
DSPTRF(l) ) DSPTRF(l)
NAME
DSPTRF  compute the factorization of a real symmetric matrix A stored in packed format
using the BunchKaufman diagonal pivoting method
SYNOPSIS
SUBROUTINE DSPTRF( UPLO, N, AP, IPIV, INFO )
CHARACTER UPLO
INTEGER INFO, N
INTEGER IPIV( * )
DOUBLE PRECISION AP( * )
PURPOSE
DSPTRF computes the factorization of a real symmetric matrix A stored in packed format
using the BunchKaufman diagonal pivoting method:
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.
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.
AP (input/output) DOUBLE PRECISION array, dimension (N*(N+1)/2)
On entry, the upper or lower triangle of the symmetric matrix A, packed columnwise
in a linear array. The jth column of A is stored in the array AP as follows: if
UPLO = 'U', AP(i + (j1)*j/2) = A(i,j) for 1<=i<=j; if UPLO = 'L', AP(i +
(j1)*(2nj)/2) = A(i,j) for j<=i<=n.
On exit, the block diagonal matrix D and the multipliers used to obtain the factor
U or L, stored as a packed triangular matrix overwriting A (see below for further
details).
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.
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
596  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 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 DSPTRF(l) 
