Special methods

ESCUELA DE INGENIERÍA DE PETROLEOS

RUBEN DARIO ARISMENDI RUEDA


CHAPTER 4: ‘Iterative Methods to solve lineal ecuation systems’


There are some kinds of Methods that are used to solve this lineal systems.

1- THOMAS

2-CHOLESKY


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.


THIS METHOD WILL BE MORE CLEAR IF IS EXPLAINED WITH AN EXAMPLE.Basically this method uses the LU factorization.

EXAMPLE.

MATRIX ‘A’ R


THEN L*U=A

MATRIX ‘L’

1,11,

−−− =

kk

kkk U

aL

kkkkkkk

kkk

ULbU

cU

,11,,

1,1

−−

−−

−==

MATRIX ‘U’


L*D=R; By simple substitution the vector D is found.

MATRIX ‘L’

* =

D R


U*X=D; By simple substitution the vector ‘X’ is found and the system will be solved.

MATRIX ‘U’

* =

X D


SOLUTION


2-CHOLESKY.When is a simetric and definide matrix

TLU =Then

bxLL

bAxT =

=


∑−

=

−=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


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


1. (k=1)

2. (k=2)


3. (k=3) and (i=1)

(k=3) and (i=2)


Cholesky decomposition is

2,4495

6,1237 4,1833

22,454 20,916 6,1106

L =


Bibliography:

•Numerical Methods for Engineers . Steven C. Chapra•Prf. Eduardo Carrillo's presentation ''METODOS NUMERICOS EN INGENIERIA DE PETROLEOS''.

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Special methods