choleski
TRANSCRIPT
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NON-CONVENTIONAL METHODS TO SOLVE L.E
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` Algebraic methods like matrixes are used to solve
linear equation systems, furthermore, this method is
used to solve some another non-linear system inwhich we need to give a solution.
As a consequence of using matrixes the methods to
find solutions are the result of algebraic solution for
matrixes.
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SolutionMethods
Direct Methods
EliminacinGaussiana
Gauss conPivoteo
Gauss-Jordan
SistemasEspeciales
IterativeMethods
Jacobi
Gauss-Seidel
Gauss-Seidelwith relaxation
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A way to write matrices that is commonly to find at any place:
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!
-
y
-
m
i
n
i
mnmm
inii
n
n
c
c
c
c
y
y
y
y
fff
fff
fff
fff
2
1
2
1
21
21
22221
11211
.
/
.
//
.
.
Fy c
Fy = c
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` This method emerge as a simplification of LU factorization but only if wehave a tri-diagonal matrix .
-
!
-
y
-
n
n
n
n
nn
nnn
r
r
rr
r
x
x
xx
x
ba
cba
cbacba
cb
1
3
2
1
1
3
2
1
111
313
222
11
//111
A x r
Note that a simply
form to identify when
to use this method iswhen your matrix is
banded.
We are going to
solve the system as
usual as LU for othermatrices.
WE ALSO CAN SOLVE THIS METHOD AS A SIMPLIFICATION OF GAUSSIAN
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!
-
y
-
nn
nnn
nn
nnnn
nn
nn
ba
cba
cba
cba
cb
U
UU
UU
UU
UU
L
L
L
L
111
333
222
11
,
,11,1
3433
2322
1211
1,
2,1
32
21
1
1
1
1
1
111111
As what is usual on LU we are going to say that A = LU and using Doolitlewhere Lii=1 for i=1 till n, we finally have:
L U A
ote that the Lower atri and the U er were si lify as LU ethod re uireote that the Lower atri and the U er were si lify as LU ethod re uire
ut what we o tain for oth of the are two diagonal of nu ers. en e theut what we o tain for oth of the are two diagonal of nu ers. en e the
way to sol e had een si lified in order to find a solution; s e ially L.way to sol e had een si lified in order to find a solution; s e ially L.
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00
,
1
,11,,
1,1
1,1
1,
111
!!
!
!
!
!
n
nnnnnnn
nnn
nn
nnn
cy
Donde
ULbU
cU
U
aL
bU
Based on the matrix product showed before
we obtain these expressions
kkkkkkk
kkk
kk
k
kk
ULbU
cU
U
a
L
,11,,
1,1
1,11,
!
!
!
Now scanning from k=2 till n we finally have
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!
-
y
-
n
n
n
n
nn
nn
r
r
r
r
r
d
d
d
d
d
L
L
L
L
1
3
2
1
1
3
2
1
1,
2,1
32
21
1
1
1
1
1
//11
11,
11
2
!
!
!
kkkkkdLrd
tillkFrom
rd
If LUx=r and Ux=d then Ld=r, hence:
Ld r
Base on a regressive
substitution
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-
!
-
-
n
n
n
n
nn
nnnn
d
d
d
dd
x
x
x
xx
U
UU
UU
UUUU
1
3
2
1
1
3
2
1
,11,1
3433
2322
1211
//11
U x d
Finally we solveFinally we solve UxUx=d based on the regressive=d based on the regressive
substitutionsubstitution
nn
n
n
U
dx
Where
,
,
!
kk
n
kj
jkjk
kU
xUd
x
tillnkTo
,
1
,11
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Is a decomposition of a symmetric, positive-definite matrix into the product
of a lower triangle matrix and its conjugate transpose. When is applicable
this method is twice as efficient as LU decomposition for solving systems
TLU !
HENCE
bxLL
bAx
T !
!
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-
-
y
-
nnnnnnnn
nnnnnnnn
nnnnnnnn
nnn
nnn
nn
nnnn
nnnnnn
nnn
nnn
nnnnnnnn
nnnnnn
nnnn
aaaaa
aaaaa
aaaaa
aaaaa
aaaaa
L
LL
LLL
LLLL
LLLLL
LLLLL
LLLL
LLL
LL
L
,1,2,2,1,
,11,12,12,11,1
,21,22,22,21,2
,21,22,22221
,11,12,11211
,
1,1,1
2,2,12,2
2,2,12,222
1,1,11,22111
,1,2,2,1,
1,12,12,11,1
2,22,21,2
2221
11
.
.
.
.
.
.
.
.
.
.
L LT
A
A =LLT
What was mention before shows that:
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From the product of the nth row of L and the nth columnFrom the product of the nth row of L and the nth column LLTT of weof we
obtain that:obtain that:
!
!
!
!
!
!
1
1
2
,
1
1
2
,2
2
1,2
2,2
2,2
1,2
22
1,
2
2,
2
2,
2
1,
n
j
jnnnnn
n
j
jnnnnn
nnnnnnnnnn
nnnnnnnnnn
a
a
aa
.
.
Once again
scanning fromk=1 till n we
obtain
!
!1
1
2
,
k
j
jkkkkk LaL
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!
!
!
!
2
1
,1,1,1,
1,1
2,12,2,12,1,11,1,
1,
1,1,11,2,12,2,12,1,11,
n
j
jnjnnnnn
nn
nnnnnnnnnnnn
nnnnnnnnnnnnnn
LLaL
L
LLLLLLaL
aLLLLLLLL
.
.
In the other way if we multiply the nth row of L with the (nIn the other way if we multiply the nth row of L with the (n--1) column of1) column ofLLTT wewe
will have:will have:
11
1
1
,,,,
ee
!
!
kidonde
LLaLi
j
jijkikik
scanning fr till n tain
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` LinearLinear leastleast squaressquares:: Systems of the form Ax = b with A symmetricand positive definite arise quite often in applications. For instance,the normal equations in linear least squares problems are of thisform.
` MonteMonte CarloCarlo SimulationSimulation:: The Cholesky decomposition is commonlyused in the Monte Carlo method for simulating systems with multiplecorrelated variables: The matrix of inter-variable correlations isdecomposed, to give the lower-triangularL.
` NonNon--linearlinear optimizationoptimization:: Non-linear multi-variate functions may beminimized over their parameters using variants of Newton's methodcalled quasi-Newton methods.
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` CHAPRA, Steven C. y CANALE, Raymond P.:
Mtodos Numricos ara Ingenieros. McGraw Hill
2002.
` http://en.wikipedia.org/wiki/Cholesky_decomposition#Applications
` http://math.fullerton.edu/mathews/n2003/Cholesky
Mod.html