Global Positioning System Reference
In-Depth Information
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
Because
N
is positive definite, it is written as the product of a lower triangular matrix
L
and an upper triangular matrix
L
T
:
LL
T
N
=
(A.94)
It
is readily seen that if
E
is an orthonormal matrix having the property (A.45) then
th
e new matrix
B
LL
T
.
The lower and upper triangular matrices have several useful properties. For ex-
am
ple, the eigenvalues of a triangular matrix equal the diagonal elements, and the
de
terminant of the triangular matrix equals the product of the diagonal elements. Be-
cause the determinant of a matrix product is equal to the product of the determinants
of the factors, it follows that
N
is singular if one of the diagonal elements of
L
is
zero. This fact can be used advantageously during the computation of
L
to eliminate
parameters that cause a singularity.
The Cholesky algorithm provides the instruction for computing the lower triangu-
lar matrix
L
. The elements of
L
are
LE
is also a Cholesky factor because
BB
T
LEE
T
L
T
=
=
=
[35
Lin
—
1
——
Lon
PgE
/
1
n
jj
−
j
−
1
l
jm
for
k
=
j
m
=
1
n
jk
−
l
jk
=
(A.95)
k
−
1
2
1
l
kk
l
jm
l
km
for
k<j
m
=
1
[35
0
for
k>j
where 1
u
. The Cholesky algorithm preserves the pattern of
leading zeros in the rows and columns of
N
, as can be readily verified. For example,
if the first
x
elements in row
y
of
N
are zero, then the first
x
elements in row
y
of
L
are also zero. Taking advantage of this fact speeds up the computation of
L
for a
large system that exhibits significant patterns of leading zeros. The algorithm (A.95)
begins with the element
l
11
. Subsequently, the columns (or rows) can be computed
sequentially from 1 to
u
, whereby previously computed columns (or rows) remain
unchanged while the next one is computed.
The inverse of the submatrix
Q
22
—see (A.77)—can be conveniently expressed as
a function of the submatrix
L
22
. Using
L
,
≤
j
≤
u
and 1
≤
k
≤
L
11
O
L
=
(A.96)
L
21
L
22
to
express
N
in terms of submatrices,
L
11
L
11
L
11
L
21
N
=
(A.97)
L
21
L
11
L
21
L
21
+
L
22
L
22
