SSPSVX(l)	     LAPACK driver routine (version 1.1)	    SSPSVX(l)

NAME
  SSPSVX - use the diagonal pivoting factorization A = U*D*U**T	or A =
  L*D*L**T to compute the solution to a	real system of linear equations	A * X
  = B, where A is an N-by-N symmetric matrix stored in packed format and X
  and B	are N-by-NRHS matrices

SYNOPSIS

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

      CHARACTER	     FACT, UPLO

      INTEGER	     INFO, LDB,	LDX, N,	NRHS

      REAL	     RCOND

      INTEGER	     IPIV( * ),	IWORK( * )

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

PURPOSE
  SSPSVX uses the diagonal pivoting factorization A = U*D*U**T or A =
  L*D*L**T to compute the solution to a	real system of linear equations	A * X
  = B, where A is an N-by-N symmetric matrix stored in packed format and X
  and B	are N-by-NRHS matrices.

  Error	bounds on the solution and a condition estimate	are also provided.

DESCRIPTION
  The following	steps are performed:

  1. If	FACT = 'N', the	diagonal pivoting method is used to factor A as
	A = U *	D * U**T,  if UPLO = 'U', or
	A = L *	D * L**T,  if UPLO = 'L',
     where U (or L) is a product of permutation	and unit upper (lower)
     triangular	matrices and D is symmetric and	block diagonal with
     1-by-1 and	2-by-2 diagonal	blocks.

  2. The factored form of A is used to estimate	the condition number
     of	the matrix A.  If the reciprocal of the	condition number is
     less than machine precision, steps	3 and 4	are skipped.

  3. The system	of equations is	solved for X using the factored	form
     of	A.

  4. Iterative refinement is applied to	improve	the computed solution
     matrix and	calculate error	bounds and backward error estimates
     for it.

ARGUMENTS

  FACT	  (input) CHARACTER*1
	  Specifies whether or not the factored	form of	A has been supplied
	  on entry.  = 'F':  On	entry, AFP and IPIV contain the	factored form
	  of A.	 AP, AFP and IPIV will not be modified.	 = 'N':	 The matrix A
	  will be copied to AFP	and factored.

  UPLO	  (input) CHARACTER*1
	  = '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
	  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)*(2n-j)/2) = A(i,j) for
	  j<=i<=n.  See	below for further details.

  AFP	  (input or output) REAL array,	dimension
	  (N*(N+1)/2) If FACT =	'F', then AFP is an input argument and on
	  entry	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 in the	same storage format as A.

	  If FACT = 'N', then AFP is an	output argument	and on exit 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 in the
	  same storage format as A.

  IPIV	  (input or output) INTEGER array, dimension (N)
	  If FACT = 'F', then IPIV is an input argument	and on entry contains
	  details of the interchanges and the block structure of D, as deter-
	  mined	by SSPTRF.  If IPIV(k) > 0, then rows and columns k and
	  IPIV(k) were interchanged and	D(k,k) is a 1-by-1 diagonal block.
	  If UPLO = 'U'	and IPIV(k) = IPIV(k-1)	< 0, then rows and columns
	  k-1 and -IPIV(k) were	interchanged and D(k-1:k,k-1:k)	is a 2-by-2
	  diagonal block.  If UPLO = 'L' and IPIV(k) = IPIV(k+1) < 0, then
	  rows and columns k+1 and -IPIV(k) were interchanged and
	  D(k:k+1,k:k+1) is a 2-by-2 diagonal block.

	  If FACT = 'N', then IPIV is an output	argument and on	exit contains
	  details of the interchanges and the block structure of D, as deter-
	  mined	by SSPTRF.

  B	  (input) REAL 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) REAL	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).

  RCOND	  (output) REAL
	  The estimate of the reciprocal condition number of the matrix	A.
	  If RCOND is less than	the machine precision (in particular, if
	  RCOND	= 0), the matrix is singular to	working	precision.  This con-
	  dition is indicated by a return code of INFO > 0, and	the solution
	  and error bounds are not computed.

  FERR	  (output) REAL	array, dimension (NRHS)
	  The estimated	forward	error bounds for each solution vector X(j)
	  (the j-th column of the solution matrix X).  If XTRUE	is the true
	  solution, FERR(j) bounds the magnitude of the	largest	entry in
	  (X(j)	- XTRUE) divided by the	magnitude of the largest entry in
	  X(j).	 The quality of	the error bound	depends	on the quality of the
	  estimate of norm(inv(A)) computed in the code; if the	estimate of
	  norm(inv(A)) is accurate, the	error bound is guaranteed.

  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 entry	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
	  > 0 and <= N:	if INFO	= i, D(i,i) is exactly zero.  The factoriza-
	  tion has been	completed, but the block diagonal matrix D is exactly
	  singular, so the solution and	error bounds could not be computed.
	  = N+1: the block diagonal matrix D is	nonsingular, but RCOND is
	  less than machine precision.	The factorization has been completed,
	  but the matrix is singular to	working	precision, so the solution
	  and error bounds have	not been computed.

FURTHER	DETAILS
  The packed storage scheme is illustrated by the following example when N =
  4, UPLO = 'U':

  Two-dimensional storage of the symmetric matrix A:

     a11 a12 a13 a14
	 a22 a23 a24
	     a33 a34	 (aij =	aji)
		 a44

  Packed storage of the	upper triangle of A:

  AP = [ a11, a12, a22,	a13, a23, a33, a14, a24, a34, a44 ]