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 equations
   A * X = B, where A is an N-by-N symmetric positive definite matrix and X and B are N-by-NRHS 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 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) DOUBLE PRECISION array,

dimension (LDA,N) On entry, 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. 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 N-by-NRHS 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 N-by-NRHS 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 right-hand 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 machine-specific 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 i-th 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. =========