1. Jorge Eduardo Celis
Cod: 2073412
Methods for Solving Linear EquationsSpecial Systems
Thomas Method
Cholesky method

2. Thomas Method
This method emerges as a simplification of an LU factorization of a tridiagonal matrix.
r
A
x
Ax=r
3. Saying that A = LU and applying Doolittle where Lii = 1 for i = 1 to n, we get:
L
U
A
Thomas Method
4. Based on the matrix product shown above gives the following expressions:
As far as making the sweep from k = 2 to n leads to the following:
5. IF LUx=ry Ux=d THEM Ld=r :
d
r
L
So from a progressive replacement
6. Finally we solve Ux = d from backward substitution :
x
U
d
7. EXAMPLE
Solve the following system using the method of Thomas
Solution:Vectors are identified, bcyr as follows:
8. We obtain the following equalities :
9. Now once known L and U Ld = r is solved by a progressive replacement:
L
d
r
10. Finally Ux = d is solved by replacing regressive
d
U
x
Por lo que el vector solucin sera:
11. Cholesky method
As LU factorization method is applicable to a positive definite symmetric matrix and where
Them:
12. LT
L
A =LLT
A
13. From the product of the n-th row of L by the n-th column of LT we have:
Making the sweep from k = 1 to n has to :
14. On the other hand if we multiply the n-th row of L by the column (n-1) LT we have:
By scanning for k = 1 to n we have
15. EXAMPLE
Apply Cholesky methodology to decompose the following symmetric matrix :
ANSWER
k= 1 s:
16. k= 2 :
k= 3:
Finally, as a result of decomposition was found that:
17. Bibliography
Material de mtodos numricos de la universidad del sur de florida (NationalScienceFoundation),
CHAPRA, Steven C. y CANALE, Raymond P.: Mtodos Numricos para Ingenieros. McGraw Hill 2002.
PPTX EDUARDO CARRILLO, PHD.