SSYTF2(3f)

NAME
SSYTF2 - the factorization of a real symmetric matrix A using the Bunch-Kaufman diagonal pivoting method

SYNOPSIS

SUBROUTINE SSYTF2(
    UPLO, N, A, LDA, IPIV, INFO )

CHARACTER
    UPLO

INTEGER
    INFO, LDA, N

INTEGER
    IPIV( * )

REAL
    A( LDA, * )

PURPOSE
SSYTF2 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) REAL 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.

INFO    (output) INTEGER
    = 0: successful exit
    < 0: if INFO = -k, the k-th argument had an illegal 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 division by zero will occur if it is used to solve a system of equations.

FURTHER DETAILS
09-29-06 - patch from
  Bobby Cheng, MathWorks
  Replace l.204 and l.372
       IF( MAX( ABSAKK, COLMAX ).EQ.ZERO ) THEN
  by
       IF( (MAX( ABSAKK, COLMAX ).EQ.ZERO) .OR. SISNAN(ABSAKK) ) THEN 01-01-96 - Based on modifications by
  J. Lewis, Boeing Computer Services Company
  A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA 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) 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).