ZCPOSV(3)      LAPACK routine of NEC Numeric Library Collection      ZCPOSV(3)



NAME
       ZCPOSV

SYNOPSIS
       SUBROUTINE ZCPOSV (UPLO, N, NRHS, A, LDA, B, LDB, X, LDX, WORK, SWORK,
           RWORK, ITER, INFO)



PURPOSE
            ZCPOSV computes the solution to a complex system of linear equations
               A * X = B,
            where A is an N-by-N Hermitian positive definite matrix and X and B
            are N-by-NRHS matrices.

            ZCPOSV first attempts to factorize the matrix in COMPLEX and use this
            factorization within an iterative refinement procedure to produce a
            solution with COMPLEX*16 normwise backward error quality (see below).
            If the approach fails the method switches to a COMPLEX*16
            factorization and solve.

            The iterative refinement is not going to be a winning strategy if
            the ratio COMPLEX performance over COMPLEX*16 performance is too
            small. A reasonable strategy should take the number of right-hand
            sides and the size of the matrix into account. This might be done
            with a call to ILAENV in the future. Up to now, we always try
            iterative refinement.

            The iterative refinement process is stopped if
                ITER > ITERMAX
            or for all the RHS we have:
                RNRM < SQRT(N)*XNRM*ANRM*EPS*BWDMAX
            where
                o ITER is the number of the current iteration in the iterative
                  refinement process
                o RNRM is the infinity-norm of the residual
                o XNRM is the infinity-norm of the solution
                o ANRM is the infinity-operator-norm of the matrix A
                o EPS is the machine epsilon returned by DLAMCH('Epsilon')
            The value ITERMAX and BWDMAX are fixed to 30 and 1.0D+00
            respectively.




ARGUMENTS
           UPLO      (input)
                     UPLO is CHARACTER*1
                     = 'U':  Upper triangle of A is stored;
                     = 'L':  Lower triangle of A is stored.

           N         (input)
                     N is INTEGER
                     The number of linear equations, i.e., the order of the
                     matrix A.  N >= 0.

           NRHS      (input)
                     NRHS is INTEGER
                     The number of right hand sides, i.e., the number of columns
                     of the matrix B.  NRHS >= 0.

           A         (input/output)
                     A is COMPLEX*16 array,
                     dimension (LDA,N)
                     On entry, 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.

                     Note that the imaginary parts of the diagonal
                     elements need not be set and are assumed to be zero.

                     On exit, if iterative refinement has been successfully used
                     (INFO.EQ.0 and ITER.GE.0, see description below), then A is
                     unchanged, if double precision factorization has been used
                     (INFO.EQ.0 and ITER.LT.0, see description below), then the
                     array A contains the factor U or L from the Cholesky
                     factorization A = U**H*U or A = L*L**H.

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

           B         (input)
                     B is COMPLEX*16 array, dimension (LDB,NRHS)
                     The N-by-NRHS right hand side matrix B.

           LDB       (input)
                     LDB is INTEGER
                     The leading dimension of the array B.  LDB >= max(1,N).

           X         (output)
                     X is COMPLEX*16 array, dimension (LDX,NRHS)
                     If INFO = 0, the N-by-NRHS solution matrix X.

           LDX       (input)
                     LDX is INTEGER
                     The leading dimension of the array X.  LDX >= max(1,N).

           WORK      (output)
                     WORK is COMPLEX*16 array, dimension (N*NRHS)
                     This array is used to hold the residual vectors.

           SWORK     (output)
                     SWORK is COMPLEX array, dimension (N*(N+NRHS))
                     This array is used to use the single precision matrix and the
                     right-hand sides or solutions in single precision.

           RWORK     (output)
                     RWORK is DOUBLE PRECISION array, dimension (N)

           ITER      (output)
                     ITER is INTEGER
                     < 0: iterative refinement has failed, COMPLEX*16
                          factorization has been performed
                          -1 : the routine fell back to full precision for
                               implementation- or machine-specific reasons
                          -2 : narrowing the precision induced an overflow,
                               the routine fell back to full precision
                          -3 : failure of CPOTRF
                          -31: stop the iterative refinement after the 30th
                               iterations
                     > 0: iterative refinement has been sucessfully used.
                          Returns the number of iterations

           INFO      (output)
                     INFO is INTEGER
                     = 0:  successful exit
                     < 0:  if INFO = -i, the i-th argument had an illegal value
                     > 0:  if INFO = i, the leading minor of order i of
                           (COMPLEX*16) A is not positive definite, so the
                           factorization could not be completed, and the solution
                           has not been computed.



LAPACK routine                  31 October 2017                      ZCPOSV(3)