la_porfsx_extended(3)

NAME
la_porfsx_extended - la_porfsx_extended: step in porfsx

SYNOPSIS

  Functions

subroutine cla_porfsx_extended (prec_type, uplo, n, nrhs, a, lda, af, ldaf, colequ, c, b, ldb, y, ldy, berr_out, n_norms, err_bnds_norm, err_bnds_comp, res, ayb, dy, y_tail, rcond, ithresh, rthresh, dz_ub, ignore_cwise, info)
CLA_PORFSX_EXTENDED improves the computed solution to a system of linear equations for symmetric or Hermitian positive-definite matrices by performing extra-precise iterative refinement and provides error bounds and backward error estimates for the solution.
subroutine dla_porfsx_extended (prec_type, uplo, n, nrhs, a, lda, af, ldaf, colequ, c, b, ldb, y, ldy, berr_out, n_norms, err_bnds_norm, err_bnds_comp, res, ayb, dy, y_tail, rcond, ithresh, rthresh, dz_ub, ignore_cwise, info)
DLA_PORFSX_EXTENDED improves the computed solution to a system of linear equations for symmetric or Hermitian positive-definite matrices by performing extra-precise iterative refinement and provides error bounds and backward error estimates for the solution.
subroutine sla_porfsx_extended (prec_type, uplo, n, nrhs, a, lda, af, ldaf, colequ, c, b, ldb, y, ldy, berr_out, n_norms, err_bnds_norm, err_bnds_comp, res, ayb, dy, y_tail, rcond, ithresh, rthresh, dz_ub, ignore_cwise, info)
SLA_PORFSX_EXTENDED improves the computed solution to a system of linear equations for symmetric or Hermitian positive-definite matrices by performing extra-precise iterative refinement and provides error bounds and backward error estimates for the solution.
subroutine zla_porfsx_extended (prec_type, uplo, n, nrhs, a, lda, af, ldaf, colequ, c, b, ldb, y, ldy, berr_out, n_norms, err_bnds_norm, err_bnds_comp, res, ayb, dy, y_tail, rcond, ithresh, rthresh, dz_ub, ignore_cwise, info)
ZLA_PORFSX_EXTENDED improves the computed solution to a system of linear equations for symmetric or Hermitian positive-definite matrices by performing extra-precise iterative refinement and provides error bounds and backward error estimates for the solution.

Detailed Description

Function Documentation

  subroutine cla_porfsx_extended (integer prec_type, character uplo, integer n, integer nrhs, complex, dimension( lda, * ) a, integer lda, complex, dimension( ldaf, * ) af, integer ldaf, logical colequ, real, dimension( * ) c, complex, dimension( ldb, * ) b, integer ldb, complex, dimension( ldy, * ) y, integer ldy, real, dimension( * ) berr_out, integer n_norms, real, dimension( nrhs, * ) err_bnds_norm, real, dimension( nrhs, * ) err_bnds_comp, complex, dimension( * ) res, real, dimension( * ) ayb, complex, dimension( * ) dy, complex, dimension( * ) y_tail, real rcond, integer ithresh, real rthresh, real dz_ub, logical ignore_cwise, integer info)

CLA_PORFSX_EXTENDED improves the computed solution to a system of linear equations for symmetric or Hermitian positive-definite matrices by performing extra-precise iterative refinement and provides error bounds and backward error estimates for the solution.

Purpose:

    !>
    !> CLA_PORFSX_EXTENDED improves the computed solution to a system of
    !> linear equations by performing extra-precise iterative refinement
    !> and provides error bounds and backward error estimates for the solution.
    !> This subroutine is called by CPORFSX to perform iterative refinement.
    !> In addition to normwise error bound, the code provides maximum
    !> componentwise error bound if possible. See comments for ERR_BNDS_NORM
    !> and ERR_BNDS_COMP for details of the error bounds. Note that this
    !> subroutine is only responsible for setting the second fields of
    !> ERR_BNDS_NORM and ERR_BNDS_COMP.
    !>

Parameters
    PREC_TYPE

    !>          PREC_TYPE is INTEGER
    !>     Specifies the intermediate precision to be used in refinement.
    !>     The value is defined by ILAPREC(P) where P is a CHARACTER and P
    !>          = 'S':  Single
    !>          = 'D':  Double
    !>          = 'I':  Indigenous
    !>          = 'X' or 'E':  Extra
    !>

    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
    !>     matrix B.
    !>

    A

    !>          A is COMPLEX array, dimension (LDA,N)
    !>     On entry, the N-by-N matrix A.
    !>

    LDA

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

    AF

    !>          AF is COMPLEX array, dimension (LDAF,N)
    !>     The triangular factor U or L from the Cholesky factorization
    !>     A = U**T*U or A = L*L**T, as computed by CPOTRF.
    !>

    LDAF

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

    COLEQU

    !>          COLEQU is LOGICAL
    !>     If .TRUE. then column equilibration was done to A before calling
    !>     this routine. This is needed to compute the solution and error
    !>     bounds correctly.
    !>

    C

    !>          C is REAL array, dimension (N)
    !>     The column scale factors for A. If COLEQU = .FALSE., C
    !>     is not accessed. If C is input, each element of C should be a power
    !>     of the radix to ensure a reliable solution and error estimates.
    !>     Scaling by powers of the radix does not cause rounding errors unless
    !>     the result underflows or overflows. Rounding errors during scaling
    !>     lead to refining with a matrix that is not equivalent to the
    !>     input matrix, producing error estimates that may not be
    !>     reliable.
    !>

    B

    !>          B is COMPLEX array, dimension (LDB,NRHS)
    !>     The right-hand-side matrix B.
    !>

    LDB

    !>          LDB is INTEGER
    !>     The leading dimension of the array B.  LDB >= max(1,N).
    !>

    Y

    !>          Y is COMPLEX array, dimension (LDY,NRHS)
    !>     On entry, the solution matrix X, as computed by CPOTRS.
    !>     On exit, the improved solution matrix Y.
    !>

    LDY

    !>          LDY is INTEGER
    !>     The leading dimension of the array Y.  LDY >= max(1,N).
    !>

    BERR_OUT

    !>          BERR_OUT is REAL array, dimension (NRHS)
    !>     On exit, BERR_OUT(j) contains the componentwise relative backward
    !>     error for right-hand-side j from the formula
    !>         max(i) ( abs(RES(i)) / ( abs(op(A_s))*abs(Y) + abs(B_s) )(i) )
    !>     where abs(Z) is the componentwise absolute value of the matrix
    !>     or vector Z. This is computed by CLA_LIN_BERR.
    !>

    N_NORMS

    !>          N_NORMS is INTEGER
    !>     Determines which error bounds to return (see ERR_BNDS_NORM
    !>     and ERR_BNDS_COMP).
    !>     If N_NORMS >= 1 return normwise error bounds.
    !>     If N_NORMS >= 2 return componentwise error bounds.
    !>

    ERR_BNDS_NORM

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

    ERR_BNDS_COMP

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

    RES

    !>          RES is COMPLEX array, dimension (N)
    !>     Workspace to hold the intermediate residual.
    !>

    AYB

    !>          AYB is REAL array, dimension (N)
    !>     Workspace.
    !>

    DY

    !>          DY is COMPLEX array, dimension (N)
    !>     Workspace to hold the intermediate solution.
    !>

    Y_TAIL

    !>          Y_TAIL is COMPLEX array, dimension (N)
    !>     Workspace to hold the trailing bits of the intermediate solution.
    !>

    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.
    !>

    ITHRESH

    !>          ITHRESH is INTEGER
    !>     The maximum number of residual computations allowed for
    !>     refinement. The default is 10. For '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.
    !>

    RTHRESH

    !>          RTHRESH is REAL
    !>     Determines when to stop refinement if the error estimate stops
    !>     decreasing. Refinement will stop when the next solution no longer
    !>     satisfies norm(dx_{i+1}) < RTHRESH * norm(dx_i) where norm(Z) is
    !>     the infinity norm of Z. RTHRESH satisfies 0 < RTHRESH <= 1. The
    !>     default value is 0.5. For 'aggressive' set to 0.9 to permit
    !>     convergence on extremely ill-conditioned matrices. See LAWN 165
    !>     for more details.
    !>

    DZ_UB

    !>          DZ_UB is REAL
    !>     Determines when to start considering componentwise convergence.
    !>     Componentwise convergence is only considered after each component
    !>     of the solution Y is stable, which we define as the relative
    !>     change in each component being less than DZ_UB. The default value
    !>     is 0.25, requiring the first bit to be stable. See LAWN 165 for
    !>     more details.
    !>

    IGNORE_CWISE

    !>          IGNORE_CWISE is LOGICAL
    !>     If .TRUE. then ignore componentwise convergence. Default value
    !>     is .FALSE..
    !>

    INFO

    !>          INFO is INTEGER
    !>       = 0:  Successful exit.
    !>       < 0:  if INFO = -i, the ith argument to CPOTRS had an illegal
    !>             value
    !>

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

  subroutine dla_porfsx_extended (integer prec_type, character uplo, integer n, integer nrhs, double precision, dimension( lda, * ) a, integer lda, double precision, dimension( ldaf, * ) af, integer ldaf, logical colequ, double precision, dimension( * ) c, double precision, dimension( ldb, * ) b, integer ldb, double precision, dimension( ldy, * ) y, integer ldy, double precision, dimension( * ) berr_out, integer n_norms, double precision, dimension( nrhs, * ) err_bnds_norm, double precision, dimension( nrhs, * ) err_bnds_comp, double precision, dimension( * ) res, double precision, dimension(*) ayb, double precision, dimension( * ) dy, double precision, dimension( * ) y_tail, double precision rcond, integer ithresh, double precision rthresh, double precision dz_ub, logical ignore_cwise, integer info)

DLA_PORFSX_EXTENDED improves the computed solution to a system of linear equations for symmetric or Hermitian positive-definite matrices by performing extra-precise iterative refinement and provides error bounds and backward error estimates for the solution.

Purpose:

    !>
    !> DLA_PORFSX_EXTENDED improves the computed solution to a system of
    !> linear equations by performing extra-precise iterative refinement
    !> and provides error bounds and backward error estimates for the solution.
    !> This subroutine is called by DPORFSX to perform iterative refinement.
    !> In addition to normwise error bound, the code provides maximum
    !> componentwise error bound if possible. See comments for ERR_BNDS_NORM
    !> and ERR_BNDS_COMP for details of the error bounds. Note that this
    !> subroutine is only responsible for setting the second fields of
    !> ERR_BNDS_NORM and ERR_BNDS_COMP.
    !>

Parameters
    PREC_TYPE

    !>          PREC_TYPE is INTEGER
    !>     Specifies the intermediate precision to be used in refinement.
    !>     The value is defined by ILAPREC(P) where P is a CHARACTER and P
    !>          = 'S':  Single
    !>          = 'D':  Double
    !>          = 'I':  Indigenous
    !>          = 'X' or 'E':  Extra
    !>

    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
    !>     matrix B.
    !>

    A

    !>          A is DOUBLE PRECISION array, dimension (LDA,N)
    !>     On entry, the N-by-N matrix A.
    !>

    LDA

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

    AF

    !>          AF is DOUBLE PRECISION array, dimension (LDAF,N)
    !>     The triangular factor U or L from the Cholesky factorization
    !>     A = U**T*U or A = L*L**T, as computed by DPOTRF.
    !>

    LDAF

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

    COLEQU

    !>          COLEQU is LOGICAL
    !>     If .TRUE. then column equilibration was done to A before calling
    !>     this routine. This is needed to compute the solution and error
    !>     bounds correctly.
    !>

    C

    !>          C is DOUBLE PRECISION array, dimension (N)
    !>     The column scale factors for A. If COLEQU = .FALSE., C
    !>     is not accessed. If C is input, each element of C should be a power
    !>     of the radix to ensure a reliable solution and error estimates.
    !>     Scaling by powers of the radix does not cause rounding errors unless
    !>     the result underflows or overflows. Rounding errors during scaling
    !>     lead to refining with a matrix that is not equivalent to the
    !>     input matrix, producing error estimates that may not be
    !>     reliable.
    !>

    B

    !>          B is DOUBLE PRECISION array, dimension (LDB,NRHS)
    !>     The right-hand-side matrix B.
    !>

    LDB

    !>          LDB is INTEGER
    !>     The leading dimension of the array B.  LDB >= max(1,N).
    !>

    Y

    !>          Y is DOUBLE PRECISION array, dimension (LDY,NRHS)
    !>     On entry, the solution matrix X, as computed by DPOTRS.
    !>     On exit, the improved solution matrix Y.
    !>

    LDY

    !>          LDY is INTEGER
    !>     The leading dimension of the array Y.  LDY >= max(1,N).
    !>

    BERR_OUT

    !>          BERR_OUT is DOUBLE PRECISION array, dimension (NRHS)
    !>     On exit, BERR_OUT(j) contains the componentwise relative backward
    !>     error for right-hand-side j from the formula
    !>         max(i) ( abs(RES(i)) / ( abs(op(A_s))*abs(Y) + abs(B_s) )(i) )
    !>     where abs(Z) is the componentwise absolute value of the matrix
    !>     or vector Z. This is computed by DLA_LIN_BERR.
    !>

    N_NORMS

    !>          N_NORMS is INTEGER
    !>     Determines which error bounds to return (see ERR_BNDS_NORM
    !>     and ERR_BNDS_COMP).
    !>     If N_NORMS >= 1 return normwise error bounds.
    !>     If N_NORMS >= 2 return componentwise error bounds.
    !>

    ERR_BNDS_NORM

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

    ERR_BNDS_COMP

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

    RES

    !>          RES is DOUBLE PRECISION array, dimension (N)
    !>     Workspace to hold the intermediate residual.
    !>

    AYB

    !>          AYB is DOUBLE PRECISION array, dimension (N)
    !>     Workspace. This can be the same workspace passed for Y_TAIL.
    !>

    DY

    !>          DY is DOUBLE PRECISION array, dimension (N)
    !>     Workspace to hold the intermediate solution.
    !>

    Y_TAIL

    !>          Y_TAIL is DOUBLE PRECISION array, dimension (N)
    !>     Workspace to hold the trailing bits of the intermediate solution.
    !>

    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.
    !>

    ITHRESH

    !>          ITHRESH is INTEGER
    !>     The maximum number of residual computations allowed for
    !>     refinement. The default is 10. For '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.
    !>

    RTHRESH

    !>          RTHRESH is DOUBLE PRECISION
    !>     Determines when to stop refinement if the error estimate stops
    !>     decreasing. Refinement will stop when the next solution no longer
    !>     satisfies norm(dx_{i+1}) < RTHRESH * norm(dx_i) where norm(Z) is
    !>     the infinity norm of Z. RTHRESH satisfies 0 < RTHRESH <= 1. The
    !>     default value is 0.5. For 'aggressive' set to 0.9 to permit
    !>     convergence on extremely ill-conditioned matrices. See LAWN 165
    !>     for more details.
    !>

    DZ_UB

    !>          DZ_UB is DOUBLE PRECISION
    !>     Determines when to start considering componentwise convergence.
    !>     Componentwise convergence is only considered after each component
    !>     of the solution Y is stable, which we define as the relative
    !>     change in each component being less than DZ_UB. The default value
    !>     is 0.25, requiring the first bit to be stable. See LAWN 165 for
    !>     more details.
    !>

    IGNORE_CWISE

    !>          IGNORE_CWISE is LOGICAL
    !>     If .TRUE. then ignore componentwise convergence. Default value
    !>     is .FALSE..
    !>

    INFO

    !>          INFO is INTEGER
    !>       = 0:  Successful exit.
    !>       < 0:  if INFO = -i, the ith argument to DPOTRS had an illegal
    !>             value
    !>

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

  subroutine sla_porfsx_extended (integer prec_type, character uplo, integer n, integer nrhs, real, dimension( lda, * ) a, integer lda, real, dimension( ldaf, * ) af, integer ldaf, logical colequ, real, dimension( * ) c, real, dimension( ldb, * ) b, integer ldb, real, dimension( ldy, * ) y, integer ldy, real, dimension( * ) berr_out, integer n_norms, real, dimension( nrhs, * ) err_bnds_norm, real, dimension( nrhs, * ) err_bnds_comp, real, dimension( * ) res, real, dimension(*) ayb, real, dimension( * ) dy, real, dimension( * ) y_tail, real rcond, integer ithresh, real rthresh, real dz_ub, logical ignore_cwise, integer info)

SLA_PORFSX_EXTENDED improves the computed solution to a system of linear equations for symmetric or Hermitian positive-definite matrices by performing extra-precise iterative refinement and provides error bounds and backward error estimates for the solution.

Purpose:

    !>
    !> SLA_PORFSX_EXTENDED improves the computed solution to a system of
    !> linear equations by performing extra-precise iterative refinement
    !> and provides error bounds and backward error estimates for the solution.
    !> This subroutine is called by SPORFSX to perform iterative refinement.
    !> In addition to normwise error bound, the code provides maximum
    !> componentwise error bound if possible. See comments for ERR_BNDS_NORM
    !> and ERR_BNDS_COMP for details of the error bounds. Note that this
    !> subroutine is only responsible for setting the second fields of
    !> ERR_BNDS_NORM and ERR_BNDS_COMP.
    !>

Parameters
    PREC_TYPE

    !>          PREC_TYPE is INTEGER
    !>     Specifies the intermediate precision to be used in refinement.
    !>     The value is defined by ILAPREC(P) where P is a CHARACTER and P
    !>          = 'S':  Single
    !>          = 'D':  Double
    !>          = 'I':  Indigenous
    !>          = 'X' or 'E':  Extra
    !>

    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
    !>     matrix B.
    !>

    A

    !>          A is REAL array, dimension (LDA,N)
    !>     On entry, the N-by-N matrix A.
    !>

    LDA

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

    AF

    !>          AF is REAL array, dimension (LDAF,N)
    !>     The triangular factor U or L from the Cholesky factorization
    !>     A = U**T*U or A = L*L**T, as computed by SPOTRF.
    !>

    LDAF

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

    COLEQU

    !>          COLEQU is LOGICAL
    !>     If .TRUE. then column equilibration was done to A before calling
    !>     this routine. This is needed to compute the solution and error
    !>     bounds correctly.
    !>

    C

    !>          C is REAL array, dimension (N)
    !>     The column scale factors for A. If COLEQU = .FALSE., C
    !>     is not accessed. If C is input, each element of C should be a power
    !>     of the radix to ensure a reliable solution and error estimates.
    !>     Scaling by powers of the radix does not cause rounding errors unless
    !>     the result underflows or overflows. Rounding errors during scaling
    !>     lead to refining with a matrix that is not equivalent to the
    !>     input matrix, producing error estimates that may not be
    !>     reliable.
    !>

    B

    !>          B is REAL array, dimension (LDB,NRHS)
    !>     The right-hand-side matrix B.
    !>

    LDB

    !>          LDB is INTEGER
    !>     The leading dimension of the array B.  LDB >= max(1,N).
    !>

    Y

    !>          Y is REAL array, dimension (LDY,NRHS)
    !>     On entry, the solution matrix X, as computed by SPOTRS.
    !>     On exit, the improved solution matrix Y.
    !>

    LDY

    !>          LDY is INTEGER
    !>     The leading dimension of the array Y.  LDY >= max(1,N).
    !>

    BERR_OUT

    !>          BERR_OUT is REAL array, dimension (NRHS)
    !>     On exit, BERR_OUT(j) contains the componentwise relative backward
    !>     error for right-hand-side j from the formula
    !>         max(i) ( abs(RES(i)) / ( abs(op(A_s))*abs(Y) + abs(B_s) )(i) )
    !>     where abs(Z) is the componentwise absolute value of the matrix
    !>     or vector Z. This is computed by SLA_LIN_BERR.
    !>

    N_NORMS

    !>          N_NORMS is INTEGER
    !>     Determines which error bounds to return (see ERR_BNDS_NORM
    !>     and ERR_BNDS_COMP).
    !>     If N_NORMS >= 1 return normwise error bounds.
    !>     If N_NORMS >= 2 return componentwise error bounds.
    !>

    ERR_BNDS_NORM

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

    ERR_BNDS_COMP

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

    RES

    !>          RES is REAL array, dimension (N)
    !>     Workspace to hold the intermediate residual.
    !>

    AYB

    !>          AYB is REAL array, dimension (N)
    !>     Workspace. This can be the same workspace passed for Y_TAIL.
    !>

    DY

    !>          DY is REAL array, dimension (N)
    !>     Workspace to hold the intermediate solution.
    !>

    Y_TAIL

    !>          Y_TAIL is REAL array, dimension (N)
    !>     Workspace to hold the trailing bits of the intermediate solution.
    !>

    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.
    !>

    ITHRESH

    !>          ITHRESH is INTEGER
    !>     The maximum number of residual computations allowed for
    !>     refinement. The default is 10. For '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.
    !>

    RTHRESH

    !>          RTHRESH is REAL
    !>     Determines when to stop refinement if the error estimate stops
    !>     decreasing. Refinement will stop when the next solution no longer
    !>     satisfies norm(dx_{i+1}) < RTHRESH * norm(dx_i) where norm(Z) is
    !>     the infinity norm of Z. RTHRESH satisfies 0 < RTHRESH <= 1. The
    !>     default value is 0.5. For 'aggressive' set to 0.9 to permit
    !>     convergence on extremely ill-conditioned matrices. See LAWN 165
    !>     for more details.
    !>

    DZ_UB

    !>          DZ_UB is REAL
    !>     Determines when to start considering componentwise convergence.
    !>     Componentwise convergence is only considered after each component
    !>     of the solution Y is stable, which we define as the relative
    !>     change in each component being less than DZ_UB. The default value
    !>     is 0.25, requiring the first bit to be stable. See LAWN 165 for
    !>     more details.
    !>

    IGNORE_CWISE

    !>          IGNORE_CWISE is LOGICAL
    !>     If .TRUE. then ignore componentwise convergence. Default value
    !>     is .FALSE..
    !>

    INFO

    !>          INFO is INTEGER
    !>       = 0:  Successful exit.
    !>       < 0:  if INFO = -i, the ith argument to SPOTRS had an illegal
    !>             value
    !>

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

  subroutine zla_porfsx_extended (integer prec_type, character uplo, integer n, integer nrhs, complex*16, dimension( lda, * ) a, integer lda, complex*16, dimension( ldaf, * ) af, integer ldaf, logical colequ, double precision, dimension( * ) c, complex*16, dimension( ldb, * ) b, integer ldb, complex*16, dimension( ldy, * ) y, integer ldy, double precision, dimension( * ) berr_out, integer n_norms, double precision, dimension( nrhs, * ) err_bnds_norm, double precision, dimension( nrhs, * ) err_bnds_comp, complex*16, dimension( * ) res, double precision, dimension( * ) ayb, complex*16, dimension( * ) dy, complex*16, dimension( * ) y_tail, double precision rcond, integer ithresh, double precision rthresh, double precision dz_ub, logical ignore_cwise, integer info)

ZLA_PORFSX_EXTENDED improves the computed solution to a system of linear equations for symmetric or Hermitian positive-definite matrices by performing extra-precise iterative refinement and provides error bounds and backward error estimates for the solution.

Purpose:

    !>
    !> ZLA_PORFSX_EXTENDED improves the computed solution to a system of
    !> linear equations by performing extra-precise iterative refinement
    !> and provides error bounds and backward error estimates for the solution.
    !> This subroutine is called by ZPORFSX to perform iterative refinement.
    !> In addition to normwise error bound, the code provides maximum
    !> componentwise error bound if possible. See comments for ERR_BNDS_NORM
    !> and ERR_BNDS_COMP for details of the error bounds. Note that this
    !> subroutine is only responsible for setting the second fields of
    !> ERR_BNDS_NORM and ERR_BNDS_COMP.
    !>

Parameters
    PREC_TYPE

    !>          PREC_TYPE is INTEGER
    !>     Specifies the intermediate precision to be used in refinement.
    !>     The value is defined by ILAPREC(P) where P is a CHARACTER and P
    !>          = 'S':  Single
    !>          = 'D':  Double
    !>          = 'I':  Indigenous
    !>          = 'X' or 'E':  Extra
    !>

    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
    !>     matrix B.
    !>

    A

    !>          A is COMPLEX*16 array, dimension (LDA,N)
    !>     On entry, the N-by-N matrix A.
    !>

    LDA

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

    AF

    !>          AF is COMPLEX*16 array, dimension (LDAF,N)
    !>     The triangular factor U or L from the Cholesky factorization
    !>     A = U**T*U or A = L*L**T, as computed by ZPOTRF.
    !>

    LDAF

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

    COLEQU

    !>          COLEQU is LOGICAL
    !>     If .TRUE. then column equilibration was done to A before calling
    !>     this routine. This is needed to compute the solution and error
    !>     bounds correctly.
    !>

    C

    !>          C is DOUBLE PRECISION array, dimension (N)
    !>     The column scale factors for A. If COLEQU = .FALSE., C
    !>     is not accessed. If C is input, each element of C should be a power
    !>     of the radix to ensure a reliable solution and error estimates.
    !>     Scaling by powers of the radix does not cause rounding errors unless
    !>     the result underflows or overflows. Rounding errors during scaling
    !>     lead to refining with a matrix that is not equivalent to the
    !>     input matrix, producing error estimates that may not be
    !>     reliable.
    !>

    B

    !>          B is COMPLEX*16 array, dimension (LDB,NRHS)
    !>     The right-hand-side matrix B.
    !>

    LDB

    !>          LDB is INTEGER
    !>     The leading dimension of the array B.  LDB >= max(1,N).
    !>

    Y

    !>          Y is COMPLEX*16 array, dimension (LDY,NRHS)
    !>     On entry, the solution matrix X, as computed by ZPOTRS.
    !>     On exit, the improved solution matrix Y.
    !>

    LDY

    !>          LDY is INTEGER
    !>     The leading dimension of the array Y.  LDY >= max(1,N).
    !>

    BERR_OUT

    !>          BERR_OUT is DOUBLE PRECISION array, dimension (NRHS)
    !>     On exit, BERR_OUT(j) contains the componentwise relative backward
    !>     error for right-hand-side j from the formula
    !>         max(i) ( abs(RES(i)) / ( abs(op(A_s))*abs(Y) + abs(B_s) )(i) )
    !>     where abs(Z) is the componentwise absolute value of the matrix
    !>     or vector Z. This is computed by ZLA_LIN_BERR.
    !>

    N_NORMS

    !>          N_NORMS is INTEGER
    !>     Determines which error bounds to return (see ERR_BNDS_NORM
    !>     and ERR_BNDS_COMP).
    !>     If N_NORMS >= 1 return normwise error bounds.
    !>     If N_NORMS >= 2 return componentwise error bounds.
    !>

    ERR_BNDS_NORM

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

    ERR_BNDS_COMP

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

    RES

    !>          RES is COMPLEX*16 array, dimension (N)
    !>     Workspace to hold the intermediate residual.
    !>

    AYB

    !>          AYB is DOUBLE PRECISION array, dimension (N)
    !>     Workspace.
    !>

    DY

    !>          DY is COMPLEX*16 PRECISION array, dimension (N)
    !>     Workspace to hold the intermediate solution.
    !>

    Y_TAIL

    !>          Y_TAIL is COMPLEX*16 array, dimension (N)
    !>     Workspace to hold the trailing bits of the intermediate solution.
    !>

    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.
    !>

    ITHRESH

    !>          ITHRESH is INTEGER
    !>     The maximum number of residual computations allowed for
    !>     refinement. The default is 10. For '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.
    !>

    RTHRESH

    !>          RTHRESH is DOUBLE PRECISION
    !>     Determines when to stop refinement if the error estimate stops
    !>     decreasing. Refinement will stop when the next solution no longer
    !>     satisfies norm(dx_{i+1}) < RTHRESH * norm(dx_i) where norm(Z) is
    !>     the infinity norm of Z. RTHRESH satisfies 0 < RTHRESH <= 1. The
    !>     default value is 0.5. For 'aggressive' set to 0.9 to permit
    !>     convergence on extremely ill-conditioned matrices. See LAWN 165
    !>     for more details.
    !>

    DZ_UB

    !>          DZ_UB is DOUBLE PRECISION
    !>     Determines when to start considering componentwise convergence.
    !>     Componentwise convergence is only considered after each component
    !>     of the solution Y is stable, which we define as the relative
    !>     change in each component being less than DZ_UB. The default value
    !>     is 0.25, requiring the first bit to be stable. See LAWN 165 for
    !>     more details.
    !>

    IGNORE_CWISE

    !>          IGNORE_CWISE is LOGICAL
    !>     If .TRUE. then ignore componentwise convergence. Default value
    !>     is .FALSE..
    !>

    INFO

    !>          INFO is INTEGER
    !>       = 0:  Successful exit.
    !>       < 0:  if INFO = -i, the ith argument to ZPOTRS had an illegal
    !>             value
    !>

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Author

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