zhpsvx.f(3)

NAME
zhpsvx.f -

SYNOPSIS

  Functions/Subroutines
subroutine ZHPSVX (FACT, UPLO, N, NRHS, AP, AFP, IPIV, B, LDB, X, LDX, RCOND, FERR, BERR, WORK, RWORK, INFO)
ZHPSVX computes the solution to system of linear equations A * X = B for OTHER matrices

Function/Subroutine Documentation

  subroutine ZHPSVX (CHARACTERFACT, CHARACTERUPLO, INTEGERN, INTEGERNRHS, COMPLEX*16, dimension( * )AP, COMPLEX*16, dimension( * )AFP, INTEGER, dimension( * )IPIV, 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)

ZHPSVX computes the solution to system of linear equations A * X = B for OTHER matrices

Purpose:

     ZHPSVX uses the diagonal pivoting factorization A = U*D*U**H or
     A = L*D*L**H to compute the solution to a complex system of linear
     equations A * X = B, where A is an N-by-N Hermitian matrix stored
     in packed format 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 = 'N', the diagonal pivoting method is used to factor A as
           A = U * D * U**H,  if UPLO = 'U', or
           A = L * D * L**H,  if UPLO = 'L',
        where U (or L) is a product of permutation and unit upper (lower)
        triangular matrices and D is Hermitian and block diagonal with
        1-by-1 and 2-by-2 diagonal blocks.
     2. If some D(i,i)=0, so that D is exactly singular, 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.
     3. The system of equations is solved for X using the factored form
        of A.
     4. Iterative refinement is applied to improve the computed solution
        matrix and calculate error bounds and backward error estimates
        for it.

Parameters:
    FACT

              FACT is CHARACTER*1
              Specifies whether or not the factored form of A has been
              supplied on entry.
              = 'F':  On entry, AFP and IPIV contain the factored form of
                      A.  AFP and IPIV will not be modified.
              = 'N':  The matrix A will be copied to AFP 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.

    AP

              AP is COMPLEX*16 array, dimension (N*(N+1)/2)
              The upper or lower triangle of the Hermitian matrix A, packed
              columnwise in a linear array.  The j-th column of A is stored
              in the array AP as follows:
              if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
              if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.
              See below for further details.

    AFP

              AFP is COMPLEX*16 array, dimension (N*(N+1)/2)
              If FACT = 'F', then AFP is an input argument and on entry
              contains the block diagonal matrix D and the multipliers used
              to obtain the factor U or L from the factorization
              A = U*D*U**H or A = L*D*L**H as computed by ZHPTRF, stored as
              a packed triangular matrix in the same storage format as A.
              If FACT = 'N', then AFP is an output argument and on exit
              contains the block diagonal matrix D and the multipliers used
              to obtain the factor U or L from the factorization
              A = U*D*U**H or A = L*D*L**H as computed by ZHPTRF, stored as
              a packed triangular matrix in the same storage format as A.

    IPIV

              IPIV is INTEGER array, dimension (N)
              If FACT = 'F', then IPIV is an input argument and on entry
              contains details of the interchanges and the block structure
              of D, as determined by ZHPTRF.
              If IPIV(k) > 0, then rows and columns k and IPIV(k) were
              interchanged and D(k,k) is a 1-by-1 diagonal block.
              If UPLO = 'U' and IPIV(k) = IPIV(k-1) < 0, then rows and
              columns k-1 and -IPIV(k) were interchanged and D(k-1:k,k-1:k)
              is a 2-by-2 diagonal block.  If UPLO = 'L' and IPIV(k) =
              IPIV(k+1) < 0, then rows and columns k+1 and -IPIV(k) were
              interchanged and D(k:k+1,k:k+1) is a 2-by-2 diagonal block.
              If FACT = 'N', then IPIV is an output argument and on exit
              contains details of the interchanges and the block structure
              of D, as determined by ZHPTRF.

    B

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

    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.  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:  D(i,i) is exactly zero.  The factorization
                           has been completed but the factor D is exactly
                           singular, so the solution and error bounds could
                           not be computed. RCOND = 0 is returned.
                    = N+1: D 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

Further Details:

      The packed storage scheme is illustrated by the following example
      when N = 4, UPLO = 'U':
      Two-dimensional storage of the Hermitian matrix A:
         a11 a12 a13 a14
             a22 a23 a24
                 a33 a34     (aij = conjg(aji))
                     a44
      Packed storage of the upper triangle of A:
      AP = [ a11, a12, a22, a13, a23, a33, a14, a24, a34, a44 ]

Definition at line 277 of file zhpsvx.f'.

Author

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