Provided by: liblapack-doc_3.3.1-1_all bug

NAME

       LAPACK-3  -  uses  the LU factorization to compute the solution to a real system of linear
       equations  A * X = B,

SYNOPSIS

       SUBROUTINE DGESVX( FACT, TRANS, N, NRHS, A, LDA, AF, LDAF, IPIV, EQUED, R, C, B,  LDB,  X,
                          LDX, RCOND, FERR, BERR, WORK, IWORK, INFO )

           CHARACTER      EQUED, FACT, TRANS

           INTEGER        INFO, LDA, LDAF, LDB, LDX, N, NRHS

           DOUBLE         PRECISION RCOND

           INTEGER        IPIV( * ), IWORK( * )

           DOUBLE         PRECISION  A(  LDA, * ), AF( LDAF, * ), B( LDB, * ), BERR( * ), C( * ),
                          FERR( * ), R( * ), WORK( * ), X( LDX, * )

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<INFO<=N, then
                WORK(1) contains the reciprocal pivot growth factor for the
                leading INFO columns of A.

        IWORK   (workspace) INTEGER array, dimension (N)

        INFO    (output) 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.

 LAPACK driver routine (version 3.2)        April 2011                            DGESVX(3lapack)