Global Positioning System Reference
In-Depth Information
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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)