posvxx(3)

NAME
posvxx - posvxx: factor and solve, extra precise

SYNOPSIS

  Functions
subroutine cposvxx (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)
CPOSVXX computes the solution to system of linear equations A * X = B for PO matrices
subroutine dposvxx (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, iwork, info)
DPOSVXX computes the solution to system of linear equations A * X = B for PO matrices
subroutine sposvxx (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, iwork, info)
SPOSVXX computes the solution to system of linear equations A * X = B for PO matrices
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

Detailed Description

Function Documentation

  subroutine cposvxx (character fact, character uplo, integer n, integer nrhs, complex, dimension( lda, * ) a, integer lda, complex, dimension( ldaf, * ) af, integer ldaf, character equed, real, dimension( * ) s, 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)

CPOSVXX computes the solution to system of linear equations A * X = B for PO matrices

Purpose:

        CPOSVXX uses the Cholesky factorization A = U**T*U or A = L*L**T
        to compute the solution to a complex system of linear equations
        A * X = B, where A is an N-by-N Hermitian positive definite matrix
        and X and B are N-by-NRHS matrices.
        If requested, both normwise and maximum componentwise error bounds
        are returned. CPOSVXX 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.
        CPOSVXX 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
        CPOSVXX 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 CPOSVXX would itself produce.

Description:

        The following steps are performed:
        1. If FACT = 'E', real 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 principal minor of order i is not positive,
        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 array, dimension (LDA,N)
         On entry, the Hermitian 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 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 REAL 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 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 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 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 '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 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 'Trust/don't trust' boolean. Trust the answer if the
                  reciprocal condition number is less than the threshold
                  sqrt(n) * slamch('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) * 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 '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 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 'Trust/don't trust' boolean. Trust the answer if the
                  reciprocal condition number is less than the threshold
                  sqrt(n) * slamch('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) * 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 '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 <= 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 492 of file cposvxx.f.

  subroutine dposvxx (character fact, character uplo, integer n, integer nrhs, double precision, dimension( lda, * ) a, integer lda, double precision, dimension( ldaf, * ) af, integer ldaf, character equed, double precision, dimension( * ) s, 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)

DPOSVXX computes the solution to system of linear equations A * X = B for PO matrices

Purpose:

        DPOSVXX uses the Cholesky factorization A = U**T*U or A = L*L**T
        to compute the solution to a double precision 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. DPOSVXX 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.
        DPOSVXX 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
        DPOSVXX 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 DPOSVXX 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 principal minor of order i is not positive,
        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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 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 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 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(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 < 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 <= 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 490 of file dposvxx.f.

  subroutine sposvxx (character fact, character uplo, integer n, integer nrhs, real, dimension( lda, * ) a, integer lda, real, dimension( ldaf, * ) af, integer ldaf, character equed, real, dimension( * ) s, 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)

SPOSVXX computes the solution to system of linear equations A * X = B for PO matrices

Purpose:

        SPOSVXX uses the Cholesky factorization A = U**T*U or A = L*L**T
        to compute the solution to a real 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. SPOSVXX 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.
        SPOSVXX 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
        SPOSVXX 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 SPOSVXX would itself produce.

Description:

        The following steps are performed:
        1. If FACT = 'E', real 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 principal minor of order i is not positive,
        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 REAL 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 REAL 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 REAL 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 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 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 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(S))*X.

    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 '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 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 'Trust/don't trust' boolean. Trust the answer if the
                  reciprocal condition number is less than the threshold
                  sqrt(n) * slamch('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) * 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 '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 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 'Trust/don't trust' boolean. Trust the answer if the
                  reciprocal condition number is less than the threshold
                  sqrt(n) * slamch('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) * 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 '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 <= 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 493 of file sposvxx.f.

  subroutine zposvxx (character fact, character uplo, integer n, integer nrhs, complex*16, dimension( lda, * ) a, integer lda, complex*16, dimension( ldaf, * ) af, integer ldaf, character equed, double precision, dimension( * ) s, 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)

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 Hermitian 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 principal minor of order i is not positive,
        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 Hermitian 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 < 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 <= 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 489 of file zposvxx.f.

Author

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