gbsvxx(3)

NAME
gbsvxx - gbsvxx: factor and solve, extra precise

SYNOPSIS

  Functions
subroutine cgbsvxx (fact, trans, n, kl, ku, nrhs, ab, ldab, afb, ldafb, ipiv, equed, r, c, b, ldb, x, ldx, rcond, rpvgrw, berr, n_err_bnds, err_bnds_norm, err_bnds_comp, nparams, params, work, rwork, info)
CGBSVXX computes the solution to system of linear equations A * X = B for GB matrices
subroutine dgbsvxx (fact, trans, n, kl, ku, nrhs, ab, ldab, afb, ldafb, ipiv, equed, r, c, b, ldb, x, ldx, rcond, rpvgrw, berr, n_err_bnds, err_bnds_norm, err_bnds_comp, nparams, params, work, iwork, info)
DGBSVXX computes the solution to system of linear equations A * X = B for GB matrices
subroutine sgbsvxx (fact, trans, n, kl, ku, nrhs, ab, ldab, afb, ldafb, ipiv, equed, r, c, b, ldb, x, ldx, rcond, rpvgrw, berr, n_err_bnds, err_bnds_norm, err_bnds_comp, nparams, params, work, iwork, info)
SGBSVXX computes the solution to system of linear equations A * X = B for GB matrices
subroutine zgbsvxx (fact, trans, n, kl, ku, nrhs, ab, ldab, afb, ldafb, ipiv, equed, r, c, b, ldb, x, ldx, rcond, rpvgrw, berr, n_err_bnds, err_bnds_norm, err_bnds_comp, nparams, params, work, rwork, info)
ZGBSVXX computes the solution to system of linear equations A * X = B for GB matrices

Detailed Description

Function Documentation

  subroutine cgbsvxx (character fact, character trans, integer n, integer kl, integer ku, integer nrhs, complex, dimension( ldab, * ) ab, integer ldab, complex, dimension( ldafb, * ) afb, integer ldafb, 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 rpvgrw, real, dimension( * ) berr, integer n_err_bnds, real, dimension( nrhs, * ) err_bnds_norm, real, dimension( nrhs, * ) err_bnds_comp, integer nparams, real, dimension( * ) params, complex, dimension( * ) work, real, dimension( * ) rwork, integer info)

CGBSVXX computes the solution to system of linear equations A * X = B for GB matrices

Purpose:

    !>
    !>    CGBSVXX 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.
    !>
    !>    If requested, both normwise and maximum componentwise error bounds
    !>    are returned. CGBSVXX will return a solution with a tiny
    !>    guaranteed error (O(eps) where eps is the working machine
    !>    precision) unless the matrix is very ill-conditioned, in which
    !>    case a warning is returned. Relevant condition numbers also are
    !>    calculated and returned.
    !>
    !>    CGBSVXX accepts user-provided factorizations and equilibration
    !>    factors; see the definitions of the FACT and EQUED options.
    !>    Solving with refinement and using a factorization from a previous
    !>    CGBSVXX call will also produce a solution with either O(eps)
    !>    errors or warnings, but we cannot make that claim for general
    !>    user-provided factorizations and equilibration factors if they
    !>    differ from what CGBSVXX would itself produce.
    !>

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 (see
    !>    argument RCOND). If the reciprocal of the condition number is less
    !>    than machine precision, 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. By default (unless PARAMS(LA_LINRX_ITREF_I) is set to zero),
    !>    the routine will use iterative refinement to try to get a small
    !>    error and error bounds.  Refinement calculates the residual to at
    !>    least twice the working precision.
    !>
    !>    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.
    !>

    !>     Some optional parameters are bundled in the PARAMS array.  These
    !>     settings determine how refinement is performed, but often the
    !>     defaults are acceptable.  If the defaults are acceptable, users
    !>     can pass NPARAMS = 0 which prevents the source code from accessing
    !>     the PARAMS argument.
    !>

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 = Transpose)
    !>

    N

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

    KL

    !>          KL is INTEGER
    !>     The number of subdiagonals within the band of A.  KL >= 0.
    !>

    KU

    !>          KU is INTEGER
    !>     The number of superdiagonals within the band of A.  KU >= 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.
    !>

    AB

    !>          AB is COMPLEX array, dimension (LDAB,N)
    !>     On entry, the matrix A in band storage, in rows 1 to KL+KU+1.
    !>     The j-th column of A is stored in the j-th column of the
    !>     array AB as follows:
    !>     AB(KU+1+i-j,j) = A(i,j) for max(1,j-KU)<=i<=min(N,j+kl)
    !>
    !>     If FACT = 'F' and EQUED is not 'N', then AB must have been
    !>     equilibrated by the scaling factors in R and/or C.  AB 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).
    !>

    LDAB

    !>          LDAB is INTEGER
    !>     The leading dimension of the array AB.  LDAB >= KL+KU+1.
    !>

    AFB

    !>          AFB is COMPLEX array, dimension (LDAFB,N)
    !>     If FACT = 'F', then AFB is an input argument and on entry
    !>     contains details of the LU factorization of the band matrix
    !>     A, as computed by CGBTRF.  U is stored as an upper triangular
    !>     band matrix with KL+KU superdiagonals in rows 1 to KL+KU+1,
    !>     and the multipliers used during the factorization are stored
    !>     in rows KL+KU+2 to 2*KL+KU+1.  If EQUED .ne. 'N', then AFB 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).
    !>

    LDAFB

    !>          LDAFB is INTEGER
    !>     The leading dimension of the array AFB.  LDAFB >= 2*KL+KU+1.
    !>

    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.
    !>     If R is output, each element of R is a power of the radix.
    !>     If R is input, each element of R should be a power of the radix
    !>     to ensure a reliable solution and error estimates. Scaling by
    !>     powers of the radix does not cause rounding errors unless the
    !>     result underflows or overflows. Rounding errors during scaling
    !>     lead to refining with a matrix that is not equivalent to the
    !>     input matrix, producing error estimates that may not be
    !>     reliable.
    !>

    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.
    !>     If C is output, each element of C is a power of the radix.
    !>     If C is input, each element of C should be a power of the radix
    !>     to ensure a reliable solution and error estimates. Scaling by
    !>     powers of the radix does not cause rounding errors unless the
    !>     result underflows or overflows. Rounding errors during scaling
    !>     lead to refining with a matrix that is not equivalent to the
    !>     input matrix, producing error estimates that may not be
    !>     reliable.
    !>

    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, 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
    !>     Reciprocal scaled condition number.  This is an estimate of the
    !>     reciprocal Skeel condition number of the matrix A after
    !>     equilibration (if done).  If this is less than the machine
    !>     precision (in particular, if it is zero), the matrix is singular
    !>     to working precision.  Note that the error may still be small even
    !>     if this number is very small and the matrix appears ill-
    !>     conditioned.
    !>

    RPVGRW

    !>          RPVGRW is REAL
    !>     Reciprocal pivot growth.  On exit, this contains the reciprocal
    !>     pivot growth factor norm(A)/norm(U). The
    !>     norm is used.  If this 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, estimated condition numbers,
    !>     and error bounds could be unreliable. If factorization fails with
    !>     0<INFO<=N, then this contains the reciprocal pivot growth factor
    !>     for the leading INFO columns of A.  In CGESVX, this quantity is
    !>     returned in RWORK(1).
    !>

    BERR

    !>          BERR is REAL array, dimension (NRHS)
    !>     Componentwise relative backward error.  This is 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).
    !>

    N_ERR_BNDS

    !>          N_ERR_BNDS is INTEGER
    !>     Number of error bounds to return for each right hand side
    !>     and each type (normwise or componentwise).  See ERR_BNDS_NORM and
    !>     ERR_BNDS_COMP below.
    !>

    ERR_BNDS_NORM

    !>          ERR_BNDS_NORM is REAL array, dimension (NRHS, N_ERR_BNDS)
    !>     For each right-hand side, this array contains information about
    !>     various error bounds and condition numbers corresponding to the
    !>     normwise relative error, which is defined as follows:
    !>
    !>     Normwise relative error in the ith solution vector:
    !>             max_j (abs(XTRUE(j,i) - X(j,i)))
    !>            ------------------------------
    !>                  max_j abs(X(j,i))
    !>
    !>     The array is indexed by the type of error information as described
    !>     below. There currently are up to three pieces of information
    !>     returned.
    !>
    !>     The first index in ERR_BNDS_NORM(i,:) corresponds to the ith
    !>     right-hand side.
    !>
    !>     The second index in ERR_BNDS_NORM(:,err) contains the following
    !>     three fields:
    !>     err = 1  boolean. Trust the answer if the
    !>              reciprocal condition number is less than the threshold
    !>              sqrt(n) * slamch('Epsilon').
    !>
    !>     err = 2  error bound: The estimated forward error,
    !>              almost certainly within a factor of 10 of the true error
    !>              so long as the next entry is greater than the threshold
    !>              sqrt(n) * slamch('Epsilon'). This error bound should only
    !>              be trusted if the previous boolean is true.
    !>
    !>     err = 3  Reciprocal condition number: Estimated normwise
    !>              reciprocal condition number.  Compared with the threshold
    !>              sqrt(n) * slamch('Epsilon') to determine if the error
    !>              estimate is . These reciprocal condition
    !>              numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some
    !>              appropriately scaled matrix Z.
    !>              Let Z = S*A, where S scales each row by a power of the
    !>              radix so all absolute row sums of Z are approximately 1.
    !>
    !>     See Lapack Working Note 165 for further details and extra
    !>     cautions.
    !>

    ERR_BNDS_COMP

    !>          ERR_BNDS_COMP is REAL array, dimension (NRHS, N_ERR_BNDS)
    !>     For each right-hand side, this array contains information about
    !>     various error bounds and condition numbers corresponding to the
    !>     componentwise relative error, which is defined as follows:
    !>
    !>     Componentwise relative error in the ith solution vector:
    !>                    abs(XTRUE(j,i) - X(j,i))
    !>             max_j ----------------------
    !>                         abs(X(j,i))
    !>
    !>     The array is indexed by the right-hand side i (on which the
    !>     componentwise relative error depends), and the type of error
    !>     information as described below. There currently are up to three
    !>     pieces of information returned for each right-hand side. If
    !>     componentwise accuracy is not requested (PARAMS(3) = 0.0), then
    !>     ERR_BNDS_COMP is not accessed.  If N_ERR_BNDS < 3, then at most
    !>     the first (:,N_ERR_BNDS) entries are returned.
    !>
    !>     The first index in ERR_BNDS_COMP(i,:) corresponds to the ith
    !>     right-hand side.
    !>
    !>     The second index in ERR_BNDS_COMP(:,err) contains the following
    !>     three fields:
    !>     err = 1  boolean. Trust the answer if the
    !>              reciprocal condition number is less than the threshold
    !>              sqrt(n) * slamch('Epsilon').
    !>
    !>     err = 2  error bound: The estimated forward error,
    !>              almost certainly within a factor of 10 of the true error
    !>              so long as the next entry is greater than the threshold
    !>              sqrt(n) * slamch('Epsilon'). This error bound should only
    !>              be trusted if the previous boolean is true.
    !>
    !>     err = 3  Reciprocal condition number: Estimated componentwise
    !>              reciprocal condition number.  Compared with the threshold
    !>              sqrt(n) * slamch('Epsilon') to determine if the error
    !>              estimate is . These reciprocal condition
    !>              numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some
    !>              appropriately scaled matrix Z.
    !>              Let Z = S*(A*diag(x)), where x is the solution for the
    !>              current right-hand side and S scales each row of
    !>              A*diag(x) by a power of the radix so all absolute row
    !>              sums of Z are approximately 1.
    !>
    !>     See Lapack Working Note 165 for further details and extra
    !>     cautions.
    !>

    NPARAMS

    !>          NPARAMS is INTEGER
    !>     Specifies the number of parameters set in PARAMS.  If <= 0, the
    !>     PARAMS array is never referenced and default values are used.
    !>

    PARAMS

    !>          PARAMS is REAL array, dimension NPARAMS
    !>     Specifies algorithm parameters.  If an entry is < 0.0, then
    !>     that entry will be filled with default value used for that
    !>     parameter.  Only positions up to NPARAMS are accessed; defaults
    !>     are used for higher-numbered parameters.
    !>
    !>       PARAMS(LA_LINRX_ITREF_I = 1) : Whether to perform iterative
    !>            refinement or not.
    !>         Default: 1.0
    !>            = 0.0:  No refinement is performed, and no error bounds are
    !>                    computed.
    !>            = 1.0:  Use the double-precision refinement algorithm,
    !>                    possibly with doubled-single computations if the
    !>                    compilation environment does not support DOUBLE
    !>                    PRECISION.
    !>              (other values are reserved for future use)
    !>
    !>       PARAMS(LA_LINRX_ITHRESH_I = 2) : Maximum number of residual
    !>            computations allowed for refinement.
    !>         Default: 10
    !>         Aggressive: Set to 100 to permit convergence using approximate
    !>                     factorizations or factorizations other than LU. If
    !>                     the factorization uses a technique other than
    !>                     Gaussian elimination, the guarantees in
    !>                     err_bnds_norm and err_bnds_comp may no longer be
    !>                     trustworthy.
    !>
    !>       PARAMS(LA_LINRX_CWISE_I = 3) : Flag determining if the code
    !>            will attempt to find a solution with small componentwise
    !>            relative error in the double-precision algorithm.  Positive
    !>            is true, 0.0 is false.
    !>         Default: 1.0 (attempt componentwise convergence)
    !>

    WORK

    !>          WORK is COMPLEX array, dimension (2*N)
    !>

    RWORK

    !>          RWORK is REAL array, dimension (2*N)
    !>

    INFO

    !>          INFO is INTEGER
    !>       = 0:  Successful exit. The solution to every right-hand side is
    !>         guaranteed.
    !>       < 0:  If INFO = -i, the i-th argument had an illegal value
    !>       > 0 and <= N:  U(INFO,INFO) 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+J: The solution corresponding to the Jth right-hand side is
    !>         not guaranteed. The solutions corresponding to other right-
    !>         hand sides K with K > J may not be guaranteed as well, but
    !>         only the first such right-hand side is reported. If a small
    !>         componentwise error is not requested (PARAMS(3) = 0.0) then
    !>         the Jth right-hand side is the first with a normwise error
    !>         bound that is not guaranteed (the smallest J such
    !>         that ERR_BNDS_NORM(J,1) = 0.0). By default (PARAMS(3) = 1.0)
    !>         the Jth right-hand side is the first with either a normwise or
    !>         componentwise error bound that is not guaranteed (the smallest
    !>         J such that either ERR_BNDS_NORM(J,1) = 0.0 or
    !>         ERR_BNDS_COMP(J,1) = 0.0). See the definition of
    !>         ERR_BNDS_NORM(:,1) and ERR_BNDS_COMP(:,1). To get information
    !>         about all of the right-hand sides check ERR_BNDS_NORM or
    !>         ERR_BNDS_COMP.
    !>

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 558 of file cgbsvxx.f.

  subroutine dgbsvxx (character fact, character trans, integer n, integer kl, integer ku, integer nrhs, double precision, dimension( ldab, * ) ab, integer ldab, double precision, dimension( ldafb, * ) afb, integer ldafb, 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 rpvgrw, double precision, dimension( * ) berr, integer n_err_bnds, double precision, dimension( nrhs, * ) err_bnds_norm, double precision, dimension( nrhs, * ) err_bnds_comp, integer nparams, double precision, dimension( * ) params, double precision, dimension( * ) work, integer, dimension( * ) iwork, integer info)

DGBSVXX computes the solution to system of linear equations A * X = B for GB matrices

Purpose:

    !>
    !>    DGBSVXX uses the LU factorization to compute the solution to a
    !>    double precision system of linear equations  A * X = B,  where A is an
    !>    N-by-N matrix and X and B are N-by-NRHS matrices.
    !>
    !>    If requested, both normwise and maximum componentwise error bounds
    !>    are returned. DGBSVXX will return a solution with a tiny
    !>    guaranteed error (O(eps) where eps is the working machine
    !>    precision) unless the matrix is very ill-conditioned, in which
    !>    case a warning is returned. Relevant condition numbers also are
    !>    calculated and returned.
    !>
    !>    DGBSVXX accepts user-provided factorizations and equilibration
    !>    factors; see the definitions of the FACT and EQUED options.
    !>    Solving with refinement and using a factorization from a previous
    !>    DGBSVXX call will also produce a solution with either O(eps)
    !>    errors or warnings, but we cannot make that claim for general
    !>    user-provided factorizations and equilibration factors if they
    !>    differ from what DGBSVXX would itself produce.
    !>

Description:

    !>
    !>    The following steps are performed:
    !>
    !>    1. If FACT = 'E', double precision 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 (see
    !>    argument RCOND). If the reciprocal of the condition number is less
    !>    than machine precision, 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. By default (unless PARAMS(LA_LINRX_ITREF_I) is set to zero),
    !>    the routine will use iterative refinement to try to get a small
    !>    error and error bounds.  Refinement calculates the residual to at
    !>    least twice the working precision.
    !>
    !>    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.
    !>

    !>     Some optional parameters are bundled in the PARAMS array.  These
    !>     settings determine how refinement is performed, but often the
    !>     defaults are acceptable.  If the defaults are acceptable, users
    !>     can pass NPARAMS = 0 which prevents the source code from accessing
    !>     the PARAMS argument.
    !>

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 = Transpose)
    !>

    N

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

    KL

    !>          KL is INTEGER
    !>     The number of subdiagonals within the band of A.  KL >= 0.
    !>

    KU

    !>          KU is INTEGER
    !>     The number of superdiagonals within the band of A.  KU >= 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.
    !>

    AB

    !>          AB is DOUBLE PRECISION array, dimension (LDAB,N)
    !>     On entry, the matrix A in band storage, in rows 1 to KL+KU+1.
    !>     The j-th column of A is stored in the j-th column of the
    !>     array AB as follows:
    !>     AB(KU+1+i-j,j) = A(i,j) for max(1,j-KU)<=i<=min(N,j+kl)
    !>
    !>     If FACT = 'F' and EQUED is not 'N', then AB must have been
    !>     equilibrated by the scaling factors in R and/or C.  AB 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).
    !>

    LDAB

    !>          LDAB is INTEGER
    !>     The leading dimension of the array AB.  LDAB >= KL+KU+1.
    !>

    AFB

    !>          AFB is DOUBLE PRECISION array, dimension (LDAFB,N)
    !>     If FACT = 'F', then AFB is an input argument and on entry
    !>     contains details of the LU factorization of the band matrix
    !>     A, as computed by DGBTRF.  U is stored as an upper triangular
    !>     band matrix with KL+KU superdiagonals in rows 1 to KL+KU+1,
    !>     and the multipliers used during the factorization are stored
    !>     in rows KL+KU+2 to 2*KL+KU+1.  If EQUED .ne. 'N', then AFB 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).
    !>

    LDAFB

    !>          LDAFB is INTEGER
    !>     The leading dimension of the array AFB.  LDAFB >= 2*KL+KU+1.
    !>

    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.
    !>     If R is output, each element of R is a power of the radix.
    !>     If R is input, each element of R should be a power of the radix
    !>     to ensure a reliable solution and error estimates. Scaling by
    !>     powers of the radix does not cause rounding errors unless the
    !>     result underflows or overflows. Rounding errors during scaling
    !>     lead to refining with a matrix that is not equivalent to the
    !>     input matrix, producing error estimates that may not be
    !>     reliable.
    !>

    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.
    !>     If C is output, each element of C is a power of the radix.
    !>     If C is input, each element of C should be a power of the radix
    !>     to ensure a reliable solution and error estimates. Scaling by
    !>     powers of the radix does not cause rounding errors unless the
    !>     result underflows or overflows. Rounding errors during scaling
    !>     lead to refining with a matrix that is not equivalent to the
    !>     input matrix, producing error estimates that may not be
    !>     reliable.
    !>

    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, 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
    !>     Reciprocal scaled condition number.  This is an estimate of the
    !>     reciprocal Skeel condition number of the matrix A after
    !>     equilibration (if done).  If this is less than the machine
    !>     precision (in particular, if it is zero), the matrix is singular
    !>     to working precision.  Note that the error may still be small even
    !>     if this number is very small and the matrix appears ill-
    !>     conditioned.
    !>

    RPVGRW

    !>          RPVGRW is DOUBLE PRECISION
    !>     Reciprocal pivot growth.  On exit, this contains the reciprocal
    !>     pivot growth factor norm(A)/norm(U). The
    !>     norm is used.  If this 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, estimated condition numbers,
    !>     and error bounds could be unreliable. If factorization fails with
    !>     0<INFO<=N, then this contains the reciprocal pivot growth factor
    !>     for the leading INFO columns of A.  In DGESVX, this quantity is
    !>     returned in WORK(1).
    !>

    BERR

    !>          BERR is DOUBLE PRECISION array, dimension (NRHS)
    !>     Componentwise relative backward error.  This is 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).
    !>

    N_ERR_BNDS

    !>          N_ERR_BNDS is INTEGER
    !>     Number of error bounds to return for each right hand side
    !>     and each type (normwise or componentwise).  See ERR_BNDS_NORM and
    !>     ERR_BNDS_COMP below.
    !>

    ERR_BNDS_NORM

    !>          ERR_BNDS_NORM is DOUBLE PRECISION array, dimension (NRHS, N_ERR_BNDS)
    !>     For each right-hand side, this array contains information about
    !>     various error bounds and condition numbers corresponding to the
    !>     normwise relative error, which is defined as follows:
    !>
    !>     Normwise relative error in the ith solution vector:
    !>             max_j (abs(XTRUE(j,i) - X(j,i)))
    !>            ------------------------------
    !>                  max_j abs(X(j,i))
    !>
    !>     The array is indexed by the type of error information as described
    !>     below. There currently are up to three pieces of information
    !>     returned.
    !>
    !>     The first index in ERR_BNDS_NORM(i,:) corresponds to the ith
    !>     right-hand side.
    !>
    !>     The second index in ERR_BNDS_NORM(:,err) contains the following
    !>     three fields:
    !>     err = 1  boolean. Trust the answer if the
    !>              reciprocal condition number is less than the threshold
    !>              sqrt(n) * dlamch('Epsilon').
    !>
    !>     err = 2  error bound: The estimated forward error,
    !>              almost certainly within a factor of 10 of the true error
    !>              so long as the next entry is greater than the threshold
    !>              sqrt(n) * dlamch('Epsilon'). This error bound should only
    !>              be trusted if the previous boolean is true.
    !>
    !>     err = 3  Reciprocal condition number: Estimated normwise
    !>              reciprocal condition number.  Compared with the threshold
    !>              sqrt(n) * dlamch('Epsilon') to determine if the error
    !>              estimate is . These reciprocal condition
    !>              numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some
    !>              appropriately scaled matrix Z.
    !>              Let Z = S*A, where S scales each row by a power of the
    !>              radix so all absolute row sums of Z are approximately 1.
    !>
    !>     See Lapack Working Note 165 for further details and extra
    !>     cautions.
    !>

    ERR_BNDS_COMP

    !>          ERR_BNDS_COMP is DOUBLE PRECISION array, dimension (NRHS, N_ERR_BNDS)
    !>     For each right-hand side, this array contains information about
    !>     various error bounds and condition numbers corresponding to the
    !>     componentwise relative error, which is defined as follows:
    !>
    !>     Componentwise relative error in the ith solution vector:
    !>                    abs(XTRUE(j,i) - X(j,i))
    !>             max_j ----------------------
    !>                         abs(X(j,i))
    !>
    !>     The array is indexed by the right-hand side i (on which the
    !>     componentwise relative error depends), and the type of error
    !>     information as described below. There currently are up to three
    !>     pieces of information returned for each right-hand side. If
    !>     componentwise accuracy is not requested (PARAMS(3) = 0.0), then
    !>     ERR_BNDS_COMP is not accessed.  If N_ERR_BNDS < 3, then at most
    !>     the first (:,N_ERR_BNDS) entries are returned.
    !>
    !>     The first index in ERR_BNDS_COMP(i,:) corresponds to the ith
    !>     right-hand side.
    !>
    !>     The second index in ERR_BNDS_COMP(:,err) contains the following
    !>     three fields:
    !>     err = 1  boolean. Trust the answer if the
    !>              reciprocal condition number is less than the threshold
    !>              sqrt(n) * dlamch('Epsilon').
    !>
    !>     err = 2  error bound: The estimated forward error,
    !>              almost certainly within a factor of 10 of the true error
    !>              so long as the next entry is greater than the threshold
    !>              sqrt(n) * dlamch('Epsilon'). This error bound should only
    !>              be trusted if the previous boolean is true.
    !>
    !>     err = 3  Reciprocal condition number: Estimated componentwise
    !>              reciprocal condition number.  Compared with the threshold
    !>              sqrt(n) * dlamch('Epsilon') to determine if the error
    !>              estimate is . These reciprocal condition
    !>              numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some
    !>              appropriately scaled matrix Z.
    !>              Let Z = S*(A*diag(x)), where x is the solution for the
    !>              current right-hand side and S scales each row of
    !>              A*diag(x) by a power of the radix so all absolute row
    !>              sums of Z are approximately 1.
    !>
    !>     See Lapack Working Note 165 for further details and extra
    !>     cautions.
    !>

    NPARAMS

    !>          NPARAMS is INTEGER
    !>     Specifies the number of parameters set in PARAMS.  If <= 0, the
    !>     PARAMS array is never referenced and default values are used.
    !>

    PARAMS

    !>          PARAMS is DOUBLE PRECISION array, dimension (NPARAMS)
    !>     Specifies algorithm parameters.  If an entry is < 0.0, then
    !>     that entry will be filled with default value used for that
    !>     parameter.  Only positions up to NPARAMS are accessed; defaults
    !>     are used for higher-numbered parameters.
    !>
    !>       PARAMS(LA_LINRX_ITREF_I = 1) : Whether to perform iterative
    !>            refinement or not.
    !>         Default: 1.0D+0
    !>            = 0.0:  No refinement is performed, and no error bounds are
    !>                    computed.
    !>            = 1.0:  Use the extra-precise refinement algorithm.
    !>              (other values are reserved for future use)
    !>
    !>       PARAMS(LA_LINRX_ITHRESH_I = 2) : Maximum number of residual
    !>            computations allowed for refinement.
    !>         Default: 10
    !>         Aggressive: Set to 100 to permit convergence using approximate
    !>                     factorizations or factorizations other than LU. If
    !>                     the factorization uses a technique other than
    !>                     Gaussian elimination, the guarantees in
    !>                     err_bnds_norm and err_bnds_comp may no longer be
    !>                     trustworthy.
    !>
    !>       PARAMS(LA_LINRX_CWISE_I = 3) : Flag determining if the code
    !>            will attempt to find a solution with small componentwise
    !>            relative error in the double-precision algorithm.  Positive
    !>            is true, 0.0 is false.
    !>         Default: 1.0 (attempt componentwise convergence)
    !>

    WORK

    !>          WORK is DOUBLE PRECISION array, dimension (4*N)
    !>

    IWORK

    !>          IWORK is INTEGER array, dimension (N)
    !>

    INFO

    !>          INFO is INTEGER
    !>       = 0:  Successful exit. The solution to every right-hand side is
    !>         guaranteed.
    !>       < 0:  If INFO = -i, the i-th argument had an illegal value
    !>       > 0 and <= N:  U(INFO,INFO) 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+J: The solution corresponding to the Jth right-hand side is
    !>         not guaranteed. The solutions corresponding to other right-
    !>         hand sides K with K > J may not be guaranteed as well, but
    !>         only the first such right-hand side is reported. If a small
    !>         componentwise error is not requested (PARAMS(3) = 0.0) then
    !>         the Jth right-hand side is the first with a normwise error
    !>         bound that is not guaranteed (the smallest J such
    !>         that ERR_BNDS_NORM(J,1) = 0.0). By default (PARAMS(3) = 1.0)
    !>         the Jth right-hand side is the first with either a normwise or
    !>         componentwise error bound that is not guaranteed (the smallest
    !>         J such that either ERR_BNDS_NORM(J,1) = 0.0 or
    !>         ERR_BNDS_COMP(J,1) = 0.0). See the definition of
    !>         ERR_BNDS_NORM(:,1) and ERR_BNDS_COMP(:,1). To get information
    !>         about all of the right-hand sides check ERR_BNDS_NORM or
    !>         ERR_BNDS_COMP.
    !>

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 555 of file dgbsvxx.f.

  subroutine sgbsvxx (character fact, character trans, integer n, integer kl, integer ku, integer nrhs, real, dimension( ldab, * ) ab, integer ldab, real, dimension( ldafb, * ) afb, integer ldafb, 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 rpvgrw, real, dimension( * ) berr, integer n_err_bnds, real, dimension( nrhs, * ) err_bnds_norm, real, dimension( nrhs, * ) err_bnds_comp, integer nparams, real, dimension( * ) params, real, dimension( * ) work, integer, dimension( * ) iwork, integer info)

SGBSVXX computes the solution to system of linear equations A * X = B for GB matrices

Purpose:

    !>
    !>    SGBSVXX 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.
    !>
    !>    If requested, both normwise and maximum componentwise error bounds
    !>    are returned. SGBSVXX will return a solution with a tiny
    !>    guaranteed error (O(eps) where eps is the working machine
    !>    precision) unless the matrix is very ill-conditioned, in which
    !>    case a warning is returned. Relevant condition numbers also are
    !>    calculated and returned.
    !>
    !>    SGBSVXX accepts user-provided factorizations and equilibration
    !>    factors; see the definitions of the FACT and EQUED options.
    !>    Solving with refinement and using a factorization from a previous
    !>    SGBSVXX call will also produce a solution with either O(eps)
    !>    errors or warnings, but we cannot make that claim for general
    !>    user-provided factorizations and equilibration factors if they
    !>    differ from what SGBSVXX would itself produce.
    !>

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 (see
    !>    argument RCOND). If the reciprocal of the condition number is less
    !>    than machine precision, 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. By default (unless PARAMS(LA_LINRX_ITREF_I) is set to zero),
    !>    the routine will use iterative refinement to try to get a small
    !>    error and error bounds.  Refinement calculates the residual to at
    !>    least twice the working precision.
    !>
    !>    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.
    !>

    !>     Some optional parameters are bundled in the PARAMS array.  These
    !>     settings determine how refinement is performed, but often the
    !>     defaults are acceptable.  If the defaults are acceptable, users
    !>     can pass NPARAMS = 0 which prevents the source code from accessing
    !>     the PARAMS argument.
    !>

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 = Transpose)
    !>

    N

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

    KL

    !>          KL is INTEGER
    !>     The number of subdiagonals within the band of A.  KL >= 0.
    !>

    KU

    !>          KU is INTEGER
    !>     The number of superdiagonals within the band of A.  KU >= 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.
    !>

    AB

    !>          AB is REAL array, dimension (LDAB,N)
    !>     On entry, the matrix A in band storage, in rows 1 to KL+KU+1.
    !>     The j-th column of A is stored in the j-th column of the
    !>     array AB as follows:
    !>     AB(KU+1+i-j,j) = A(i,j) for max(1,j-KU)<=i<=min(N,j+kl)
    !>
    !>     If FACT = 'F' and EQUED is not 'N', then AB must have been
    !>     equilibrated by the scaling factors in R and/or C.  AB 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).
    !>

    LDAB

    !>          LDAB is INTEGER
    !>     The leading dimension of the array AB.  LDAB >= KL+KU+1.
    !>

    AFB

    !>          AFB is REAL array, dimension (LDAFB,N)
    !>     If FACT = 'F', then AFB is an input argument and on entry
    !>     contains details of the LU factorization of the band matrix
    !>     A, as computed by SGBTRF.  U is stored as an upper triangular
    !>     band matrix with KL+KU superdiagonals in rows 1 to KL+KU+1,
    !>     and the multipliers used during the factorization are stored
    !>     in rows KL+KU+2 to 2*KL+KU+1.  If EQUED .ne. 'N', then AFB 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).
    !>

    LDAFB

    !>          LDAFB is INTEGER
    !>     The leading dimension of the array AFB.  LDAFB >= 2*KL+KU+1.
    !>

    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.
    !>     If R is output, each element of R is a power of the radix.
    !>     If R is input, each element of R should be a power of the radix
    !>     to ensure a reliable solution and error estimates. Scaling by
    !>     powers of the radix does not cause rounding errors unless the
    !>     result underflows or overflows. Rounding errors during scaling
    !>     lead to refining with a matrix that is not equivalent to the
    !>     input matrix, producing error estimates that may not be
    !>     reliable.
    !>

    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.
    !>     If C is output, each element of C is a power of the radix.
    !>     If C is input, each element of C should be a power of the radix
    !>     to ensure a reliable solution and error estimates. Scaling by
    !>     powers of the radix does not cause rounding errors unless the
    !>     result underflows or overflows. Rounding errors during scaling
    !>     lead to refining with a matrix that is not equivalent to the
    !>     input matrix, producing error estimates that may not be
    !>     reliable.
    !>

    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, 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
    !>     Reciprocal scaled condition number.  This is an estimate of the
    !>     reciprocal Skeel condition number of the matrix A after
    !>     equilibration (if done).  If this is less than the machine
    !>     precision (in particular, if it is zero), the matrix is singular
    !>     to working precision.  Note that the error may still be small even
    !>     if this number is very small and the matrix appears ill-
    !>     conditioned.
    !>

    RPVGRW

    !>          RPVGRW is REAL
    !>     Reciprocal pivot growth.  On exit, this contains the reciprocal
    !>     pivot growth factor norm(A)/norm(U). The
    !>     norm is used.  If this 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, estimated condition numbers,
    !>     and error bounds could be unreliable. If factorization fails with
    !>     0<INFO<=N, then this contains the reciprocal pivot growth factor
    !>     for the leading INFO columns of A.  In SGESVX, this quantity is
    !>     returned in WORK(1).
    !>

    BERR

    !>          BERR is REAL array, dimension (NRHS)
    !>     Componentwise relative backward error.  This is 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).
    !>

    N_ERR_BNDS

    !>          N_ERR_BNDS is INTEGER
    !>     Number of error bounds to return for each right hand side
    !>     and each type (normwise or componentwise).  See ERR_BNDS_NORM and
    !>     ERR_BNDS_COMP below.
    !>

    ERR_BNDS_NORM

    !>          ERR_BNDS_NORM is REAL array, dimension (NRHS, N_ERR_BNDS)
    !>     For each right-hand side, this array contains information about
    !>     various error bounds and condition numbers corresponding to the
    !>     normwise relative error, which is defined as follows:
    !>
    !>     Normwise relative error in the ith solution vector:
    !>             max_j (abs(XTRUE(j,i) - X(j,i)))
    !>            ------------------------------
    !>                  max_j abs(X(j,i))
    !>
    !>     The array is indexed by the type of error information as described
    !>     below. There currently are up to three pieces of information
    !>     returned.
    !>
    !>     The first index in ERR_BNDS_NORM(i,:) corresponds to the ith
    !>     right-hand side.
    !>
    !>     The second index in ERR_BNDS_NORM(:,err) contains the following
    !>     three fields:
    !>     err = 1  boolean. Trust the answer if the
    !>              reciprocal condition number is less than the threshold
    !>              sqrt(n) * slamch('Epsilon').
    !>
    !>     err = 2  error bound: The estimated forward error,
    !>              almost certainly within a factor of 10 of the true error
    !>              so long as the next entry is greater than the threshold
    !>              sqrt(n) * slamch('Epsilon'). This error bound should only
    !>              be trusted if the previous boolean is true.
    !>
    !>     err = 3  Reciprocal condition number: Estimated normwise
    !>              reciprocal condition number.  Compared with the threshold
    !>              sqrt(n) * slamch('Epsilon') to determine if the error
    !>              estimate is . These reciprocal condition
    !>              numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some
    !>              appropriately scaled matrix Z.
    !>              Let Z = S*A, where S scales each row by a power of the
    !>              radix so all absolute row sums of Z are approximately 1.
    !>
    !>     See Lapack Working Note 165 for further details and extra
    !>     cautions.
    !>

    ERR_BNDS_COMP

    !>          ERR_BNDS_COMP is REAL array, dimension (NRHS, N_ERR_BNDS)
    !>     For each right-hand side, this array contains information about
    !>     various error bounds and condition numbers corresponding to the
    !>     componentwise relative error, which is defined as follows:
    !>
    !>     Componentwise relative error in the ith solution vector:
    !>                    abs(XTRUE(j,i) - X(j,i))
    !>             max_j ----------------------
    !>                         abs(X(j,i))
    !>
    !>     The array is indexed by the right-hand side i (on which the
    !>     componentwise relative error depends), and the type of error
    !>     information as described below. There currently are up to three
    !>     pieces of information returned for each right-hand side. If
    !>     componentwise accuracy is not requested (PARAMS(3) = 0.0), then
    !>     ERR_BNDS_COMP is not accessed.  If N_ERR_BNDS < 3, then at most
    !>     the first (:,N_ERR_BNDS) entries are returned.
    !>
    !>     The first index in ERR_BNDS_COMP(i,:) corresponds to the ith
    !>     right-hand side.
    !>
    !>     The second index in ERR_BNDS_COMP(:,err) contains the following
    !>     three fields:
    !>     err = 1  boolean. Trust the answer if the
    !>              reciprocal condition number is less than the threshold
    !>              sqrt(n) * slamch('Epsilon').
    !>
    !>     err = 2  error bound: The estimated forward error,
    !>              almost certainly within a factor of 10 of the true error
    !>              so long as the next entry is greater than the threshold
    !>              sqrt(n) * slamch('Epsilon'). This error bound should only
    !>              be trusted if the previous boolean is true.
    !>
    !>     err = 3  Reciprocal condition number: Estimated componentwise
    !>              reciprocal condition number.  Compared with the threshold
    !>              sqrt(n) * slamch('Epsilon') to determine if the error
    !>              estimate is . These reciprocal condition
    !>              numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some
    !>              appropriately scaled matrix Z.
    !>              Let Z = S*(A*diag(x)), where x is the solution for the
    !>              current right-hand side and S scales each row of
    !>              A*diag(x) by a power of the radix so all absolute row
    !>              sums of Z are approximately 1.
    !>
    !>     See Lapack Working Note 165 for further details and extra
    !>     cautions.
    !>

    NPARAMS

    !>          NPARAMS is INTEGER
    !>     Specifies the number of parameters set in PARAMS.  If <= 0, the
    !>     PARAMS array is never referenced and default values are used.
    !>

    PARAMS

    !>          PARAMS is REAL array, dimension NPARAMS
    !>     Specifies algorithm parameters.  If an entry is < 0.0, then
    !>     that entry will be filled with default value used for that
    !>     parameter.  Only positions up to NPARAMS are accessed; defaults
    !>     are used for higher-numbered parameters.
    !>
    !>       PARAMS(LA_LINRX_ITREF_I = 1) : Whether to perform iterative
    !>            refinement or not.
    !>         Default: 1.0
    !>            = 0.0:  No refinement is performed, and no error bounds are
    !>                    computed.
    !>            = 1.0:  Use the double-precision refinement algorithm,
    !>                    possibly with doubled-single computations if the
    !>                    compilation environment does not support DOUBLE
    !>                    PRECISION.
    !>              (other values are reserved for future use)
    !>
    !>       PARAMS(LA_LINRX_ITHRESH_I = 2) : Maximum number of residual
    !>            computations allowed for refinement.
    !>         Default: 10
    !>         Aggressive: Set to 100 to permit convergence using approximate
    !>                     factorizations or factorizations other than LU. If
    !>                     the factorization uses a technique other than
    !>                     Gaussian elimination, the guarantees in
    !>                     err_bnds_norm and err_bnds_comp may no longer be
    !>                     trustworthy.
    !>
    !>       PARAMS(LA_LINRX_CWISE_I = 3) : Flag determining if the code
    !>            will attempt to find a solution with small componentwise
    !>            relative error in the double-precision algorithm.  Positive
    !>            is true, 0.0 is false.
    !>         Default: 1.0 (attempt componentwise convergence)
    !>

    WORK

    !>          WORK is REAL array, dimension (4*N)
    !>

    IWORK

    !>          IWORK is INTEGER array, dimension (N)
    !>

    INFO

    !>          INFO is INTEGER
    !>       = 0:  Successful exit. The solution to every right-hand side is
    !>         guaranteed.
    !>       < 0:  If INFO = -i, the i-th argument had an illegal value
    !>       > 0 and <= N:  U(INFO,INFO) 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+J: The solution corresponding to the Jth right-hand side is
    !>         not guaranteed. The solutions corresponding to other right-
    !>         hand sides K with K > J may not be guaranteed as well, but
    !>         only the first such right-hand side is reported. If a small
    !>         componentwise error is not requested (PARAMS(3) = 0.0) then
    !>         the Jth right-hand side is the first with a normwise error
    !>         bound that is not guaranteed (the smallest J such
    !>         that ERR_BNDS_NORM(J,1) = 0.0). By default (PARAMS(3) = 1.0)
    !>         the Jth right-hand side is the first with either a normwise or
    !>         componentwise error bound that is not guaranteed (the smallest
    !>         J such that either ERR_BNDS_NORM(J,1) = 0.0 or
    !>         ERR_BNDS_COMP(J,1) = 0.0). See the definition of
    !>         ERR_BNDS_NORM(:,1) and ERR_BNDS_COMP(:,1). To get information
    !>         about all of the right-hand sides check ERR_BNDS_NORM or
    !>         ERR_BNDS_COMP.
    !>

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 558 of file sgbsvxx.f.

  subroutine zgbsvxx (character fact, character trans, integer n, integer kl, integer ku, integer nrhs, complex*16, dimension( ldab, * ) ab, integer ldab, complex*16, dimension( ldafb, * ) afb, integer ldafb, 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 rpvgrw, double precision, dimension( * ) berr, integer n_err_bnds, double precision, dimension( nrhs, * ) err_bnds_norm, double precision, dimension( nrhs, * ) err_bnds_comp, integer nparams, double precision, dimension( * ) params, complex*16, dimension( * ) work, double precision, dimension( * ) rwork, integer info)

ZGBSVXX computes the solution to system of linear equations A * X = B for GB matrices

Purpose:

    !>
    !>    ZGBSVXX uses the LU factorization to compute the solution to a
    !>    complex*16 system of linear equations  A * X = B,  where A is an
    !>    N-by-N matrix and X and B are N-by-NRHS matrices.
    !>
    !>    If requested, both normwise and maximum componentwise error bounds
    !>    are returned. ZGBSVXX will return a solution with a tiny
    !>    guaranteed error (O(eps) where eps is the working machine
    !>    precision) unless the matrix is very ill-conditioned, in which
    !>    case a warning is returned. Relevant condition numbers also are
    !>    calculated and returned.
    !>
    !>    ZGBSVXX accepts user-provided factorizations and equilibration
    !>    factors; see the definitions of the FACT and EQUED options.
    !>    Solving with refinement and using a factorization from a previous
    !>    ZGBSVXX call will also produce a solution with either O(eps)
    !>    errors or warnings, but we cannot make that claim for general
    !>    user-provided factorizations and equilibration factors if they
    !>    differ from what ZGBSVXX would itself produce.
    !>

Description:

    !>
    !>    The following steps are performed:
    !>
    !>    1. If FACT = 'E', double precision 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 (see
    !>    argument RCOND). If the reciprocal of the condition number is less
    !>    than machine precision, 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. By default (unless PARAMS(LA_LINRX_ITREF_I) is set to zero),
    !>    the routine will use iterative refinement to try to get a small
    !>    error and error bounds.  Refinement calculates the residual to at
    !>    least twice the working precision.
    !>
    !>    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.
    !>

    !>     Some optional parameters are bundled in the PARAMS array.  These
    !>     settings determine how refinement is performed, but often the
    !>     defaults are acceptable.  If the defaults are acceptable, users
    !>     can pass NPARAMS = 0 which prevents the source code from accessing
    !>     the PARAMS argument.
    !>

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 = Transpose)
    !>

    N

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

    KL

    !>          KL is INTEGER
    !>     The number of subdiagonals within the band of A.  KL >= 0.
    !>

    KU

    !>          KU is INTEGER
    !>     The number of superdiagonals within the band of A.  KU >= 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.
    !>

    AB

    !>          AB is COMPLEX*16 array, dimension (LDAB,N)
    !>     On entry, the matrix A in band storage, in rows 1 to KL+KU+1.
    !>     The j-th column of A is stored in the j-th column of the
    !>     array AB as follows:
    !>     AB(KU+1+i-j,j) = A(i,j) for max(1,j-KU)<=i<=min(N,j+kl)
    !>
    !>     If FACT = 'F' and EQUED is not 'N', then AB must have been
    !>     equilibrated by the scaling factors in R and/or C.  AB 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).
    !>

    LDAB

    !>          LDAB is INTEGER
    !>     The leading dimension of the array AB.  LDAB >= KL+KU+1.
    !>

    AFB

    !>          AFB is COMPLEX*16 array, dimension (LDAFB,N)
    !>     If FACT = 'F', then AFB is an input argument and on entry
    !>     contains details of the LU factorization of the band matrix
    !>     A, as computed by ZGBTRF.  U is stored as an upper triangular
    !>     band matrix with KL+KU superdiagonals in rows 1 to KL+KU+1,
    !>     and the multipliers used during the factorization are stored
    !>     in rows KL+KU+2 to 2*KL+KU+1.  If EQUED .ne. 'N', then AFB 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).
    !>

    LDAFB

    !>          LDAFB is INTEGER
    !>     The leading dimension of the array AFB.  LDAFB >= 2*KL+KU+1.
    !>

    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.
    !>     If R is output, each element of R is a power of the radix.
    !>     If R is input, each element of R should be a power of the radix
    !>     to ensure a reliable solution and error estimates. Scaling by
    !>     powers of the radix does not cause rounding errors unless the
    !>     result underflows or overflows. Rounding errors during scaling
    !>     lead to refining with a matrix that is not equivalent to the
    !>     input matrix, producing error estimates that may not be
    !>     reliable.
    !>

    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.
    !>     If C is output, each element of C is a power of the radix.
    !>     If C is input, each element of C should be a power of the radix
    !>     to ensure a reliable solution and error estimates. Scaling by
    !>     powers of the radix does not cause rounding errors unless the
    !>     result underflows or overflows. Rounding errors during scaling
    !>     lead to refining with a matrix that is not equivalent to the
    !>     input matrix, producing error estimates that may not be
    !>     reliable.
    !>

    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, 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
    !>     Reciprocal scaled condition number.  This is an estimate of the
    !>     reciprocal Skeel condition number of the matrix A after
    !>     equilibration (if done).  If this is less than the machine
    !>     precision (in particular, if it is zero), the matrix is singular
    !>     to working precision.  Note that the error may still be small even
    !>     if this number is very small and the matrix appears ill-
    !>     conditioned.
    !>

    RPVGRW

    !>          RPVGRW is DOUBLE PRECISION
    !>     Reciprocal pivot growth.  On exit, this contains the reciprocal
    !>     pivot growth factor norm(A)/norm(U). The
    !>     norm is used.  If this 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, estimated condition numbers,
    !>     and error bounds could be unreliable. If factorization fails with
    !>     0<INFO<=N, then this contains the reciprocal pivot growth factor
    !>     for the leading INFO columns of A.  In ZGESVX, this quantity is
    !>     returned in RWORK(1).
    !>

    BERR

    !>          BERR is DOUBLE PRECISION array, dimension (NRHS)
    !>     Componentwise relative backward error.  This is 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).
    !>

    N_ERR_BNDS

    !>          N_ERR_BNDS is INTEGER
    !>     Number of error bounds to return for each right hand side
    !>     and each type (normwise or componentwise).  See ERR_BNDS_NORM and
    !>     ERR_BNDS_COMP below.
    !>

    ERR_BNDS_NORM

    !>          ERR_BNDS_NORM is DOUBLE PRECISION array, dimension (NRHS, N_ERR_BNDS)
    !>     For each right-hand side, this array contains information about
    !>     various error bounds and condition numbers corresponding to the
    !>     normwise relative error, which is defined as follows:
    !>
    !>     Normwise relative error in the ith solution vector:
    !>             max_j (abs(XTRUE(j,i) - X(j,i)))
    !>            ------------------------------
    !>                  max_j abs(X(j,i))
    !>
    !>     The array is indexed by the type of error information as described
    !>     below. There currently are up to three pieces of information
    !>     returned.
    !>
    !>     The first index in ERR_BNDS_NORM(i,:) corresponds to the ith
    !>     right-hand side.
    !>
    !>     The second index in ERR_BNDS_NORM(:,err) contains the following
    !>     three fields:
    !>     err = 1  boolean. Trust the answer if the
    !>              reciprocal condition number is less than the threshold
    !>              sqrt(n) * dlamch('Epsilon').
    !>
    !>     err = 2  error bound: The estimated forward error,
    !>              almost certainly within a factor of 10 of the true error
    !>              so long as the next entry is greater than the threshold
    !>              sqrt(n) * dlamch('Epsilon'). This error bound should only
    !>              be trusted if the previous boolean is true.
    !>
    !>     err = 3  Reciprocal condition number: Estimated normwise
    !>              reciprocal condition number.  Compared with the threshold
    !>              sqrt(n) * dlamch('Epsilon') to determine if the error
    !>              estimate is . These reciprocal condition
    !>              numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some
    !>              appropriately scaled matrix Z.
    !>              Let Z = S*A, where S scales each row by a power of the
    !>              radix so all absolute row sums of Z are approximately 1.
    !>
    !>     See Lapack Working Note 165 for further details and extra
    !>     cautions.
    !>

    ERR_BNDS_COMP

    !>          ERR_BNDS_COMP is DOUBLE PRECISION array, dimension (NRHS, N_ERR_BNDS)
    !>     For each right-hand side, this array contains information about
    !>     various error bounds and condition numbers corresponding to the
    !>     componentwise relative error, which is defined as follows:
    !>
    !>     Componentwise relative error in the ith solution vector:
    !>                    abs(XTRUE(j,i) - X(j,i))
    !>             max_j ----------------------
    !>                         abs(X(j,i))
    !>
    !>     The array is indexed by the right-hand side i (on which the
    !>     componentwise relative error depends), and the type of error
    !>     information as described below. There currently are up to three
    !>     pieces of information returned for each right-hand side. If
    !>     componentwise accuracy is not requested (PARAMS(3) = 0.0), then
    !>     ERR_BNDS_COMP is not accessed.  If N_ERR_BNDS < 3, then at most
    !>     the first (:,N_ERR_BNDS) entries are returned.
    !>
    !>     The first index in ERR_BNDS_COMP(i,:) corresponds to the ith
    !>     right-hand side.
    !>
    !>     The second index in ERR_BNDS_COMP(:,err) contains the following
    !>     three fields:
    !>     err = 1  boolean. Trust the answer if the
    !>              reciprocal condition number is less than the threshold
    !>              sqrt(n) * dlamch('Epsilon').
    !>
    !>     err = 2  error bound: The estimated forward error,
    !>              almost certainly within a factor of 10 of the true error
    !>              so long as the next entry is greater than the threshold
    !>              sqrt(n) * dlamch('Epsilon'). This error bound should only
    !>              be trusted if the previous boolean is true.
    !>
    !>     err = 3  Reciprocal condition number: Estimated componentwise
    !>              reciprocal condition number.  Compared with the threshold
    !>              sqrt(n) * dlamch('Epsilon') to determine if the error
    !>              estimate is . These reciprocal condition
    !>              numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some
    !>              appropriately scaled matrix Z.
    !>              Let Z = S*(A*diag(x)), where x is the solution for the
    !>              current right-hand side and S scales each row of
    !>              A*diag(x) by a power of the radix so all absolute row
    !>              sums of Z are approximately 1.
    !>
    !>     See Lapack Working Note 165 for further details and extra
    !>     cautions.
    !>

    NPARAMS

    !>          NPARAMS is INTEGER
    !>     Specifies the number of parameters set in PARAMS.  If <= 0, the
    !>     PARAMS array is never referenced and default values are used.
    !>

    PARAMS

    !>          PARAMS is DOUBLE PRECISION array, dimension NPARAMS
    !>     Specifies algorithm parameters.  If an entry is < 0.0, then
    !>     that entry will be filled with default value used for that
    !>     parameter.  Only positions up to NPARAMS are accessed; defaults
    !>     are used for higher-numbered parameters.
    !>
    !>       PARAMS(LA_LINRX_ITREF_I = 1) : Whether to perform iterative
    !>            refinement or not.
    !>         Default: 1.0D+0
    !>            = 0.0:  No refinement is performed, and no error bounds are
    !>                    computed.
    !>            = 1.0:  Use the extra-precise refinement algorithm.
    !>              (other values are reserved for future use)
    !>
    !>       PARAMS(LA_LINRX_ITHRESH_I = 2) : Maximum number of residual
    !>            computations allowed for refinement.
    !>         Default: 10
    !>         Aggressive: Set to 100 to permit convergence using approximate
    !>                     factorizations or factorizations other than LU. If
    !>                     the factorization uses a technique other than
    !>                     Gaussian elimination, the guarantees in
    !>                     err_bnds_norm and err_bnds_comp may no longer be
    !>                     trustworthy.
    !>
    !>       PARAMS(LA_LINRX_CWISE_I = 3) : Flag determining if the code
    !>            will attempt to find a solution with small componentwise
    !>            relative error in the double-precision algorithm.  Positive
    !>            is true, 0.0 is false.
    !>         Default: 1.0 (attempt componentwise convergence)
    !>

    WORK

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

    RWORK

    !>          RWORK is DOUBLE PRECISION array, dimension (2*N)
    !>

    INFO

    !>          INFO is INTEGER
    !>       = 0:  Successful exit. The solution to every right-hand side is
    !>         guaranteed.
    !>       < 0:  If INFO = -i, the i-th argument had an illegal value
    !>       > 0 and <= N:  U(INFO,INFO) 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+J: The solution corresponding to the Jth right-hand side is
    !>         not guaranteed. The solutions corresponding to other right-
    !>         hand sides K with K > J may not be guaranteed as well, but
    !>         only the first such right-hand side is reported. If a small
    !>         componentwise error is not requested (PARAMS(3) = 0.0) then
    !>         the Jth right-hand side is the first with a normwise error
    !>         bound that is not guaranteed (the smallest J such
    !>         that ERR_BNDS_NORM(J,1) = 0.0). By default (PARAMS(3) = 1.0)
    !>         the Jth right-hand side is the first with either a normwise or
    !>         componentwise error bound that is not guaranteed (the smallest
    !>         J such that either ERR_BNDS_NORM(J,1) = 0.0 or
    !>         ERR_BNDS_COMP(J,1) = 0.0). See the definition of
    !>         ERR_BNDS_NORM(:,1) and ERR_BNDS_COMP(:,1). To get information
    !>         about all of the right-hand sides check ERR_BNDS_NORM or
    !>         ERR_BNDS_COMP.
    !>

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 555 of file zgbsvxx.f.

Author

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