Graphics Programs Reference
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if size(b,2) > 1; b = b'; end
n = length(b);
fork=2:n
b(k) = b(k) - A(k,1:k-1)*b(1:k-1);
end
fork=n:-1:1
b(k) = (b(k) - A(k,k+1:n)*b(k+1:n))/A(k,k);
end
x=b;
Choleski's Decomposition
LL
T
hastwolimitations:
Choleski's decomposition
A
=
Since the matrix product
LL
T
issymmetric, Choleski's decompositionrequires
A
to be
symmetric
.
The decompositionprocess involves taking square roots of certain combinations
of the elements of
A
. It can be shown thatsquare roots of negative numberscan
be avoided onlyif
A
is
positive definite
.
Although the number of long operations in all the decompositionmethods is
about the same, Choleski's decompositionis not a particularlypopular meansof
solving simultaneousequations, mainlydue to the restrictionslistedabove.Westudy
ithere because it is invaluable in certain other applications(e.g., in the transformation
of eigenvalue problems).
Let usstart by looking at Choleski's decomposition
LL
T
A
=
(2.15)
of a3
×
3matrix:
⎡
⎣
⎤
⎦
=
⎡
⎣
⎤
⎦
⎡
⎣
⎤
⎦
A
11
A
12
A
13
A
21
A
22
A
23
A
31
A
32
A
33
L
11
00
L
11
L
21
L
31
0
L
22
L
32
00
L
33
L
21
L
22
0
L
31
L
32
L
33
After completing the matrix multiplication on the right hand side, we get
⎡
⎣
⎤
⎦
=
⎡
⎣
⎤
⎦
L
11
A
11
A
12
A
13
A
21
A
22
A
23
A
31
A
32
A
33
L
11
L
21
L
11
L
31
L
21
+
L
22
L
11
L
21
L
21
L
31
+
L
22
L
32
(2.16)
L
31
+
L
32
+
L
33
L
11
L
31
L
21
L
31
+
L
22
L
32
Note that the right-hand-sidematrix issymmetric, as pointed out before.Equating the
matrices
A
and
LL
T
element-by-element, weobtain six equations(duetosymmetry
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