DSPRFS(l)								 )								 DSPRFS(l)

NAME

       DSPRFS  - improve the computed solution to a system of linear equations when the coefficient matrix is symmetric indefinite and packed, and
       provides error bounds and backward error estimates for the solution

SYNOPSIS

       SUBROUTINE DSPRFS( UPLO, N, NRHS, AP, AFP, IPIV, B, LDB, X, LDX, FERR, BERR, WORK, IWORK, INFO )

	   CHARACTER	  UPLO

	   INTEGER	  INFO, LDB, LDX, N, NRHS

	   INTEGER	  IPIV( * ), IWORK( * )

	   DOUBLE	  PRECISION AFP( * ), AP( * ), B( LDB, * ), BERR( * ), FERR( * ), WORK( * ), X( LDX, * )

PURPOSE

       DSPRFS improves the computed solution to a system of linear equations when the coefficient matrix is symmetric indefinite and  packed,  and
       provides error bounds and backward error estimates for the solution.

ARGUMENTS

       UPLO    (input) CHARACTER*1
	       = 'U':  Upper triangle of A is stored;
	       = 'L':  Lower triangle of A is stored.

       N       (input) INTEGER
	       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 matrices B and X.  NRHS >= 0.

       AP      (input) DOUBLE PRECISION array, dimension (N*(N+1)/2)
	       The  upper or lower triangle of the symmetric matrix A, packed columnwise in a linear array.  The j-th column of A is stored in the
	       array AP as follows: if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j; if UPLO = 'L', AP(i +  (j-1)*(2*n-j)/2)  =  A(i,j)  for
	       j<=i<=n.

       AFP     (input) DOUBLE PRECISION array, dimension (N*(N+1)/2)
	       The  factored  form of the matrix A.  AFP contains the block diagonal matrix D and the multipliers used to obtain the factor U or L
	       from the factorization A = U*D*U**T or A = L*D*L**T as computed by DSPTRF, stored as a packed triangular matrix.

       IPIV    (input) INTEGER array, dimension (N)
	       Details of the interchanges and the block structure of D as determined by DSPTRF.

       B       (input) DOUBLE PRECISION array, dimension (LDB,NRHS)
	       The right hand side matrix B.

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

       X       (input/output) DOUBLE PRECISION array, dimension (LDX,NRHS)
	       On entry, the solution matrix X, as computed by DSPTRS.	On exit, the improved solution matrix X.

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

       FERR    (output) DOUBLE PRECISION array, dimension (NRHS)
	       The estimated forward error bound for each solution vector X(j) (the j-th column of the solution matrix X).  If XTRUE is  the  true
	       solution  corresponding	to  X(j),  FERR(j)  is an estimated upper bound for the magnitude of the largest element in (X(j) - XTRUE)
	       divided by the magnitude of the largest element in X(j).  The estimate is as reliable as the estimate  for  RCOND,  and	is  almost
	       always a slight overestimate of the true error.

       BERR    (output) DOUBLE PRECISION array, dimension (NRHS)
	       The componentwise relative backward error of each solution vector X(j) (i.e., the smallest relative change in any element of A or B
	       that makes X(j) an exact solution).

       WORK    (workspace) DOUBLE PRECISION array, dimension (3*N)

       IWORK   (workspace) INTEGER array, dimension (N)

       INFO    (output) INTEGER
	       = 0:  successful exit
	       < 0:  if INFO = -i, the i-th argument had an illegal value

PARAMETERS

       ITMAX is the maximum number of steps of iterative refinement.

LAPACK version 3.0						   15 June 2000 							 DSPRFS(l)