DSYTF2  -  compute  the  factorization of a real symmetric
       matrix A using the Bunch-Kaufman diagonal pivoting method


SYNOPSIS

       SUBROUTINE DSYTF2( UPLO, N, A, LDA, IPIV, INFO )

           CHARACTER      UPLO

           INTEGER        INFO, LDA, N

           INTEGER        IPIV( * )

           DOUBLE         PRECISION A( LDA, * )


PURPOSE

       DSYTF2 computes the  factorization  of  a  real  symmetric
       matrix A using the Bunch-Kaufman diagonal pivoting method:
          A = U*D*U'  or  A = L*D*L'

       where U (or L) is a product of permutation and unit  upper
       (lower) triangular matrices, U' is the transpose of U, and
       D is symmetric and block diagonal with 1-by-1  and  2-by-2
       diagonal blocks.

       This  is  the  unblocked version of the algorithm, calling
       Level 2 BLAS.


ARGUMENTS

       UPLO    (input) CHARACTER*1
               Specifies whether the upper  or  lower  triangular
               part of the symmetric matrix A is stored:
               = 'U':  Upper triangular
               = 'L':  Lower triangular

       N       (input) INTEGER
               The order of the matrix A.  N >= 0.

       A       (input/output) DOUBLE PRECISION array, dimension
               (LDA,N)
               On entry, the symmetric matrix A.  If UPLO =  'U',
               the leading n-by-n upper triangular part of A con­
               tains the upper triangular part of the  matrix  A,
               and the strictly lower triangular part of A is not
               referenced.  If UPLO =  'L',  the  leading  n-by-n
               lower triangular part of A contains the lower tri­
               angular part of the matrix  A,  and  the  strictly
               upper triangular part of A is not referenced.

               On  exit, the block diagonal matrix D and the mul­
               tipliers used to obtain the factor  U  or  L  (see
               below for further details).

               The  leading  dimension  of  the  array A.  LDA >=
               max(1,N).

       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
               -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 = -k, the k-th argument had an ille­
               gal value
               > 0: if INFO = k, D(k,k)  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

       1-96 - 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

       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) 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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