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NAME

       zgbrfsx.f -

SYNOPSIS

   Functions/Subroutines
       subroutine zgbrfsx (TRANS, EQUED, N, KL, KU, NRHS, AB, LDAB, AFB, LDAFB, IPIV, R, C, B,
           LDB, X, LDX, RCOND, BERR, N_ERR_BNDS, ERR_BNDS_NORM, ERR_BNDS_COMP, NPARAMS, PARAMS,
           WORK, RWORK, INFO)
           ZGBRFSX

Function/Subroutine Documentation

   subroutine zgbrfsx (characterTRANS, characterEQUED, integerN, integerKL, integerKU,
       integerNRHS, complex*16, dimension( ldab, * )AB, integerLDAB, complex*16, dimension(
       ldafb, * )AFB, integerLDAFB, integer, dimension( * )IPIV, double precision, dimension( *
       )R, double precision, dimension( * )C, complex*16, dimension( ldb, * )B, integerLDB,
       complex*16, dimension( ldx , * )X, integerLDX, double precisionRCOND, double precision,
       dimension( * )BERR, integerN_ERR_BNDS, double precision, dimension( nrhs, *
       )ERR_BNDS_NORM, double precision, dimension( nrhs, * )ERR_BNDS_COMP, integerNPARAMS,
       double precision, dimension( * )PARAMS, complex*16, dimension( * )WORK, double precision,
       dimension( * )RWORK, integerINFO)
       ZGBRFSX

       Purpose:

               ZGBRFSX improves the computed solution to a system of linear
               equations and provides error bounds and backward error estimates
               for the solution.  In addition to normwise error bound, the code
               provides maximum componentwise error bound if possible.  See
               comments for ERR_BNDS_NORM and ERR_BNDS_COMP for details of the
               error bounds.

               The original system of linear equations may have been equilibrated
               before calling this routine, as described by arguments EQUED, R
               and C below. In this case, the solution and error bounds returned
               are for the original unequilibrated system.

                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:
           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)

           EQUED

                     EQUED is CHARACTER*1
                Specifies the form of equilibration that was done to A
                before calling this routine. This is needed to compute
                the solution and error bounds correctly.
                  = 'N':  No equilibration
                  = '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).
                          The right hand side B has been changed accordingly.

           N

                     N is INTEGER
                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)
                The original band matrix A, stored 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).

           LDAB

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

           AFB

                     AFB is COMPLEX*16 array, dimension (LDAFB,N)
                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.

           LDAFB

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

           IPIV

                     IPIV is INTEGER array, dimension (N)
                The pivot indices from DGETRF; for 1<=i<=N, row i of the
                matrix was interchanged with row IPIV(i).

           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)
                The right hand side matrix 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)
                On entry, the solution matrix X, as computed by DGETRS.
                On exit, the improved solution matrix X.

           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.

           BERR

                     BERR is COMPLEX*16 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 "Trust/don't trust" boolean. Trust the answer if the
                         reciprocal condition number is less than the threshold
                         sqrt(n) * dlamch('Epsilon').

                err = 2 "Guaranteed" 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 "guaranteed". 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 .LT. 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 "Trust/don't trust" boolean. Trust the answer if the
                         reciprocal condition number is less than the threshold
                         sqrt(n) * dlamch('Epsilon').

                err = 2 "Guaranteed" 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 "guaranteed". 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 .LE. 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 .LT. 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 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*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.

       Date:
           April 2012

       Definition at line 437 of file zgbrfsx.f.

Author

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