SSPTRF(l)								 )								 SSPTRF(l)

NAME

       SSPTRF - compute the factorization of a real symmetric matrix A stored in packed format using the Bunch-Kaufman diagonal pivoting method

SYNOPSIS

       SUBROUTINE SSPTRF( UPLO, N, AP, IPIV, INFO )

	   CHARACTER	  UPLO

	   INTEGER	  INFO, N

	   INTEGER	  IPIV( * )

	   REAL 	  AP( * )

PURPOSE

       SSPTRF computes the factorization of a real symmetric matrix A stored in packed format using the Bunch-Kaufman diagonal pivoting method:
	  A = U*D*U**T	or  A = L*D*L**T

       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.

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.

       AP      (input/output) REAL array, dimension (N*(N+1)/2)
	       On entry, 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.

	       On exit, the block diagonal matrix D and the multipliers used to obtain the factor U or L, stored as  a	packed	triangular  matrix
	       overwriting A (see below for further details).

       IPIV    (output) INTEGER array, dimension (N)
	       Details	of  the  interchanges and the block structure of D.  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.

       INFO    (output) INTEGER
	       = 0: successful exit
	       < 0: if INFO = -i, the i-th argument had an illegal value
	       > 0: if INFO = i, D(i,i) is exactly zero.  The factorization has been completed, but the block diagonal matrix D is exactly  singu-
	       lar, and division by zero will occur if it is used to solve a system of equations.

FURTHER DETAILS

       5-96 - Based on modifications by J. Lewis, Boeing Computer Services
	      Company

       If UPLO = 'U', then A = U*D*U', where
	  U = P(n)*U(n)* ... *P(k)U(k)* ...,
       i.e., U is a product of terms P(k)*U(k), where k decreases from n to 1 in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1 and
       2-by-2 diagonal blocks D(k).  P(k) is a permutation matrix as defined by IPIV(k), and U(k) is a unit upper triangular matrix, such that	if
       the diagonal block D(k) is of order s (s = 1 or 2), then

		  (   I    v	0   )	k-s
	  U(k) =  (   0    I	0   )	s
		  (   0    0	I   )	n-k
		     k-s   s   n-k

       If  s  = 1, D(k) overwrites A(k,k), and v overwrites A(1:k-1,k).  If s = 2, the upper triangle of D(k) overwrites A(k-1,k-1), A(k-1,k), and
       A(k,k), and v overwrites A(1:k-2,k-1:k).

       If UPLO = 'L', then A = L*D*L', where
	  L = P(1)*L(1)* ... *P(k)*L(k)* ...,
       i.e., L is a product of terms P(k)*L(k), where k increases from 1 to n in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1 and
       2-by-2  diagonal blocks D(k).  P(k) is a permutation matrix as defined by IPIV(k), and L(k) is a unit lower triangular matrix, such that if
       the diagonal block D(k) is of order s (s = 1 or 2), then

		  (   I    0	 0   )	k-1
	  L(k) =  (   0    I	 0   )	s
		  (   0    v	 I   )	n-k-s+1
		     k-1   s  n-k-s+1

       If s = 1, D(k) overwrites A(k,k), and v overwrites A(k+1:n,k).  If s = 2, the lower triangle  of  D(k)  overwrites  A(k,k),  A(k+1,k),  and
       A(k+1,k+1), and v overwrites A(k+2:n,k:k+1).

LAPACK version 3.0						   15 June 2000 							 SSPTRF(l)