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dsytrf


 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 )

 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 triangular
              part of the matrix A, and the strictly upper tri-
              angular part of A is not referenced.

              On exit, the block diagonal matrix D and the multi-
              pliers 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 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.

      WORK    (workspace) 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.

      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 fac-
              torization has been completed, but the block diago-
              nal matrix D is exactly singular, and division 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).