Cholesky method and Thomas
- 1. Jorge Eduardo Celis Cod: 2073412 Methods for Solving Linear EquationsSpecial SystemsThomas MethodCholesky method
- 2. Thomas MethodThis method emerges as a simplification of an LU factorization of a tridiagonal matrix.rAxAx=r
- 3. Saying that A = LU and applying Doolittle where Lii = 1 for i = 1 to n, we get:LUAThomas 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 : drLSo from a progressive replacement
- 6. Finally we solve Ux = d from backward substitution :xUd
- 7. EXAMPLESolve the following system using the method of ThomasSolution: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 dr
- 10. Finally Ux = d is solved by replacing regressivedU xPor lo que el vector solución sería:
- 11. Cholesky methodAs LU factorization method is applicable to a positive definite symmetric matrix and whereThem:
- 12. LTLA =LLTA
- 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. EXAMPLEApply 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. BibliographyMaterial 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.