SSPRFS(l)								 )								 SSPRFS(l)

NAME

       SSPRFS  - 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 SSPRFS( 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( * )

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

PURPOSE

       SSPRFS 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) REAL 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) REAL 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 SSPTRF, 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 SSPTRF.

       B       (input) REAL 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) REAL array, dimension (LDX,NRHS)
	       On entry, the solution matrix X, as computed by SSPTRS.	On exit, the improved solution matrix X.

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

       FERR    (output) REAL 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) REAL 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) REAL 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 							 SSPRFS(l)