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NAME

       zposvxx.f -

SYNOPSIS

   Functions/Subroutines
       subroutine zposvxx (FACT, UPLO, N, NRHS, A, LDA, AF, LDAF, EQUED, S, B, LDB, X, LDX,
           RCOND, RPVGRW, BERR, N_ERR_BNDS, ERR_BNDS_NORM, ERR_BNDS_COMP, NPARAMS, PARAMS, WORK,
           RWORK, INFO)
            ZPOSVXX computes the solution to system of linear equations A * X = B for PO matrices

Function/Subroutine Documentation

   subroutine zposvxx (characterFACT, characterUPLO, integerN, integerNRHS, complex*16,
       dimension( lda, * )A, integerLDA, complex*16, dimension( ldaf, * )AF, integerLDAF,
       characterEQUED, double precision, dimension( * )S, complex*16, dimension( ldb, * )B,
       integerLDB, complex*16, dimension( ldx, * )X, integerLDX, double precisionRCOND, double
       precisionRPVGRW, 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)
        ZPOSVXX computes the solution to system of linear equations A * X = B for PO matrices

       Purpose:

               ZPOSVXX uses the Cholesky factorization A = U**T*U or A = L*L**T
               to compute the solution to a complex*16 system of linear equations
               A * X = B, where A is an N-by-N symmetric positive definite matrix
               and X and B are N-by-NRHS matrices.

               If requested, both normwise and maximum componentwise error bounds
               are returned. ZPOSVXX 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.

               ZPOSVXX 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
               ZPOSVXX 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 ZPOSVXX would itself produce.

       Description:

               The following steps are performed:

               1. If FACT = 'E', double precision scaling factors are computed to equilibrate
               the system:

                 diag(S)*A*diag(S)     *inv(diag(S))*X = diag(S)*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(S)*A*diag(S) and B by diag(S)*B.

               2. If FACT = 'N' or 'E', the Cholesky decomposition is used to
               factor the matrix A (after equilibration if FACT = 'E') as
                  A = U**T* U,  if UPLO = 'U', or
                  A = L * L**T,  if UPLO = 'L',
               where U is an upper triangular matrix and L is a lower triangular
               matrix.

               3. If the leading i-by-i principal minor is not positive definite,
               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(S) 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 contains the factored form of A.
                          If EQUED is not 'N', the matrix A has been
                          equilibrated with scaling factors given by S.
                          A and AF 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.

           UPLO

                     UPLO is CHARACTER*1
                  = 'U':  Upper triangle of A is stored;
                  = 'L':  Lower triangle of A is stored.

           N

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

           NRHS

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

           A

                     A is COMPLEX*16 array, dimension (LDA,N)
                On entry, the symmetric matrix A, except if FACT = 'F' and EQUED =
                'Y', then A must contain the equilibrated matrix
                diag(S)*A*diag(S).  If UPLO = 'U', the leading N-by-N upper
                triangular part of A contains the upper triangular part of the
                matrix A, and the strictly lower triangular part of A is not
                referenced.  If UPLO = 'L', the leading N-by-N lower triangular
                part of A contains the lower triangular part of the matrix A, and
                the strictly upper triangular part of A is not referenced.  A is
                not modified if FACT = 'F' or 'N', or if FACT = 'E' and EQUED =
                'N' on exit.

                On exit, if FACT = 'E' and EQUED = 'Y', A is overwritten by
                diag(S)*A*diag(S).

           LDA

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

           AF

                     AF is COMPLEX*16 array, dimension (LDAF,N)
                If FACT = 'F', then AF is an input argument and on entry
                contains the triangular factor U or L from the Cholesky
                factorization A = U**T*U or A = L*L**T, in the same storage
                format as A.  If EQUED .ne. 'N', then AF is the factored
                form of the equilibrated matrix diag(S)*A*diag(S).

                If FACT = 'N', then AF is an output argument and on exit
                returns the triangular factor U or L from the Cholesky
                factorization A = U**T*U or A = L*L**T of the original
                matrix A.

                If FACT = 'E', then AF is an output argument and on exit
                returns the triangular factor U or L from the Cholesky
                factorization A = U**T*U or A = L*L**T of the equilibrated
                matrix A (see the description of A for the form of the
                equilibrated matrix).

           LDAF

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

           EQUED

                     EQUED is CHARACTER*1
                Specifies the form of equilibration that was done.
                  = 'N':  No equilibration (always true if FACT = 'N').
                  = 'Y':  Both row and column equilibration, i.e., A has been
                          replaced by diag(S) * A * diag(S).
                EQUED is an input argument if FACT = 'F'; otherwise, it is an
                output argument.

           S

                     S is DOUBLE PRECISION array, dimension (N)
                The row scale factors for A.  If EQUED = 'Y', A is multiplied on
                the left and right by diag(S).  S is an input argument if FACT =
                'F'; otherwise, S is an output argument.  If FACT = 'F' and EQUED
                = 'Y', each element of S must be positive.  If S is output, each
                element of S is a power of the radix. If S is input, each element
                of S 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 EQUED = 'Y', B is overwritten by diag(S)*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(S))*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.

           RPVGRW

                     RPVGRW is DOUBLE PRECISION
                Reciprocal pivot growth.  On exit, this contains the reciprocal
                pivot growth factor norm(A)/norm(U). The "max absolute element"
                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.

           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 "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 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.

       Date:
           April 2012

       Definition at line 491 of file zposvxx.f.

Author

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