SRC/zhpsvx.f(3)

NAME
SRC/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 (character fact, character uplo, integer n, integer nrhs, complex*16, dimension( * ) ap, complex*16, dimension( * ) afp, integer, dimension( * ) ipiv, complex*16, dimension( ldb, * ) b, integer ldb, complex*16, dimension( ldx, * ) x, integer ldx, double precision rcond, double precision, dimension( * ) ferr, double precision, dimension( * ) berr, complex*16, dimension( * ) work, double precision, dimension( * ) rwork, integer info)

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.

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 275 of file zhpsvx.f.

Author

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