gesvx(3)

NAME
gesvx - gesvx: factor and solve, expert

SYNOPSIS

  Functions
subroutine cgesvx (fact, trans, n, nrhs, a, lda, af, ldaf, ipiv, equed, r, c, b, ldb, x, ldx, rcond, ferr, berr, work, rwork, info)
CGESVX computes the solution to system of linear equations A * X = B for GE matrices
subroutine dgesvx (fact, trans, n, nrhs, a, lda, af, ldaf, ipiv, equed, r, c, b, ldb, x, ldx, rcond, ferr, berr, work, iwork, info)
DGESVX computes the solution to system of linear equations A * X = B for GE matrices
subroutine sgesvx (fact, trans, n, nrhs, a, lda, af, ldaf, ipiv, equed, r, c, b, ldb, x, ldx, rcond, ferr, berr, work, iwork, info)
SGESVX computes the solution to system of linear equations A * X = B for GE matrices
subroutine zgesvx (fact, trans, n, nrhs, a, lda, af, ldaf, ipiv, equed, r, c, b, ldb, x, ldx, rcond, ferr, berr, work, rwork, info)
ZGESVX computes the solution to system of linear equations A * X = B for GE matrices

Detailed Description

Function Documentation

  subroutine cgesvx (character fact, character trans, integer n, integer nrhs, complex, dimension( lda, * ) a, integer lda, complex, dimension( ldaf, * ) af, integer ldaf, integer, dimension( * ) ipiv, character equed, real, dimension( * ) r, real, dimension( * ) c, complex, dimension( ldb, * ) b, integer ldb, complex, dimension( ldx, * ) x, integer ldx, real rcond, real, dimension( * ) ferr, real, dimension( * ) berr, complex, dimension( * ) work, real, dimension( * ) rwork, integer info)

CGESVX computes the solution to system of linear equations A * X = B for GE matrices

Purpose:

     CGESVX uses the LU factorization to compute the solution to a complex
     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.

Parameters
    FACT

              FACT is 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

              TRANS is 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  (Conjugate transpose)

    N

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

    NRHS

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

    A

              A is COMPLEX 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

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

    AF

              AF is COMPLEX 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 CGETRF.  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

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

    IPIV

              IPIV is 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 CGETRF; 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

              EQUED is 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

              R is REAL 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

              C is REAL 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

              B is COMPLEX 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

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

    X

              X is COMPLEX 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

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

    RCOND

              RCOND is REAL
              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

              FERR is REAL 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

              BERR is REAL 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

              WORK is COMPLEX array, dimension (2*N)

    RWORK

              RWORK is REAL array, dimension (MAX(1,2*N))
              On exit, RWORK(1) contains the reciprocal pivot growth
              factor norm(A)/norm(U). The 'max absolute element' norm is
              used. If RWORK(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
              RWORK(1) contains the reciprocal pivot growth factor for the
              leading INFO columns of A.

    INFO

              INFO is 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.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 347 of file cgesvx.f.

  subroutine dgesvx (character fact, character trans, integer n, integer nrhs, double precision, dimension( lda, * ) a, integer lda, double precision, dimension( ldaf, * ) af, integer ldaf, integer, dimension( * ) ipiv, character equed, double precision, dimension( * ) r, double precision, dimension( * ) c, double precision, dimension( ldb, * ) b, integer ldb, double precision, dimension( ldx, * ) x, integer ldx, double precision rcond, double precision, dimension( * ) ferr, double precision, dimension( * ) berr, double precision, dimension( * ) work, integer, dimension( * ) iwork, integer info)

DGESVX computes the solution to system of linear equations A * X = B for GE matrices

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.

Parameters
    FACT

              FACT is 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

              TRANS is 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

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

    NRHS

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

    A

              A is 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

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

    AF

              AF is 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

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

    IPIV

              IPIV is 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

              EQUED is 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

              R is 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

              C is 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

              B is 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

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

    X

              X is 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

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

    RCOND

              RCOND is 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

              FERR is 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

              BERR is 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

              WORK is DOUBLE PRECISION array, dimension (MAX(1,4*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

              IWORK is INTEGER array, dimension (N)

    INFO

              INFO is 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.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 346 of file dgesvx.f.

  subroutine sgesvx (character fact, character trans, integer n, integer nrhs, real, dimension( lda, * ) a, integer lda, real, dimension( ldaf, * ) af, integer ldaf, integer, dimension( * ) ipiv, character equed, real, dimension( * ) r, real, dimension( * ) c, real, dimension( ldb, * ) b, integer ldb, real, dimension( ldx, * ) x, integer ldx, real rcond, real, dimension( * ) ferr, real, dimension( * ) berr, real, dimension( * ) work, integer, dimension( * ) iwork, integer info)

SGESVX computes the solution to system of linear equations A * X = B for GE matrices

Purpose:

     SGESVX 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.

Parameters
    FACT

              FACT is 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

              TRANS is 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

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

    NRHS

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

    A

              A is REAL 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

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

    AF

              AF is REAL 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 SGETRF.  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

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

    IPIV

              IPIV is 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 SGETRF; 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

              EQUED is 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

              R is REAL 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

              C is REAL 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

              B is REAL 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

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

    X

              X is REAL 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

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

    RCOND

              RCOND is REAL
              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

              FERR is REAL 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

              BERR is REAL 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

              WORK is REAL array, dimension (MAX(1,4*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

              IWORK is INTEGER array, dimension (N)

    INFO

              INFO is 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.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 346 of file sgesvx.f.

  subroutine zgesvx (character fact, character trans, integer n, integer nrhs, complex*16, dimension( lda, * ) a, integer lda, complex*16, dimension( ldaf, * ) af, integer ldaf, integer, dimension( * ) ipiv, character equed, double precision, dimension( * ) r, double precision, dimension( * ) c, complex*16, dimension( ldb, * ) b, integer ldb, complex*16, dimension( ldx, * ) x, integer ldx, double precision rcond, double precision, dimension( * ) ferr, double precision, dimension( * ) berr, complex*16, dimension( * ) work, double precision, dimension( * ) rwork, integer info)

ZGESVX computes the solution to system of linear equations A * X = B for GE matrices

Purpose:

     ZGESVX uses the LU factorization to compute the solution to a complex
     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.

Parameters
    FACT

              FACT is 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

              TRANS is 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  (Conjugate transpose)

    N

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

    NRHS

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

    A

              A is COMPLEX*16 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

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

    AF

              AF is COMPLEX*16 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 ZGETRF.  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

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

    IPIV

              IPIV is 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 ZGETRF; 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

              EQUED is 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

              R is 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

              C is 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

              B is COMPLEX*16 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

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

    X

              X is COMPLEX*16 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

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

    RCOND

              RCOND is 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

              FERR is 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

              BERR is 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

              WORK is COMPLEX*16 array, dimension (2*N)

    RWORK

              RWORK is DOUBLE PRECISION array, dimension (MAX(1,2*N))
              On exit, RWORK(1) contains the reciprocal pivot growth
              factor norm(A)/norm(U). The 'max absolute element' norm is
              used. If RWORK(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
              RWORK(1) contains the reciprocal pivot growth factor for the
              leading INFO columns of A.

    INFO

              INFO is 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.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 347 of file zgesvx.f.

Author

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