sppsvx.f(3)

NAME
sppsvx.f -

SYNOPSIS

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

Function/Subroutine Documentation

  subroutine sppsvx (characterFACT, characterUPLO, integerN, integerNRHS, real, dimension( * )AP, real, dimension( * )AFP, characterEQUED, real, dimension( * )S, real, dimension( ldb, * )B, integerLDB, real, dimension( ldx, * )X, integerLDX, realRCOND, real, dimension( * )FERR, real, dimension( * )BERR, real, dimension( * )WORK, integer, dimension( * )IWORK, integerINFO)

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

Purpose:

     SPPSVX uses the Cholesky factorization A = U**T*U or A = L*L**T to
     compute the solution to a real system of linear equations
        A * X = B,
     where A is an N-by-N symmetric positive definite 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 = '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**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 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, AFP contains the factored form of A.
                      If EQUED = 'Y', the matrix A has been equilibrated
                      with scaling factors given by S.  AP and AFP will not
                      be modified.
              = 'N':  The matrix A will be copied to AFP and factored.
              = 'E':  The matrix A will be equilibrated if necessary, then
                      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 REAL array, dimension (N*(N+1)/2)
              On entry, the upper or lower triangle of the symmetric matrix
              A, packed columnwise in a linear array, except if FACT = 'F'
              and EQUED = 'Y', then A must contain the equilibrated matrix
              diag(S)*A*diag(S).  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)*(2n-j)/2) = A(i,j) for j<=i<=n.
              See below for further details.  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).

    AFP

              AFP is REAL array, dimension
                                (N*(N+1)/2)
              If FACT = 'F', then AFP 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 AFP is the factored
              form of the equilibrated matrix A.
              If FACT = 'N', then AFP 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 AFP 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 AP for the form of the
              equilibrated matrix).

    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 REAL 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 REAL 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 REAL 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 REAL
              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 REAL 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 REAL 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 REAL array, dimension (3*N)

    IWORK

              IWORK is INTEGER 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

Further Details:

      The packed storage scheme is illustrated by the following example
      when N = 4, UPLO = 'U':
      Two-dimensional storage of the symmetric 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 312 of file sppsvx.f.

Author

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