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



NAME
       DSPOSV

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



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

           B         (input)
                     B is DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION array, dimension (N,NRHS)
                     This array is used to hold the residual vectors.

           SWORK     (output)
                     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      (output)
                     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      (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 (DOUBLE
                           PRECISION) A is not positive definite, so the
                           factorization could not be completed, and the solution
                           has not been computed.



LAPACK routine                  31 October 2017                      DSPOSV(3)