
DSYTRF(l) ) DSYTRF(l)
NAME
DSYTRF  compute the factorization of a real symmetric matrix A using the BunchKaufman
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 BunchKaufman
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 1by1 and 2by2 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 NbyN upper trian
gular 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
NbyN 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 1by1 diagonal
block. If UPLO = 'U' and IPIV(k) = IPIV(k1) < 0, then rows and columns k1 and
IPIV(k) were interchanged and D(k1:k,k1:k) is a 2by2 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 2by2 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.
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 ith 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 1by1 and 2by2 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 ) ks
U(k) = ( 0 I 0 ) s
( 0 0 I ) nk
ks s nk
If s = 1, D(k) overwrites A(k,k), and v overwrites A(1:k1,k). If s = 2, the upper trian
gle of D(k) overwrites A(k1,k1), A(k1,k), and A(k,k), and v overwrites A(1:k2,k1: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 1by1 and 2by2 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 ) k1
L(k) = ( 0 I 0 ) s
( 0 v I ) nks+1
k1 s nks+1
If s = 1, D(k) overwrites A(k,k), and v overwrites A(k+1:n,k). If s = 2, the lower trian
gle 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).
LAPACK version 3.0 15 June 2000 DSYTRF(l) 
