1. SPECIAL METHODS

2. THOMAS METHOD This method emerges as a simplification of an LU factorization of a tridiagonal matrix. We know that a positive definite matrixA has a unique symmetric square root F such that : 𝑭 ² = 𝑭𝑭′ = 𝑨 Now if we do not insist on symmetry, there is a very large set of (non symmetric) matrices G such that : 𝑮𝑮′ = 𝑨 and which may also be regarded as "square roots" of A. The positive definite matrix A is then said to be factored into the "square" of its square root.

3. One of these factorizations is of particular interest, both from theoretical and practical standpoints: Cholesky Decomposition, which is expressed as follows. THOMAS METHOD

4. Based on the matrix product shown above gives the following expressions: THOMAS METHOD

5. As far as making the sweep from k = 2 to n leads to the following: THOMAS METHOD

6. If Lux and Ux = r = d then Ld = r, therefore: THOMAS METHOD

7. Finally we solve Ux = d from a backward place: THOMAS METHOD

8. EXAMPLE Solve the following system using the method of Thomas: Vectors are identified a, b, c and r as follows:

9. We obtain the following equalities: EXAMPLE

10. Now once known L and U, Ld = r is solved by a progressive replacement: EXAMPLE

11. Finally Ux = d is solved by replacing regressive: EXAMPLE

12. So the solution vector is: EXAMPLE

13. If A is only positive semi-definite, the diagonal elements of L can only be said to be non negative. The Cholesky factorization can be symbolically represented by : CHOLESKY METHODS LT L A =LLT A

14. The Cholesky factorization is the prefered numerical method for calculating : The inverse, and the determinant of a positive definite matrix (in particular of a covariance matrix), as well as for the simulation of a random multivariate normalvariable. CHOLESKY METHODS

15. From the product of the n-th row of L by the n-th column of LT we have: CHOLESKY METHODS

16. Making the sweep from k = 1 to n we have: CHOLESKY METHODS

17. On the other hand if we multiply the n-th row of L by the column (n-1) LT we have: CHOLESKY METHODS

18. By scanning for k = 1 to n we have: CHOLESKY METHODS

19. EXAMPLE Apply Cholesky for symmetric matrix decomposed as follows: For k = 1:

20. EXAMPLE For k = 2: For k = 3:

21. EXAMPLE Finally, as a result of decomposition was found that:

22. BIBLIOGRAPHY http://www.google.com.co/imgres?imgurl=http://userweb.cs.utexas.edu/~plapack/icpp98/img38.gif&imgrefurl=http://www.cs.utexas.edu/~plapack/icpp98/node2.html&usg=__lcX8ioMpUm91zLexBn6t8JE72_Q=&h=823&w=751&sz=20&hl=es&start=3&um=1&itbs=1&tbnid=cWGkKDGz2Os9EM:&tbnh=144&tbnw=131&prev=/images%3Fq%3Dcholesky%26um%3D1%26hl%3Des%26ndsp%3D20%26tbs%3Disch:1