dsytrf(3F)

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 contains 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 triangular part of A is not
             referenced.

             On exit, the block diagonal matrix D and the multipliers 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/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 performance 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 factorization has
             been completed, but the block diagonal 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).