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 sym­
               metric 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  mul­
               tipliers  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 struc­
               ture 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

               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  ille­
               gal value
               >  0:  if  INFO  = i, D(i,i) is exactly zero.  The
               factorization has been completed,  but  the  block
               diagonal  matrix  D is exactly singular, and divi­
               sion 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)  over­
       writes  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

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


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