## MB01RY

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

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

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.

Specification

          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,*)

Arguments

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'.

Input/Output Parameters

      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).

Workspace

      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'.

Error Indicator

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

Method

      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.

Further Comments

      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'.

Example

Program Text

      None

Program Data

      None

Program Results

      None

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