MB01RY

Computation of a triangle of matrix expression alpha R + beta H B or alpha R + beta B H, H upper Hessenberg matrix

Purpose

  To compute either the upper or lower triangular part of one of the
  matrix formulas
     _
     R = alpha*R + beta*op( H )*B,                               (1)
     _
     R = alpha*R + beta*B*op( H ),                               (2)
                                                 _
  where alpha and beta are scalars, H, B, R, and R are m-by-m
  matrices, H is an upper Hessenberg matrix, and op( H ) is one of

     op( H ) = H   or   op( H ) = H',  the transpose of H.

  The result is overwritten on R.

      SUBROUTINE MB01RY( SIDE, UPLO, TRANS, M, ALPHA, BETA, R, LDR, H,
     $                   LDH, B, LDB, DWORK, INFO )
C     .. Scalar Arguments ..
      CHARACTER         SIDE, TRANS, UPLO
      INTEGER           INFO, LDB, LDH, LDR, M
      DOUBLE PRECISION  ALPHA, BETA
C     .. Array Arguments ..
      DOUBLE PRECISION  B(LDB,*), DWORK(*), H(LDH,*), R(LDR,*)

Mode Parameters

  SIDE    CHARACTER*1
          Specifies whether the Hessenberg matrix H appears on the
          left or right in the matrix product as follows:
                  _
          = 'L':  R = alpha*R + beta*op( H )*B;
                  _
          = 'R':  R = alpha*R + beta*B*op( H ).

  UPLO    CHARACTER*1                               _
          Specifies which triangles of the matrices R and R are
          computed and given, respectively, as follows:
          = 'U':  the upper triangular part;
          = 'L':  the lower triangular part.

  TRANS   CHARACTER*1
          Specifies the form of op( H ) to be used in the matrix
          multiplication as follows:
          = 'N':  op( H ) = H;
          = 'T':  op( H ) = H';
          = 'C':  op( H ) = H'.

  M       (input) INTEGER           _
          The order of the matrices R, R, H and B.  M >= 0.

  ALPHA   (input) DOUBLE PRECISION
          The scalar alpha. When alpha is zero then R need not be
          set before entry.

  BETA    (input) DOUBLE PRECISION
          The scalar beta. When beta is zero then H and B 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 matrix R; the strictly lower
          triangular part of the array is not referenced.
          On entry with UPLO = 'L', the leading M-by-M lower
          triangular part of this array must contain the lower
          triangular part of the matrix R; the strictly upper
          triangular part of the array is not referenced.
          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.

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

  H       (input) DOUBLE PRECISION array, dimension (LDH,M)
          On entry, the leading M-by-M upper Hessenberg part of
          this array must contain the upper Hessenberg part of the
          matrix H.
          The elements below the subdiagonal are not referenced,
          except possibly for those in the first column, which
          could be overwritten, but are restored on exit.

  LDH     INTEGER
          The leading dimension of array H.  LDH >= MAX(1,M).

  B       (input) DOUBLE PRECISION array, dimension (LDB,M)
          On entry, the leading M-by-M part of this array must
          contain the matrix B.

  LDB     INTEGER
          The leading dimension of array B.  LDB >= MAX(1,M).

  DWORK   DOUBLE PRECISION array, dimension (LDWORK)
          LDWORK >= M, if  beta <> 0 and SIDE = 'L';
          LDWORK >= 0, if  beta =  0 or  SIDE = 'R'.
          This array is not referenced when beta = 0 or SIDE = 'R'.

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

  The matrix expression is efficiently evaluated taking the
  Hessenberg/triangular structure into account. BLAS 2 operations
  are used. A block algorithm can be constructed; it can use BLAS 3
  GEMM operations for most computations, and calls of this BLAS 2
  algorithm for computing the triangles.

  The main application of this routine is when the result should
  be a symmetric matrix, e.g., when B = X*op( H )', for (1), or
  B = op( H )'*X, for (2), where B is already available and X = X'.

Program Text

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