This document discusses two methods for solving linear equations: the Thomas method and the Cholesky method. It provides examples of applying each method. The Thomas method emerges from an LU factorization of a tridiagonal matrix. It involves forward and backward substitution to solve for the vector x given Ax=b. The Cholesky method applies to positive definite symmetric matrices and factors the matrix A as A=LLT using an upper triangular matrix L. It involves solving Ly=b and LTx=y to solve Ax=b. An example shows applying the Cholesky method to decompose a symmetric matrix.

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=r y 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 solución sería:

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 métodos numéricos de la universidad del sur de florida (NationalScienceFoundation), CHAPRA, Steven C. y CANALE, Raymond P.: Métodos Numéricos para Ingenieros. McGraw Hill 2002. PPTX EDUARDO CARRILLO, PHD.