SRC/dposvxx.f(3)

NAME
SRC/dposvxx.f

SYNOPSIS

  Functions/Subroutines
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

Function/Subroutine Documentation

  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
    !>     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  boolean. Trust the answer if the
    !>              reciprocal condition number is less than the threshold
    !>              sqrt(n) * dlamch('Epsilon').
    !>
    !>     err = 2  error bound: The estimated forward error,
    !>              almost certainly within a factor of 10 of the true error
    !>              so long as the next entry is greater than the threshold
    !>              sqrt(n) * dlamch('Epsilon'). This error bound should only
    !>              be trusted if the previous boolean is true.
    !>
    !>     err = 3  Reciprocal condition number: Estimated normwise
    !>              reciprocal condition number.  Compared with the threshold
    !>              sqrt(n) * dlamch('Epsilon') to determine if the error
    !>              estimate is . These reciprocal condition
    !>              numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some
    !>              appropriately scaled matrix Z.
    !>              Let Z = S*A, where S scales each row by a power of the
    !>              radix so all absolute row sums of Z are approximately 1.
    !>
    !>     See Lapack Working Note 165 for further details and extra
    !>     cautions.
    !>

    ERR_BNDS_COMP

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

    NPARAMS

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

    PARAMS

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

    WORK

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

    IWORK

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

    INFO

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

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 490 of file dposvxx.f.

Author

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