CXML
        	    
LAPACK version 3.0
sspsvx(3)

PURPOSE
  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

SYNTAX
  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, * )

DESCRIPTION
  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. If some D(i,i)=0, so that D is exactly singular, then the routine
     returns with INFO = i. Otherwise, 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,
     INFO = N+1 is returned as a warning, but the routine still goes on
     to solve for X and compute error bounds as described below.

  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)*(2*n-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
	  determined 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
	  determined 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 or INFO = N+1, 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
	  condition is indicated by a return code of INFO > 0.

  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
	  > 0:	if INFO = i, and i is
	  <= N:	 D(i,i) is exactly zero.  The factorization has been
	  completed but the factor D is exactly singular, so the solution and
	  error bounds could not be computed. RCOND = 0 is returned.  = N+1:
	  D is nonsingular, but RCOND is less than machine precision, meaning
	  that the matrix is singular to working precision.  Nevertheless,
	  the solution and error bounds are computed because there are a
	  number of situations where the computed solution can be more
	  accurate than the value of RCOND would suggest.

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 ]