MB02OD

Solution of matrix equations A X = alpha B, or X A = alpha B, A triangular

Purpose

  To solve (if well-conditioned) one of the matrix equations

     op( A )*X = alpha*B,   or   X*op( A ) = alpha*B,

  where alpha is a scalar, X and B are m-by-n matrices, A is a unit,
  or non-unit, upper or lower triangular matrix and op( A ) is one
  of

     op( A ) = A   or   op( A ) = A'.

  An estimate of the reciprocal of the condition number of the
  triangular matrix A, in either the 1-norm or the infinity-norm, is
  also computed as

     RCOND = 1 / ( norm(A) * norm(inv(A)) ).

  and the specified matrix equation is solved only if RCOND is
  larger than a given tolerance TOL.  In that case, the matrix X is
  overwritten on B.

      SUBROUTINE MB02OD( SIDE, UPLO, TRANS, DIAG, NORM, M, N, ALPHA, A,
     $                   LDA, B, LDB, RCOND, TOL, IWORK, DWORK, INFO )
C     .. Scalar Arguments ..
      CHARACTER          DIAG, NORM, SIDE, TRANS, UPLO
      INTEGER            INFO, LDA, LDB, M, N
      DOUBLE PRECISION   ALPHA, RCOND, TOL
C     .. Array Arguments ..
      INTEGER            IWORK(*)
      DOUBLE PRECISION   A(LDA,*), B(LDB,*), DWORK(*)

Mode Parameters

  SIDE    CHARACTER*1
          Specifies whether op( A ) appears on the left or right
          of X as follows:
          = 'L':  op( A )*X = alpha*B;
          = 'R':  X*op( A ) = alpha*B.

  UPLO    CHARACTER*1
          Specifies whether the matrix A is an upper or lower
          triangular matrix as follows:
          = 'U':  A is an upper triangular matrix;
          = 'L':  A is a lower triangular matrix.

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

  DIAG    CHARACTER*1
          Specifies whether or not A is unit triangular as follows:
          = 'U':  A is assumed to be unit triangular;
          = 'N':  A is not assumed to be unit triangular.

  NORM    CHARACTER*1
          Specifies whether the 1-norm condition number or the
          infinity-norm condition number is required:
          = '1' or 'O':  1-norm;
          = 'I':         Infinity-norm.

  M       (input) INTEGER
          The number of rows of B.  M >= 0.

  N       (input) INTEGER
          The number of columns of B.  N >= 0.

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

  A       (input) DOUBLE PRECISION array, dimension (LDA,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
          triangular part of this array must contain the upper
          triangular matrix and the strictly lower triangular part
          of A is not referenced.
          On entry with UPLO = 'L', the leading k-by-k lower
          triangular part of this array must contain the lower
          triangular matrix and the strictly upper triangular part
          of A is not referenced.
          Note that when DIAG = 'U', the diagonal elements of A are
          not referenced either, but are assumed to be unity.

  LDA     INTEGER
          The leading dimension of array A.
          LDA >= max(1,M) when SIDE = 'L';
          LDA >= max(1,N) when SIDE = 'R'.

  B       (input/output) DOUBLE PRECISION array, dimension (LDB,N)
          On entry, the leading M-by-N part of this array must
          contain the right-hand side matrix B.
          On exit, if INFO = 0, the leading M-by-N part of this
          array contains the solution matrix X.
          Otherwise, this array is not modified by the routine.

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

  RCOND   (output) DOUBLE PRECISION
          The reciprocal of the condition number of the matrix A,
          computed as RCOND = 1/(norm(A) * norm(inv(A))).

  TOL     DOUBLE PRECISION
          The tolerance to be used to test for near singularity of
          the matrix A. If the user sets TOL > 0, then the given
          value of TOL is used as a lower bound for the reciprocal
          condition number of that matrix; a matrix whose estimated
          condition number is less than 1/TOL is considered to be
          nonsingular. If the user sets TOL <= 0, then an implicitly
          computed, default tolerance, defined by TOLDEF = k*k*EPS,
          is used instead, where EPS is the machine precision (see
          LAPACK Library routine DLAMCH).

  IWORK   INTEGER array, dimension (k)

  DWORK   DOUBLE PRECISION array, dimension (3*k)

  INFO    INTEGER
          = 0:  successful exit;
          < 0:  if INFO = -i, the i-th argument had an illegal
                value;
          = 1:  the matrix A is numerically singular, i.e. the
                condition number estimate of A (in the specified
                norm) exceeds 1/TOL.

  An estimate of the reciprocal of the condition number of the
  triangular matrix A (in the specified norm) is computed, and if
  this estimate is larger then the given (or default) tolerance,
  the specified matrix equation is solved using Level 3 BLAS
  routine DTRSM.

  None.

                          2
  The algorithm requires k N/2 operations.

  None

Program Text

  None

  None

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