1
Colorado Center for Astrodynamics Research
The University of Colorado 1
STATISTICAL
ORBIT DETERMINATION
Cholesky Algorithm
ASEN 5070
LECTURE 23
10/25/06
2
Copyright 2006
Square Root Solution Methods
Cholesky Decomposition
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Cholesky Algorithm for Orbit Determination: University of Colorado Lecture Notes - Prof. G, Study notes of Aerospace Engineering
A portion of lecture notes from the university of colorado's astrodynamics research center on the cholesky algorithm used for statistical orbit determination. The notes cover the cholesky decomposition method, a 3x3 example, and solving for z and x using forward and backward substitutions.
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Colorado Center for Astrodynamics Research The University of Colorado^1
STATISTICAL
ORBIT DETERMINATION
Cholesky Algorithm
ASEN 5070
LECTURE 23
Copyright 2006^2
Square Root Solution Methods
Cholesky Decomposition
Copyright 2006^3
Square Root Solution Methods
The Cholesky Algorithm
Copyright 2006^4
Square Root Solution Methods
The Cholesky Algorithm
Colorado Center for Astrodynamics Research The University of Colorado
3 x 3 Example of the Cholesky Algorithm
Solve for z
from RT z = N 11 1 1 12 22 2 2 13 23 33 3 3 0 0 0 R z N R R z N R R R z N ! "! "! "
$ # $ # $
=
$ # $ # $
#% $& #% $& #% $& 11 1 1 1 1 11
R z N
z N R
= ( )
R z R z N
z N R z R
R 1 (^) 3 z 1 (^) + R 2 (^) 3 z 2 (^) + R 3 (^) 3 z 3 (^) = N 3
In general z is solved for using a forward substitution
1 1
- 7 i i i ji j ii j z N r z r ! =
$ "%
= (^) '! (^) ( & ) * %+
i = 1 ... n Colorado Center for Astrodynamics Research The University of Colorado
3 x 3 Example of the Cholesky Algorithm
Finally solve for (^) x ˆ i using a backward substitution R x ˆ = z 11 12 13 1 1 22 23 2 2 33 3 3 ˆ 0 ˆ 0 0 ˆ R R R x z R R x z R x z ! "! "! "
$ # $ # $
=
$ # $ # $
#% $& #% $& #% $& R 3 (^) 3 x ˆ 3 (^) = z 3
etc.
ˆ ˆ
n
i i ij j ii
j i
x z r x r
! " = (^) $ # % & ' (
i = n , n! 1 ... 1
Colorado Center for Astrodynamics Research The University of Colorado
3 x 3 Example of the Cholesky Algorithm
Assume 1 2 3 2 8 2 3 2 14 M ! " = % = #^ $
$
#& $'
applying Eqns. 5.2.6 yields
1 2 3 0 2 2 0 0 1 R ! " = #^ % $
$
#& $'
Note that we could use
i
ii ii ki
k
r M r
" # = ± (^) $! % & ' (
Colorado Center for Astrodynamics Research The University of Colorado
3 x 3 Example of the Cholesky Algorithm
If we chose the minus sign we will obtain – R (the negative of R above)
If we choose i.e. use the minus sign in Eq. (5.2.6) only for ,
we obtain
r 1 1 =! r 11 r 11
R
= $^! %
Hence, R is not unique
Colorado Center for Astrodynamics Research The University of Colorado Cholesky Decomposition with apriori Colorado Center for Astrodynamics Research The University of Colorado
Cholesky Decomposition with apriori
Colorado Center for Astrodynamics Research The University of Colorado
Cholesky Decomposition with apriori
Copyright 2006^16
Square Root Solution Methods
The Square Root Free Cholesky Algorithm
Copyright 2006^19 Square Root Free Cholesky Algorithm Copyright 2006^20 Square Root Free Cholesky Algorithm
Colorado Center for Astrodynamics Research The University of Colorado 3 x 3 Example of the Cholesky Square Root Free Algorithm
for 1 2 3
M UDU^ T
Using the square root free algorithm given by Eq. (5.2.11) and (5.2.12) yields
(^7 0 ) 189 0 54 0 7 0 0 14 D ! "
$
= #^ $
$
$
#% $& 1 11 3 54 14 0 1 1 7 0 0 1 U ! "
$
= #^ $
$
$
#% $& 1 2 3 2 8 2 3 2 14 UDU^ T ! " = #^ $
$
#% $&