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( * )


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).

               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,
               dimension (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.

               If  LWORK = -1, then a workspace query is assumed;
               the routine only calculates the  optimal  size  of
               the  WORK  array,  returns this value as the first
               entry of the WORK  array,  and  no  error  message
               related to LWORK is issued by XERBLA.

       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

                     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

       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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