Special methods
- 1. RUBEN DARIO ARISMENDI RUEDA
- 2. CHAPTER 4: ‘Iterative Methods to solve lineal ecuation systems’
- 3. There are some kinds of Methods that are used to solve this lineal systems. 1- THOMAS 2-CHOLESKY
- 4. THOMAS. This method is used with a special kind of Matrix that has this form. This is a special kind of matrix that has all it’s elements cero except that ones shown in the last picture.
- 5. THIS METHOD WILL BE MORE CLEAR IF IS EXPLAINED WITH AN EXAMPLE. Basically this method uses the LU factorization. EXAMPLE. MATRIX ‘A’ R
- 6. THEN L*U=A MATRIX ‘L’ MATRIX ‘U’
- 7. L*D=R; By simple substitution the vector D is found. MATRIX ‘L’ * = D R
- 8. U*X=D; By simple substitution the vector ‘X’ is found and the system will be solved. MATRIX ‘U’ * = X D
- 9. SOLUTION
- 10. 2-CHOLESKY. When is a simetric and definide matrix Then
- 11. To find the values of the matrix . The next expression is the result of the product between the n-file of L and the n-column of L T The next expression is the result of the product between the n-file of L and the (n-1)column of L T
- 12. EXAMPLE To understand this method, it will be easier with an example, that show how the Cholesky decomposition is made. MATRIX ‘A’ 6 15 55 15 55 225 55 225 979
- 13. (k=1) 2. (k=2)
- 14. 3. (k=3) and (i=1) (k=3) and (i=2)
- 15. Cholesky decomposition is L = 2,4495 6,1237 4,1833 22,454 20,916 6,1106
- 16. Bibliography: Numerical Methods for Engineers . Steven C. Chapra Prf. Eduardo Carrillo's presentation ''METODOS NUMERICOS EN INGENIERIA DE PETROLEOS''.