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

SYNOPSIS
     SUBROUTINE DSYTRF( UPLO, N, A, LDA, IPIV, WORK, LWORK,  INFO
               )

     CHARACTER UPLO

     INTEGER INFO, LDA, LWORK, N

     INTEGER IPIV( * )

     DOUBLE PRECISION A( LDA, * ), WORK( LWORK )



     #include <sunperf.h>

     void dsytrf(char uplo, int  n,  double  *da,  int  lda,  int
               *ipivot, int *info) ;

PURPOSE
     DSYTRF computes the factorization of a real symmetric matrix
     A  using  the  Bunch-Kaufman  diagonal pivoting method.  The
     form of the factorization is

        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.

     This is the blocked version of the algorithm, calling  Level
     3 BLAS.


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.

     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).

     LDA       (input) INTEGER
               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.

     WORK      (workspace/output) DOUBLE PRECISION array,  dimen-
               sion (LWORK)
               On exit, if INFO = 0, WORK(1) returns the  optimal
               LWORK.

     LWORK     (input) INTEGER
               The length of WORK.  LWORK >=1.  For best  perfor-
               mance  LWORK  >=  N*NB, where NB is the block size
               returned by ILAENV.

     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
     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).

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