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
and applying (A.83) readily gives
Q 1
22
L 22 L 22
=
(A.98)
Depending on the application one might group the parameters such that (A.98) can
be used directly, i.e., the needed inverse is a simple function of the Cholesky factors
that had been computed previously.
The diagonal elements of L are not necessarily unity. Consider a new matrix G
with elements taken from L such that g jk =
l jk /l jj and a new diagonal matrix D such
l jj ; then
that d jj =
G D
L
=
(A.99)
LL T
GDG T
[35
=
=
N
(A.100)
Because N is a positive definite matrix, the diagonal elements of G are
1.
The unknown x can be solved without explicitly inverting the matrix. Assume that
we must solve the system of equations
+
Lin
0.0
——
Lon
PgE
Nx
=
u
(A.101)
The first step in the solution of (A.101) is to substitute (A.94) for N and premultiply
with L 1
to obtain the triangular equations
L T x
L 1 u
=
(A.102)
[35
Denoting the right-hand side of (A.102) by c u and multiplying by L , we can write
L T x
=
c u
(A.103)
Lc u =
u
(A.104)
We solve c u from (A.104) starting with the first element. Using L and c u , the solution
of (A.103) yields the parameters x , starting with the last element.
In least-squares, the auxiliary quantity
=− L 1 u T L 1 u =−
c u c u
u T N 1 u
=−
l
(A.105)
is useful for computing v T Pv (see Table 4.1) to assess the quality of the adjustment.
The Cholesky algorithm provides l from c u without explicitly using the inverses of
N and L .
Computing the inverse requires a much bigger computational effort than merely
solving the system of equations. The first step is to make u solutions of the type
(A.104) to obtain the columns of C ,
LC
=
I
(A.106)
 
Search WWH ::




Custom Search