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 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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