## MB02VD

### Solution of linear equations X op(A) = B

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

Purpose

      To compute the solution to a real system of linear equations
         X * op(A) = B,
      where op(A) is either A or its transpose, A is an N-by-N matrix,
      and X and B are M-by-N matrices.
      The LU decomposition with partial pivoting and row interchanges,
      A = P * L * U, is used, where P is a permutation matrix, L is unit
      lower triangular, and U is upper triangular.

Specification

          SUBROUTINE MB02VD( TRANS, M, N, A, LDA, IPIV, B, LDB, INFO )
    C     .. Scalar Arguments ..
          CHARACTER          TRANS
          INTEGER            INFO, LDA, LDB, M, N
    C     .. Array Arguments ..
          INTEGER            IPIV( * )
          DOUBLE PRECISION   A( LDA, * ), B( LDB, * )

Arguments

Mode Parameters

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

Input/Output Parameters

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

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

      A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)
              On entry, the leading N-by-N part of this array must
              contain the coefficient matrix A.
              On exit, the leading N-by-N part of this array contains
              the factors L and U from the factorization A = P*L*U;
              the unit diagonal elements of L are not stored.

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

      IPIV    (output) INTEGER array, dimension (N)
              The pivot indices that define the permutation matrix P;
              row i of the matrix was interchanged with row IPIV(i).

      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.

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

      INFO    (output) INTEGER
              = 0:  successful exit;
              < 0:  if INFO = -i, the i-th argument had an illegal
                    value;
              > 0:  if INFO = i, U(i,i) is exactly zero.  The
                    factorization has been completed, but the factor U
                    is exactly singular, so the solution could not be
                    computed.

Method

      The LU decomposition with partial pivoting and row interchanges is
      used to factor A as
         A = P * L * U,
      where P is a permutation matrix, L is unit lower triangular, and
      U is upper triangular.  The factored form of A is then used to
      solve the system of equations X * A = B or X * A' = B.

Further Comments

      This routine enables to solve the system X * A = B or X * A' = B
      as easily and efficiently as possible; it is similar to the LAPACK
      Library routine DGESV, which solves A * X = B.

Example

Program Text

    *     MB02VD EXAMPLE PROGRAM TEXT
    *
          INTEGER          NIN, NOUT
          PARAMETER        ( NIN = 5, NOUT = 6 )
          INTEGER          MMAX, NMAX
          PARAMETER        ( MMAX = 20, NMAX = 20 )
          INTEGER          LDA, LDB
          PARAMETER        ( LDA = NMAX, LDB = MMAX )
    *     .. Local Scalars ..
          INTEGER          I, INFO, J, M, N
          CHARACTER*1      TRANS
    *     .. Local Arrays ..
          DOUBLE PRECISION A(LDA,NMAX), B(LDB,NMAX)
          INTEGER          IPIV(NMAX)
    *     .. External Subroutines ..
          EXTERNAL         MB02VD
    *     .. Executable Statements ..
    *
          WRITE ( NOUT, FMT = 99999 )
    *     Skip the heading in the data file and read in the data.
          READ ( NIN, FMT = '()' )
          READ ( NIN, FMT = * ) M, N, TRANS
          IF ( N.LT.0 .OR. N.GT.NMAX ) THEN
             WRITE ( NOUT, FMT = 99995 ) N
          ELSE
             READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,N )
             IF ( M.LT.0 .OR. M.GT.MMAX ) THEN
                WRITE ( NOUT, FMT = 99994 ) M
             ELSE
                READ ( NIN, FMT = * ) ( ( B(I,J), J = 1,N ), I = 1,M )
    *           Solve the linear system using the LU factorization.
                CALL MB02VD( TRANS, M, N, A, LDA, IPIV, B, LDB, INFO )
    *
                IF ( INFO.EQ.0 ) THEN
                   WRITE ( NOUT, FMT = 99997 )
                   DO 10 I = 1, M
                      WRITE ( NOUT, FMT = 99996 ) ( B(I,J), J = 1,N )
       10          CONTINUE
                ELSE
                   WRITE ( NOUT, FMT = 99998 ) INFO
                END IF
             END IF
          END IF
          STOP
    *
    99999 FORMAT (' MB02VD EXAMPLE PROGRAM RESULTS',/1X)
    99998 FORMAT (' INFO on exit from MB02VD = ',I2)
    99997 FORMAT (' The solution matrix is ')
    99996 FORMAT (20(1X,F8.4))
    99995 FORMAT (/' N is out of range.',/' N = ',I5)
    99994 FORMAT (/' M is out of range.',/' M = ',I5)
          END

Program Data

     MB02VD EXAMPLE PROGRAM DATA
       5    4      N
       1.    2.    6.    3.
      -2.   -1.   -1.    0.
       2.    3.    1.    5.
       1.   -1.    2.    0.
       0.    0.    0.    1.
       5.    5.    1.    5.
      -2.    1.    3.    1.
       0.    0.    4.    5.
       2.    1.    1.    3.

Program Results

     MB02VD EXAMPLE PROGRAM RESULTS

     The solution matrix is
      -0.0690   0.3333   0.2414   0.2529
      -0.1724  -1.6667   1.1034  -0.3678
       0.9655   0.6667  -0.3793  -0.8736
       0.3448   1.6667   0.7931   1.4023
      -0.2069   0.0000   0.7241   0.7586

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