SRC/dppsvx.f(3)

NAME
SRC/dppsvx.f

SYNOPSIS

  Functions/Subroutines
subroutine dppsvx (fact, uplo, n, nrhs, ap, afp, equed, s, b, ldb, x, ldx, rcond, ferr, berr, work, iwork, info)
DPPSVX computes the solution to system of linear equations A * X = B for OTHER matrices

Function/Subroutine Documentation

  subroutine dppsvx (character fact, character uplo, integer n, integer nrhs, double precision, dimension( * ) ap, double precision, dimension( * ) afp, 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, dimension( * ) ferr, double precision, dimension( * ) berr, double precision, dimension( * ) work, integer, dimension( * ) iwork, integer info)

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

Purpose:

    !>
    !> DPPSVX 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 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.  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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 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 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 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 DOUBLE PRECISION 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 principal minor of order i of A
    !>                       is not positive, 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.

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 309 of file dppsvx.f.

Author

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