MB01UX

Computation of matrix expressions alpha T A or alpha A T, over A, T quasi-triangular

Purpose

  To compute one of the matrix products

    A : = alpha*op( T ) * A, or A : = alpha*A * op( T ),

  where alpha is a scalar, A is an m-by-n matrix, T is a quasi-
  triangular matrix, and op( T ) is one of

     op( T ) = T   or   op( T ) = T',  the transpose of T.

      SUBROUTINE MB01UX( SIDE, UPLO, TRANS, M, N, ALPHA, T, LDT, A, LDA,
     $                   DWORK, LDWORK, INFO )
C     .. Scalar Arguments ..
      CHARACTER         SIDE, TRANS, UPLO
      INTEGER           INFO, LDA, LDT, LDWORK, M, N
      DOUBLE PRECISION  ALPHA
C     .. Array Arguments ..
      DOUBLE PRECISION  A(LDA,*), DWORK(*), T(LDT,*)

Mode Parameters

  SIDE    CHARACTER*1
          Specifies whether the upper quasi-triangular matrix H
          appears on the left or right in the matrix product as
          follows:
          = 'L':  A := alpha*op( T ) * A;
          = 'R':  A := alpha*A * op( T ).

  UPLO    CHARACTER*1.
          Specifies whether the matrix T is an upper or lower
          quasi-triangular matrix as follows:
          = 'U':  T is an upper quasi-triangular matrix;
          = 'L':  T is a lower quasi-triangular matrix.

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

  M       (input) INTEGER
          The number of rows of the matrix A.  M >= 0.

  N       (input) INTEGER
          The number of columns of the matrix A.  N >= 0.

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

  T       (input) DOUBLE PRECISION array, dimension (LDT,k)
          where k is M when SIDE = 'L' and is N when SIDE = 'R'.
          On entry with UPLO = 'U', the leading k-by-k upper
          Hessenberg part of this array must contain the upper
          quasi-triangular matrix T. The elements below the
          subdiagonal are not referenced.
          On entry with UPLO = 'L', the leading k-by-k lower
          Hessenberg part of this array must contain the lower
          quasi-triangular matrix T. The elements above the
          supdiagonal are not referenced.

  LDT     INTEGER
          The leading dimension of the array T.  LDT >= max(1,k),
          where k is M when SIDE = 'L' and is N when SIDE = 'R'.

  A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)
          On entry, the leading M-by-N part of this array must
          contain the matrix A.
          On exit, the leading M-by-N part of this array contains
          the computed product.

  LDA     INTEGER
          The leading dimension of the array A.  LDA >= max(1,M).

  DWORK   DOUBLE PRECISION array, dimension (LDWORK)
          On exit, if INFO = 0 and ALPHA<>0,  DWORK(1)  returns the
          optimal value of LDWORK.
          On exit, if  INFO = -12,  DWORK(1)  returns the minimum
          value of LDWORK.
          This array is not referenced when alpha = 0.

  LDWORK  The length of the array DWORK.
          LDWORK >= 1,       if alpha =  0 or MIN(M,N) = 0;
          LDWORK >= 2*(M-1), if SIDE  = 'L';
          LDWORK >= 2*(N-1), if SIDE  = 'R'.
          For maximal efficiency LDWORK should be at least
          NOFF*N + M - 1,    if SIDE  = 'L';
          NOFF*M + N - 1,    if SIDE  = 'R';
          where NOFF is the number of nonzero elements on the
          subdiagonal (if UPLO = 'U') or supdiagonal (if UPLO = 'L')
          of T.

          If LDWORK = -1, then a workspace query is assumed;
          the routine only calculates the optimal size of the
          DWORK array, returns this value as the first entry of
          the DWORK array, and no error message related to LDWORK
          is issued by XERBLA.

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

  The technique used in this routine is similiar to the technique
  used in the SLICOT [1] subroutine MB01UW developed by Vasile Sima.
  The required matrix product is computed in two steps. In the first
  step, the triangle of T specified by UPLO is used; in the second
  step, the contribution of the sub-/supdiagonal is added. If the
  workspace can accommodate parts of A, a fast BLAS 3 DTRMM
  operation is used in the first step.

  [1] Benner, P., Mehrmann, V., Sima, V., Van Huffel, S., and
      Varga, A.
      SLICOT - A subroutine library in systems and control theory.
      In: Applied and computational control, signals, and circuits,
      Vol. 1, pp. 499-539, Birkhauser, Boston, 1999.

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Program Text

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