## MB02PD

### Solution of matrix equation op(A) X = B, with error bounds and condition estimates

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

Purpose

      To solve (if well-conditioned) the matrix equations

         op( A )*X = B,

      where X and B are N-by-NRHS matrices, A is an N-by-N matrix and
      op( A ) is one of

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

      Error bounds on the solution and a condition estimate are also
      provided.

Specification

          SUBROUTINE MB02PD( FACT, TRANS, N, NRHS, A, LDA, AF, LDAF, IPIV,
         $                   EQUED, R, C, B, LDB, X, LDX, RCOND, FERR, BERR,
         $                   IWORK, DWORK, INFO )
    C     .. Scalar Arguments ..
          CHARACTER         EQUED, FACT, TRANS
          INTEGER           INFO, LDA, LDAF, LDB, LDX, N, NRHS
          DOUBLE PRECISION  RCOND
    C     .. Array Arguments ..
          INTEGER           IPIV( * ), IWORK( * )
          DOUBLE PRECISION  A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
         $                  BERR( * ), C( * ), DWORK( * ), FERR( * ),
         $                  R( * ), X( LDX, * )

Arguments

Mode Parameters

      FACT    CHARACTER*1
              Specifies whether or not the factored form of the matrix A
              is supplied on entry, and if not, whether the matrix A
              should be equilibrated before it is factored.
              = 'F':  On entry, AF and IPIV contain the factored form
                      of A. If EQUED is not 'N', the matrix A has been
                      equilibrated with scaling factors given by R
                      and C. A, AF, and IPIV are not modified.
              = 'N':  The matrix A will be copied to AF and factored.
              = 'E':  The matrix A will be equilibrated if necessary,
                      then copied to AF and factored.

      TRANS   CHARACTER*1
              Specifies the form of the system of equations as follows:
              = 'N':  A * X = B     (No transpose);
              = 'T':  A**T * X = B  (Transpose);
              = 'C':  A**H * X = B  (Transpose).

Input/Output Parameters

      N       (input) INTEGER
              The number of linear equations, i.e., the order of the
              matrix A.  N >= 0.

      NRHS    (input) INTEGER
              The number of right hand sides, i.e., the number of
              columns of the matrices B and X.  NRHS >= 0.

      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 FACT = 'F' and EQUED is not 'N',
              then A must have been equilibrated by the scaling factors
              in R and/or C.  A is not modified if FACT = 'F' or 'N',
              or if FACT = 'E' and EQUED = 'N' on exit.
              On exit, if EQUED .NE. 'N', the leading N-by-N part of
              this array contains the matrix A scaled as follows:
              EQUED = 'R':  A := diag(R) * A;
              EQUED = 'C':  A := A * diag(C);
              EQUED = 'B':  A := diag(R) * A * diag(C).

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

      AF      (input or output) DOUBLE PRECISION array, dimension
              (LDAF,N)
              If FACT = 'F', then AF is an input argument and on entry
              the leading N-by-N part of this array must contain the
              factors L and U from the factorization A = P*L*U as
              computed by DGETRF.  If EQUED .NE. 'N', then AF is the
              factored form of the equilibrated matrix A.
              If FACT = 'N', then AF is an output argument and on exit
              the leading N-by-N part of this array contains the factors
              L and U from the factorization A = P*L*U of the original
              matrix A.
              If FACT = 'E', then AF is an output argument and on exit
              the leading N-by-N part of this array contains the factors
              L and U from the factorization A = P*L*U of the
              equilibrated matrix A (see the description of A for the
              form of the equilibrated matrix).

      LDAF    (input) INTEGER
              The leading dimension of the array AF.  LDAF >= max(1,N).

      IPIV    (input or output) INTEGER array, dimension (N)
              If FACT = 'F', then IPIV is an input argument and on entry
              it must contain the pivot indices from the factorization
              A = P*L*U as computed by DGETRF; row i of the matrix was
              interchanged with row IPIV(i).
              If FACT = 'N', then IPIV is an output argument and on exit
              it contains the pivot indices from the factorization
              A = P*L*U of the original matrix A.
              If FACT = 'E', then IPIV is an output argument and on exit
              it contains the pivot indices from the factorization
              A = P*L*U of the equilibrated matrix A.

      EQUED   (input or output) CHARACTER*1
              Specifies the form of equilibration that was done as
              follows:
              = 'N':  No equilibration (always true if FACT = 'N');
              = 'R':  Row equilibration, i.e., A has been premultiplied
                      by diag(R);
              = 'C':  Column equilibration, i.e., A has been
                      postmultiplied by diag(C);
              = 'B':  Both row and column equilibration, i.e., A has
                      been replaced by diag(R) * A * diag(C).
              EQUED is an input argument if FACT = 'F'; otherwise, it is
              an output argument.

      R       (input or output) DOUBLE PRECISION array, dimension (N)
              The row scale factors for A.  If EQUED = 'R' or 'B', A is
              multiplied on the left by diag(R); if EQUED = 'N' or 'C',
              R is not accessed.  R is an input argument if FACT = 'F';
              otherwise, R is an output argument.  If FACT = 'F' and
              EQUED = 'R' or 'B', each element of R must be positive.

      C       (input or output) DOUBLE PRECISION array, dimension (N)
              The column scale factors for A.  If EQUED = 'C' or 'B',
              A is multiplied on the right by diag(C); if EQUED = 'N'
              or 'R', C is not accessed.  C is an input argument if
              FACT = 'F'; otherwise, C is an output argument.  If
              FACT = 'F' and EQUED = 'C' or 'B', each element of C must
              be positive.

      B       (input/output) DOUBLE PRECISION array, dimension
              (LDB,NRHS)
              On entry, the leading N-by-NRHS part of this array must
              contain the right-hand side matrix B.
              On exit,
              if EQUED = 'N', B is not modified;
              if TRANS = 'N' and EQUED = 'R' or 'B', the leading
              N-by-NRHS part of this array contains diag(R)*B;
              if TRANS = 'T' or 'C' and EQUED = 'C' or 'B', the leading
              N-by-NRHS part of this array contains diag(C)*B.

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

      X       (output) DOUBLE PRECISION array, dimension (LDX,NRHS)
              If INFO = 0 or INFO = N+1, the leading N-by-NRHS part of
              this array contains the solution matrix X to the original
              system of equations.  Note that A and B are modified on
              exit if EQUED .NE. 'N', and the solution to the
              equilibrated system is inv(diag(C))*X if TRANS = 'N' and
              EQUED = 'C' or 'B', or inv(diag(R))*X if TRANS = 'T' or
              'C' and EQUED = 'R' or 'B'.

      LDX     (input) INTEGER
              The leading dimension of the array X.  LDX >= max(1,N).

      RCOND   (output) DOUBLE PRECISION
              The estimate of the reciprocal condition number of the
              matrix A after equilibration (if done).  If RCOND is less
              than the machine precision (in particular, if RCOND = 0),
              the matrix is singular to working precision.  This
              condition is indicated by a return code of INFO > 0.
              For efficiency reasons, RCOND is computed only when the
              matrix A is factored, i.e., for FACT = 'N' or 'E'.  For
              FACT = 'F', RCOND is not used, but it is assumed that it
              has been computed and checked before the routine call.

      FERR    (output) DOUBLE PRECISION array, dimension (NRHS)
              The estimated forward error bound for each solution vector
              X(j) (the j-th column of the solution matrix X).
              If XTRUE is the true solution corresponding to X(j),
              FERR(j) is an estimated upper bound for the magnitude of
              the largest element in (X(j) - XTRUE) divided by the
              magnitude of the largest element in X(j).  The estimate
              is as reliable as the estimate for RCOND, and is almost
              always a slight overestimate of the true error.

      BERR    (output) DOUBLE PRECISION array, dimension (NRHS)
              The componentwise relative backward error of each solution
              vector X(j) (i.e., the smallest relative change in
              any element of A or B that makes X(j) an exact solution).

Workspace

      IWORK   INTEGER array, dimension (N)

      DWORK   DOUBLE PRECISION array, dimension (4*N)
              On entry, if FACT = 'F', DWORK(1) contains the reciprocal
              pivot growth factor norm(A)/norm(U), computed previously
              by this routine, with FACT <> 'N', for the same matrix A.
              On exit, DWORK(1) contains the reciprocal pivot growth
              factor norm(A)/norm(U). The "max absolute element" norm is
              used. If DWORK(1) is much less than 1, then the stability
              of the LU factorization of the (equilibrated) matrix A
              could be poor. This also means that the solution X,
              condition estimator RCOND, and forward error bound FERR
              could be unreliable. If factorization fails with
              0 < INFO <= N, then DWORK(1) contains the reciprocal pivot
              growth factor for the leading INFO columns of A.

Error Indicator

      INFO    INTEGER
              = 0:  successful exit;
              < 0:  if INFO = -i, the i-th argument had an illegal
                    value;
              > 0:  if INFO = i, and i is
                    <= N:  U(i,i) is exactly zero.  The factorization
                           has been completed, but the factor U is
                           exactly singular, so the solution and error
                           bounds could not be computed. RCOND = 0 is
                           returned.
                    = N+1: U is nonsingular, but RCOND is less than
                           machine precision, meaning that the matrix is
                           singular to working precision.  Nevertheless,
                           the solution and error bounds are computed
                           because there are a number of situations
                           where the computed solution can be more
                           accurate than the value of RCOND would
                           suggest.
              The positive values for INFO are set only when the
              matrix A is factored, i.e., for FACT = 'N' or 'E'.

Method

      The following steps are performed:

      1. If FACT = 'E', real scaling factors are computed to equilibrate
         the system:

         TRANS = 'N':  diag(R)*A*diag(C)     *inv(diag(C))*X = diag(R)*B
         TRANS = 'T': (diag(R)*A*diag(C))**T *inv(diag(R))*X = diag(C)*B
         TRANS = 'C': (diag(R)*A*diag(C))**H *inv(diag(R))*X = diag(C)*B

         Whether or not the system will be equilibrated depends on the
         scaling of the matrix A, but if equilibration is used, A is
         overwritten by diag(R)*A*diag(C) and B by diag(R)*B
         (if TRANS='N') or diag(C)*B (if TRANS = 'T' or 'C').

      2. If FACT = 'N' or 'E', the LU decomposition is used to factor
         the matrix A (after equilibration if FACT = 'E') as
            A = P * L * U,
         where P is a permutation matrix, L is a unit lower triangular
         matrix, and U is upper triangular.

      3. If some U(i,i)=0, so that U is exactly singular, then the
         routine returns with INFO = i. Otherwise, the factored form
         of A is used to estimate the condition number of the matrix A.
         If the reciprocal of the condition number is less than machine
         precision, INFO = N+1 is returned as a warning, but the routine
         still goes on to solve for X and compute error bounds as
         described below.

      4. The system of equations is solved for X using the factored form
         of A.

      5. Iterative refinement is applied to improve the computed
         solution matrix and calculate error bounds and backward error
         estimates for it.

      6. If equilibration was used, the matrix X is premultiplied by
         diag(C) (if TRANS = 'N') or diag(R) (if TRANS = 'T' or 'C') so
         that it solves the original system before equilibration.

References

      [1] Anderson, E., Bai, Z., Bischof, C., Demmel, J., Dongarra, J.,
          Du Croz, J., Greenbaum, A., Hammarling, S., McKenney, A.,
          Ostrouchov, S., Sorensen, D.
          LAPACK Users' Guide: Second Edition, SIAM, Philadelphia, 1995.

Further Comments

      This is a simplified version of the LAPACK Library routine DGESVX,
      useful when several sets of matrix equations with the same
      coefficient matrix  A and/or A'  should be solved.

Numerical Aspects

                                3
      The algorithm requires 0(N ) operations.

Example

Program Text

      None

Program Data

      None

Program Results

      None

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