SYNOPSIS
 SUBROUTINE DSPOSV(
 UPLO, N, NRHS, A, LDA, B, LDB, X, LDX, WORK,
 + SWORK, ITER, INFO )
 CHARACTER UPLO
 INTEGER INFO, ITER, LDA, LDB, LDX, N, NRHS
 REAL SWORK( * )
 DOUBLE PRECISION A( LDA, * ), B( LDB, * ), WORK( N, * ),
 + X( LDX, * )
PURPOSE
DSPOSV computes the solution to a real system of linear equationsA * X = B, where A is an NbyN symmetric positive definite matrix and X and B are NbyNRHS matrices.
DSPOSV first attempts to factorize the matrix in SINGLE PRECISION and use this factorization within an iterative refinement procedure to produce a solution with DOUBLE PRECISION normwise backward error quality (see below). If the approach fails the method switches to a DOUBLE PRECISION factorization and solve.
The iterative refinement is not going to be a winning strategy if the ratio SINGLE PRECISION performance over DOUBLE PRECISION performance is too small. A reasonable strategy should take the number of righthand 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 infinitynorm of the residual
o XNRM is the infinitynorm of the solution
o ANRM is the infinityoperatornorm 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) DOUBLE PRECISION array,
 dimension (LDA,N) On entry, the symmetric matrix A. If UPLO = 'U', the leading NbyN 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 NbyN 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. 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**T*U or A = L*L**T.
 LDA (input) INTEGER
 The leading dimension of the array A. LDA >= max(1,N).
 B (input) DOUBLE PRECISION array, dimension (LDB,NRHS)
 The NbyNRHS right hand side matrix B.
 LDB (input) INTEGER
 The leading dimension of the array B. LDB >= max(1,N).
 X (output) DOUBLE PRECISION array, dimension (LDX,NRHS)
 If INFO = 0, the NbyNRHS solution matrix X.
 LDX (input) INTEGER
 The leading dimension of the array X. LDX >= max(1,N).
 WORK (workspace) DOUBLE PRECISION array, dimension (N*NRHS)
 This array is used to hold the residual vectors.
 SWORK (workspace) REAL array, dimension (N*(N+NRHS))
 This array is used to use the single precision matrix and the righthand sides or solutions in single precision.
 ITER (output) INTEGER

< 0: iterative refinement has failed, double precision
factorization has been performed
1 : the routine fell back to full precision for
implementation or machinespecific reasons
2 : narrowing the precision induced an overflow,
the routine fell back to full precision
3 : failure of SPOTRF
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 ith argument had an illegal value
> 0: if INFO = i, the leading minor of order i of (DOUBLE PRECISION) A is not positive definite, so the factorization could not be completed, and the solution has not been computed. =========