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NAME

       chesvx.f -

SYNOPSIS

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

Function/Subroutine Documentation

   subroutine chesvx (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)
        CHESVX computes the solution to system of linear equations A * X = B for HE matrices

       Purpose:

            CHESVX 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 Hermitian 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**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, 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 Hermitian 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**H or A = L*D*L**H as computed by CHETRF.

                     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**H or A = L*D*L**H.

           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 CHETRF.
                     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 CHETRF.

           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 CHETRF.

                     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 chesvx.f.

Author

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