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  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.

  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  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.

  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  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.

  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  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.

Author

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