csysvx.f(3)

NAME
csysvx.f -

SYNOPSIS

  Functions/Subroutines
subroutine csysvx (FACT, UPLO, N, NRHS, A, LDA, AF, LDAF, IPIV, B, LDB, X, LDX, RCOND, FERR, BERR, WORK, LWORK, RWORK, INFO)
CSYSVX computes the solution to system of linear equations A * X = B for SY matrices

Function/Subroutine Documentation

  subroutine csysvx (characterFACT, characterUPLO, integerN, integerNRHS, complex, dimension( lda, * )A, integerLDA, complex, dimension( ldaf, * )AF, integerLDAF, integer, dimension( * )IPIV, complex, dimension( ldb, * )B, integerLDB, complex, dimension( ldx, * )X, integerLDX, realRCOND, real, dimension( * )FERR, real, dimension( * )BERR, complex, dimension( * )WORK, integerLWORK, real, dimension( * )RWORK, integerINFO)

CSYSVX computes the solution to system of linear equations A * X = B for SY matrices

Purpose:

     CSYSVX uses the diagonal pivoting factorization to compute the
     solution to a complex system of linear equations A * X = B,
     where A is an N-by-N symmetric 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 = 'N', the diagonal pivoting method is used to factor A.
        The form of the factorization is
           A = U * D * U**T,  if UPLO = 'U', or
           A = L * D * L**T,  if UPLO = 'L',
        where U (or L) is a product of permutation and unit upper (lower)
        triangular matrices, and D is symmetric 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, AF and IPIV contain the factored form
                      of A.  A, AF and IPIV will not be modified.
              = 'N':  The matrix A will be 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 array, dimension (LDA,N)
              The symmetric matrix A.  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.

    LDA

              LDA is INTEGER
              The leading dimension of the array A.  LDA >= max(1,N).

    AF

              AF is COMPLEX array, dimension (LDAF,N)
              If FACT = 'F', then AF 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**T or A = L*D*L**T as computed by CSYTRF.
              If FACT = 'N', then AF is an output argument and on exit
              returns the block diagonal matrix D and the multipliers used
              to obtain the factor U or L from the factorization
              A = U*D*U**T or A = L*D*L**T.

    LDAF

              LDAF is INTEGER
              The leading dimension of the array AF.  LDAF >= max(1,N).

    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 CSYTRF.
              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 CSYTRF.

    B

              B is COMPLEX 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 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 REAL
              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 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 COMPLEX array, dimension (MAX(1,LWORK))
              On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

    LWORK

              LWORK is INTEGER
              The length of WORK.  LWORK >= max(1,2*N), and for best
              performance, when FACT = 'N', LWORK >= max(1,2*N,N*NB), where
              NB is the optimal blocksize for CSYTRF.
              If LWORK = -1, then a workspace query is assumed; the routine
              only calculates the optimal size of the WORK array, returns
              this value as the first entry of the WORK array, and no error
              message related to LWORK is issued by XERBLA.

    RWORK

              RWORK is REAL 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

Definition at line 284 of file csysvx.f.

Author

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