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# Cholesky method and Thomas

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### Cholesky method and Thomas

1. 1. Jorge Eduardo Celis <br />Cod: 2073412 <br />Methods for Solving Linear EquationsSpecial Systems<br />Thomas Method<br />Cholesky method<br />
2. 2. Thomas Method<br />This method emerges as a simplification of an LU factorization of a tridiagonal matrix.<br />r<br />A<br />x<br />Ax=r<br />
3. 3. Saying that A = LU and applying Doolittle where Lii = 1 for i = 1 to n, we get:<br />L<br />U<br />A<br />Thomas Method<br />
4. 4. Based on the matrix product shown above gives the following expressions:<br />As far as making the sweep from k = 2 to n leads to the following:<br />
5. 5. IF LUx=r y Ux=d THEM Ld=r :<br /> d<br />r<br />L<br />So from a progressive replacement<br />
6. 6. Finally we solve Ux = d from backward substitution :<br />x<br />U<br />d<br />
7. 7. EXAMPLE<br />Solve the following system using the method of Thomas<br />Solution:Vectors are identified, bcyr as follows:<br />
8. 8. We obtain the following equalities :<br />
9. 9. Now once known L and U Ld = r is solved by a progressive replacement:<br />L<br /> d<br />r<br />
10. 10. Finally Ux = d is solved by replacing regressive<br />d<br />U<br /> x<br />Por lo que el vector solución sería:<br />
11. 11. Cholesky method<br />As LU factorization method is applicable to a positive definite symmetric matrix and where<br />Them:<br />
12. 12. LT<br />L<br />A =LLT<br />A<br />
13. 13. From the product of the n-th row of L by the n-th column of LT we have:<br />Making the sweep from k = 1 to n has to :<br />
14. 14. On the other hand if we multiply the n-th row of L by the column (n-1) LT we have:<br />By scanning for k = 1 to n we have<br />
15. 15. EXAMPLE<br />Apply Cholesky methodology to decompose the following symmetric matrix :<br />ANSWER<br /> k= 1 s:<br />
16. 16. k= 2 :<br />k= 3:<br />Finally, as a result of decomposition was found that:<br />
17. 17. Bibliography<br />Material de métodos numéricos de la universidad del sur de florida (NationalScienceFoundation),<br />CHAPRA, Steven C. y CANALE, Raymond P.: Métodos Numéricos para Ingenieros. McGraw Hill 2002.<br />PPTX EDUARDO CARRILLO, PHD.<br />