zposvx.f(3)

NAME
zposvx.f -

SYNOPSIS

  Functions/Subroutines
subroutine zposvx (FACT, UPLO, N, NRHS, A, LDA, AF, LDAF, EQUED, S, B, LDB, X, LDX, RCOND, FERR, BERR, WORK, RWORK, INFO)
ZPOSVX computes the solution to system of linear equations A * X = B for PO matrices

Function/Subroutine Documentation

  subroutine zposvx (characterFACT, characterUPLO, integerN, integerNRHS, complex*16, dimension( lda, * )A, integerLDA, complex*16, dimension( ldaf, * )AF, integerLDAF, characterEQUED, double precision, dimension( * )S, complex*16, dimension( ldb, * )B, integerLDB, complex*16, dimension( ldx, * )X, integerLDX, double precisionRCOND, double precision, dimension( * )FERR, double precision, dimension( * )BERR, complex*16, dimension( * )WORK, double precision, dimension( * )RWORK, integerINFO)

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

Purpose:

     ZPOSVX uses the Cholesky factorization A = U**H*U or A = L*L**H 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.
     Error bounds on the solution and a condition estimate are also
     provided.

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**H* U,  if UPLO = 'U', or
           A = L * L**H,  if UPLO = 'L',
        where U is an upper triangular matrix and L is a lower triangular
        matrix.
     3. If the leading i-by-i principal minor is not positive definite,
        then the routine returns with INFO = i. Otherwise, the factored
        form of A is used to estimate the condition number of the matrix
        A.  If the reciprocal of the condition number is less than machine
        precision, INFO = N+1 is returned as a warning, but 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. Iterative refinement is applied to improve the computed solution
        matrix and calculate error bounds and backward error estimates
        for it.
     6. If equilibration was used, the matrix X is premultiplied by
        diag(S) so that it solves the original system before
        equilibration.

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 = 'Y', the matrix A has been equilibrated
                      with scaling factors given by S.  A and AF will not
                      be 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**H *U or A = L*L**H, 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**H *U or A = L*L**H 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**H *U or A = L*L**H 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':  Equilibration was done, 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 scale factors for A; not accessed if EQUED = 'N'.  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.

    B

              B is COMPLEX*16 array, dimension (LDB,NRHS)
              On entry, the N-by-NRHS righthand 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 or INFO = N+1, the N-by-NRHS solution matrix X to
              the original system of equations.  Note that if EQUED = 'Y',
              A and B are modified on exit, 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
              The estimate of the reciprocal condition number of the matrix
              A after equilibration (if done).  If RCOND is less than the
              machine precision (in particular, if RCOND = 0), the matrix
              is singular to working precision.  This condition is
              indicated by a return code of INFO > 0.

    FERR

              FERR is DOUBLE PRECISION array, dimension (NRHS)
              The estimated forward error bound for each solution vector
              X(j) (the j-th column of the solution matrix X).
              If XTRUE is the true solution corresponding to X(j), FERR(j)
              is an estimated upper bound for the magnitude of the largest
              element in (X(j) - XTRUE) divided by the magnitude of the
              largest element in X(j).  The estimate is as reliable as
              the estimate for RCOND, and is almost always a slight
              overestimate of the true error.

    BERR

              BERR is DOUBLE PRECISION array, dimension (NRHS)
              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).

    WORK

              WORK is COMPLEX*16 array, dimension (2*N)

    RWORK

              RWORK is DOUBLE PRECISION array, dimension (N)

    INFO

              INFO is INTEGER
              = 0: successful exit
              < 0: if INFO = -i, the i-th argument had an illegal value
              > 0: if INFO = i, and i is
                    <= N:  the leading minor of order i of A is
                           not positive definite, so the factorization
                           could not be completed, and the solution has not
                           been computed. RCOND = 0 is returned.
                    = N+1: U is nonsingular, but RCOND is less than machine
                           precision, meaning that the matrix is singular
                           to working precision.  Nevertheless, the
                           solution and error bounds are computed because
                           there are a number of situations where the
                           computed solution can be more accurate than the
                           value of RCOND would suggest.

Author:
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Date:
    April 2012

Definition at line 305 of file zposvx.f.

Author

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