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.

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 matrix B.  NRHS >= 0.

A

          A is 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

          LDA is INTEGER


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

B

          B is DOUBLE PRECISION 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 DOUBLE PRECISION array, dimension (LDX,NRHS)


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

LDX

          LDX is INTEGER


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

WORK

          WORK is DOUBLE PRECISION array, dimension (N,NRHS)


          This array is used to hold the residual vectors.

SWORK

          SWORK is 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

          ITER is 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

          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 (DOUBLE


                PRECISION) A is not positive definite, so the


                factorization could not be completed, and the solution


                has not been computed.

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

November 2011