This information is part of the Modelica Standard Library maintained by the Modelica Association.

Lapack documentation
    Purpose
    =======

    DGESVX uses the LU factorization to compute the solution to a real
    system of linear equations
       A * X = B,
    where A is an N-by-N matrix and X and B are N-by-NRHS matrices.

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

    Description
    ===========

    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.

    Arguments
    =========

    FACT    (input) 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   (input) CHARACTER*1
            Specifies the form of the system of equations:
            = 'N':  A * X = B     (No transpose)
            = 'T':  A**T * X = B  (Transpose)
            = 'C':  A**H * X = B  (Transpose)

    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 N-by-N 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', A is scaled as follows:
            EQUED = 'R':  A := diag(R) * A
            EQUED = 'C':  A := A * diag(C)
            EQUED = 'B':  A := diag(R) * A * diag(C).

    LDA     (input) 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
            contains 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
            returns 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
            returns 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
            contains 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
            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
            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.
            = '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 N-by-NRHS right hand side matrix B.
            On exit,
            if EQUED = 'N', B is not modified;
            if TRANS = 'N' and EQUED = 'R' or 'B', B is overwritten by
            diag(R)*B;
            if TRANS = 'T' or 'C' and EQUED = 'C' or 'B', B is
            overwritten by diag(C)*B.

    LDB     (input) 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 N-by-NRHS 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.

    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).

    WORK    (workspace/output) DOUBLE PRECISION array, dimension (4*N)
            On exit, WORK(1) contains the reciprocal pivot growth
            factor norm(A)/norm(U). The "max absolute element" norm is
            used. If WORK(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 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.