## 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.

```
```  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
*     Copyright (c) 2002-2017 NICONET e.V.
*
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
```