User`s guide

LDL Factorization
5-254
5LDL Factorization
Purpose Factor a square Hermitian positive definite matrix into lower, upper, and
diagonal components.
Library Math Functions / Matrices and Linear Algebra / Matrix Factorizations
Description The LDL Factorization block uniquely factors the square Hermitian positive
definite input matrix S as
where L is a lower triangular square matrix with unity diagonal elements, D is
a diagonal matrix, and L
*
is the Hermitian (complex conjugate) transpose of L.
Only the diagonal and lower triangle of the input matrix are used, and any
imaginary component of the diagonal entries is disregarded.
The block’s output is a composite matrix with lower triangle elements l
ij
from L, diagonal elements d
ij
from D, and upper triangle elements u
ij
from L
*
.
It is always sample-based. The output format is shown below for a 5-by-5
matrix.
LDL factorization requires half the computation of Gaussian elimination
(LU decomposition), and is always stable. It is more efficient that Cholesky
factorization because it avoids computing the square roots of the diagonal
elements.
The algorithm requires that the input be square and Hermitian positive
definite. When the input is not positive definite, the block reacts with the
behavior specified by the
Non-positive definite input parameter.
S LDL
*
=
d
11
u
12
u
13
u
14
u
15
l
21
d
22
u
23
u
24
u
25
l
31
l
32
d
33
u
34
u
35
l
41
l
42
l
43
d
44
u
45
l
51
l
52
l
53
l
54
d
55
u
ij
l
ji
*
=