SYNOPSIS

SUBROUTINE ZCPOSV(

UPLO, N, NRHS, A, LDA, B, LDB, X, LDX, WORK,

    

+ SWORK, RWORK, ITER, INFO )

    

CHARACTER UPLO

    

INTEGER INFO, ITER, LDA, LDB, LDX, N, NRHS

    

DOUBLE PRECISION RWORK( * )

    

COMPLEX SWORK( * )

    

COMPLEX*16 A( LDA, * ), B( LDB, * ), WORK( N, * ),

    

+ X( LDX, * )

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) CHARACTER

= 'U': Upper triangle of A is stored;
= 'L': Lower triangle of A is stored.

N (input) INTEGER

The number of linear equations, i.e., the order of the matrix A. N >= 0.

NRHS (input) INTEGER

The number of right hand sides, i.e., the number of columns of the matrix B. NRHS >= 0.

A (input or input/ouptut) 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) INTEGER

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

B (input) COMPLEX*16 array, dimension (LDB,NRHS)

The N-by-NRHS right hand side matrix B.

LDB (input) INTEGER

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

X (output) COMPLEX*16 array, dimension (LDX,NRHS)

If INFO = 0, the N-by-NRHS solution matrix X.

LDX (input) INTEGER

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

WORK (workspace) COMPLEX*16 array, dimension (N*NRHS)

This array is used to hold the residual vectors.

SWORK (workspace) 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 (workspace) DOUBLE PRECISION array, dimension (N)

ITER (output) 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) 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. =========