## MB01WD

### Residuals of Lyapunov or Stein equations for Cholesky factored solutions

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

Purpose

      To compute the matrix formula
      _
      R = alpha*( op( A )'*op( T )'*op( T ) + op( T )'*op( T )*op( A ) )
          + beta*R,                                                  (1)

      if DICO = 'C', or
      _
      R = alpha*( op( A )'*op( T )'*op( T )*op( A ) -  op( T )'*op( T ))
          + beta*R,                                                  (2)
                                                              _
      if DICO = 'D', where alpha and beta are scalars, R, and R are
      symmetric matrices, T is a triangular matrix, A is a general or
      Hessenberg matrix, and op( M ) is one of

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

      The result is overwritten on R.

Specification

          SUBROUTINE MB01WD( DICO, UPLO, TRANS, HESS, N, ALPHA, BETA, R,
         $                   LDR, A, LDA, T, LDT, INFO )
    C     .. Scalar Arguments ..
          CHARACTER         DICO, HESS, TRANS, UPLO
          INTEGER           INFO, LDA, LDR, LDT, N
          DOUBLE PRECISION  ALPHA, BETA
    C     .. Array Arguments ..
          DOUBLE PRECISION  A(LDA,*), R(LDR,*), T(LDT,*)

Arguments

Mode Parameters

      DICO    CHARACTER*1
              Specifies the formula to be evaluated, as follows:
              = 'C':  formula (1), "continuous-time" case;
              = 'D':  formula (2), "discrete-time" case.

      UPLO    CHARACTER*1
              Specifies which triangles of the symmetric matrix R and
              triangular matrix T are given, as follows:
              = 'U':  the upper triangular parts of R and T are given;
              = 'L':  the lower triangular parts of R and T are given;

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

      HESS    CHARACTER*1
              Specifies the form of the matrix A, as follows:
              = 'F':  matrix A is full;
              = 'H':  matrix A is Hessenberg (or Schur), either upper
                      (if UPLO = 'U'), or lower (if UPLO = 'L').

Input/Output Parameters

      N       (input) INTEGER
              The order of the matrices R, A, and T.  N >= 0.

      ALPHA   (input) DOUBLE PRECISION
              The scalar alpha. When alpha is zero then the arrays A
              and T are not referenced.

      BETA    (input) DOUBLE PRECISION
              The scalar beta. When beta is zero then the array R need
              not be set before entry.

      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.
              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).

      A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)
              On entry, the leading N-by-N part of this array must
              contain the matrix A. If HESS = 'H' the elements below the
              first subdiagonal, if UPLO = 'U', or above the first
              superdiagonal, if UPLO = 'L', need not be set to zero,
              and are not referenced if DICO = 'D'.
              On exit, the leading N-by-N part of this array contains
              the following matrix product
                 alpha*T'*T*A, if TRANS = 'N', or
                 alpha*A*T*T', otherwise,
              if DICO = 'C', or
                 T*A, if TRANS = 'N', or
                 A*T, otherwise,
              if DICO = 'D' (and in this case, these products have a
              Hessenberg form, if HESS = 'H').

      LDA     INTEGER
              The leading dimension of array A.  LDA >= MAX(1,N).

      T       (input) DOUBLE PRECISION array, dimension (LDT,N)
              If UPLO = 'U', the leading N-by-N upper triangular part of
              this array must contain the upper triangular matrix T and
              the strictly lower triangular part need not be set to zero
              (and it is not referenced).
              If UPLO = 'L', the leading N-by-N lower triangular part of
              this array must contain the lower triangular matrix T and
              the strictly upper triangular part need not be set to zero
              (and it is not referenced).

      LDT     INTEGER
              The leading dimension of array T.  LDT >= MAX(1,N).

Error Indicator

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

Method

      The matrix expression (1) or (2) is efficiently evaluated taking
      the structure into account. BLAS 3 operations (DTRMM, DSYRK and
      their specializations) are used throughout.

Numerical Aspects

      If A is a full matrix, the algorithm requires approximately
       3
      N  operations, if DICO = 'C';
             3
      7/6 x N  operations, if DICO = 'D'.

Further Comments

      None

Example

Program Text

      None

Program Data

      None

Program Results

      None

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