H-Transforms: Theory and Applications (Analytical Methods and Special Functions)
Special methods
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Transcript of Special methods
ESCUELA DE INGENIERÍA DE PETROLEOS
RUBEN DARIO ARISMENDI RUEDA
ESCUELA DE INGENIERÍA DE PETROLEOS
CHAPTER 4: ‘Iterative Methods to solve lineal ecuation systems’
ESCUELA DE INGENIERÍA DE PETROLEOS
There are some kinds of Methods that are used to solve this lineal systems.
1- THOMAS
2-CHOLESKY
ESCUELA DE INGENIERÍA DE PETROLEOS
THOMAS.This method is used with a special kind of Matrix that has this form.
This is a special kind of matrix that has all it’s elements cero except that ones shown in the last picture.
ESCUELA DE INGENIERÍA DE PETROLEOS
THIS METHOD WILL BE MORE CLEAR IF IS EXPLAINED WITH AN EXAMPLE.Basically this method uses the LU factorization.
EXAMPLE.
MATRIX ‘A’ R
ESCUELA DE INGENIERÍA DE PETROLEOS
THEN L*U=A
MATRIX ‘L’
1,11,
−−− =
kk
kkk U
aL
kkkkkkk
kkk
ULbU
cU
,11,,
1,1
−−
−−
−==
MATRIX ‘U’
ESCUELA DE INGENIERÍA DE PETROLEOS
L*D=R; By simple substitution the vector D is found.
MATRIX ‘L’
* =
D R
ESCUELA DE INGENIERÍA DE PETROLEOS
U*X=D; By simple substitution the vector ‘X’ is found and the system will be solved.
MATRIX ‘U’
* =
X D
ESCUELA DE INGENIERÍA DE PETROLEOS
SOLUTION
ESCUELA DE INGENIERÍA DE PETROLEOS
2-CHOLESKY.When is a simetric and definide matrix
TLU =Then
bxLL
bAxT =
=
ESCUELA DE INGENIERÍA DE PETROLEOS
∑−
=
−=1
1
2,
k
jjkkkkk LaL
To find the values of the matrix .
•The next expression is the result of the product between the n-file of L and the n-column of LT
•The next expression is the result of the product between the n-file of L and the (n-1)column of LT
11
1
1,,,,
−≤≤
−= ∑−
=
kiwhere
LLaLi
jjijkikik
ESCUELA DE INGENIERÍA DE PETROLEOS
EXAMPLETo understand this method, it will be easier with an example, that show how the Cholesky decomposition is made.
MATRIX ‘A’
6 15 55
15 55 225
55 225 979
ESCUELA DE INGENIERÍA DE PETROLEOS
1. (k=1)
2. (k=2)
ESCUELA DE INGENIERÍA DE PETROLEOS
3. (k=3) and (i=1)
(k=3) and (i=2)
ESCUELA DE INGENIERÍA DE PETROLEOS
Cholesky decomposition is
2,4495
6,1237 4,1833
22,454 20,916 6,1106
L =
ESCUELA DE INGENIERÍA DE PETROLEOS
Bibliography:
•Numerical Methods for Engineers . Steven C. Chapra•Prf. Eduardo Carrillo's presentation ''METODOS NUMERICOS EN INGENIERIA DE PETROLEOS''.