## MB01OD

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

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

Purpose

      To compute the matrix formula

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

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

        op( M ) = M   or   op( M ) = M'.

      The result is overwritten on R.

Specification

          SUBROUTINE MB01OD( UPLO, TRANS, N, ALPHA, BETA, R, LDR, H, LDH,
         $                   X, LDX, E, LDE, DWORK, LDWORK, INFO )
    C     .. Scalar Arguments ..
          CHARACTER         TRANS, UPLO
          INTEGER           INFO, LDE, LDH, LDR, LDWORK, LDX, N
          DOUBLE PRECISION  ALPHA, BETA
    C     .. Array Arguments ..
          DOUBLE PRECISION  DWORK(*), E(LDE,*), H(LDH,*), 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( M ) to be used in the matrix
              multiplication as follows:
              = 'N':  op( M ) = M;
              = 'T':  op( M ) = M';
              = 'C':  op( M ) = M'.

Input/Output Parameters

      N       (input) INTEGER
              The order of the matrices R, H, 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 H 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).

      H       (input) DOUBLE PRECISION array, dimension (LDH,N)
              On entry, the leading N-by-N upper Hessenberg part of this
              array must contain the upper Hessenberg matrix H.
              If TRANS = 'N', the entries 3, 4,..., N of the first
              column are modified internally, but are restored on exit.
              The remaining part of this array is not referenced.

      LDH     INTEGER
              The leading dimension of array H.  LDH >= 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).

      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).

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',

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

      and if UPLO = 'L',

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

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