR     INTEGER
          The leading dimension of array R.  LDR >= MAX(1,N).

  E       (input) DOUBLE PRECISION array, dimension (LDE,N)
          On entry, the leading N-by-N upper triangular part of this
          array must contain the upper triangular matrix E.
          The remaining part of this array is not referenced.

  LDE     INTEGER
          The leading dimension of array E.  LDE >= MAX(1,N).

  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 N = 0.

  LDWORK  The length of the array DWORK.
          LDWORK >= N*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 = U + L, with U and L upper and
  lower triangular matrices, defined by

     U = triu( X ) - (1/2)*diag( X ),
     L = tril( X ) - (1/2)*diag( X ),

  where triu, tril, and diag denote the upper triangular part, lower
  triangular part, and diagonal part of X, respectively. Then,
  if UPLO = 'U',

     E*X*E' = ( E*U )*E' + E*( E*U )',  for TRANS = 'N',
     E'*X*E = E'*( U*E ) + ( U*E )'*E,  for TRANS = 'T', or 'C',

  and if UPLO = 'L',

     E*X*E' = ( E*L' )*E' + E*( E*L' )',  for TRANS = 'N',
     E'*X*E = E'*( L'*E ) + ( L'*E )'*E,  for TRANS = 'T', or 'C',

  which involve operations like in BLAS 2 and 3 (DTRMV and DSYR2K).
  This approach ensures that the matrices E*U, U*E, E*L', or L'*E
  are upper triangular.

Numerical Aspects
  The algorithm requires approximately N**3/2 operations.

Further Comments
  None
Example

Program Text

  None
Program Data
  None
Program Results
  None

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