Métodos Numéricos en Recursos Hídricos

Transcripción

1 Universidad Nacional Agraria la Molina Escuela de Postgrado Maestría en Ingeniería de Recursos Hídricos Métodos Numéricos en Recursos Hídricos Sesión 6: Álgebra Lineal Numérica (continuado) Eusebio Ingol Blanco, Ph.D., A.M.ASCE

2 Pivoting La razón principal de que la técnica anterior se llame NAIVE es que durante las fases de eliminación y de sustitución, es posible que una división por cero puede ocurrir o cerca de 0 (problema: error de redondeo) Una forma de resolver estos problemas es determinar el coeficiente con el valor absoluto más grande de la columna debajo del elemento pivote. Las filas a continuación, se pueden conmutar de manera que el elemento más grande es el elemento de pivote. Esto se llama pivoteo parcial. Si as columnas así como las filas son examinadas para el elemento más grande y después son cambiadas, el procedimiento se llama completo pivotante.

3 Programa para un Parcial Pivoteo

4 Sistemas Tridiagonales A tridiagonal system is a banded system with a bandwidth of 3: f 1 g 1 e 2 f 2 g 2 e 3 f 3 g 3 e n1 f n1 g n1 Tridiagonal systems can be solved using the same method as Gauss elimination, but with much less effort because most of the matrix elements are already 0. e n f n x 1 x 2 x 3 x n1 x n r 1 r 2 r 3 r n1 r n

5 Código Matlab

6 LU Factorization Recordemos que el paso de eliminación hacia adelante comprende la mayor parte del esfuerzo computacional Los métodos de factorización LU separan la eliminación de tiempo de la matriz [A] de las manipulaciones del lado derecho [b]. Una vez que [A] ha sido factor (o descompuesta), múltiples vectores del lado derecho pueden ser evaluados de una manera eficiente.

7 LU Factorization LU factorization involves two steps: Factorization to decompose the [A] matrix into a product of a lower triangular matrix [L] and an upper triangular matrix [U]. [L] has 1 for each entry on the diagonal. Substitution to solve for {x} Gauss elimination can be implemented using LU factorization

8 Gauss Elimination as LU Factorization [A]{x}={b} can be rewritten as [L][U]{x}={b} using LU factorization. The LU factorization algorithm requires the same total flops as for Gauss elimination. The main advantage is once [A] is decomposed, the same [L] and [U] can be used for multiple {b} vectors. MATLAB s lu function can be used to generate the [L] and [U] matrices: [L, U] = lu(a)

9 LU Factorization With Pivoting To solve [A]{x}={b}, first decompose [A] to get [L][U]{x}={b} Set up and solve [L]{d}={b}, where {d} can be found using forward substitution. Set up and solve [U]{x}={d}, where {x} can be found using backward substitution. In MATLAB: [L, U] = lu(a) d = L\b x = U\d

10 Cholesky Factorization Symmetric systems occur commonly in both mathematical and engineering/science problem contexts, and there are special solution techniques available for such systems. The Cholesky factorization is one of the most popular of these techniques, and is based on the fact that a symmetric matrix can be decomposed as [A]= [U] T [U], where T stands for transpose. The rest of the process is similar to LU decomposition and Gauss elimination, except only one matrix, [U], needs to be stored.

11 Cholesky Factorization in MATLAB MATLAB can perform a Cholesky factorization with the built-in chol command: U = chol(a) MATLAB s left division operator \ examines the system to see which method will most efficiently solve the problem. This includes trying banded solvers, back and forward substitutions, Cholesky factorization for symmetric systems. If these do not work and the system is square, Gauss elimination with partial pivoting is used.

12 Matrix Inverse Recall that if a matrix [A] is square, there is another matrix [A] -1, called the inverse of [A], for which [A][A] -1 =[A] -1 [A]=[I] The inverse can be computed in a column by column fashion by generating solutions with unit vectors as the right-hand-side constants: A x A x A 1 x 1 x 2 x 3 A x

13 Matrix Inverse Recall that LU factorization can be used to efficiently evaluate a system for multiple right-hand-side vectors - thus, it is ideal for evaluating the multiple unit vectors needed to compute the inverse.

14 Stimulus-Response Computations Many systems can be modeled as a linear combination of equations, and thus written as a matrix equation: Interactions response stimuli The system response can thus be found using the matrix inverse.

15 Vector and Matrix Norms A norm is a real-valued function that provides a measure of the size or length of multicomponent mathematical entities such as vectors and matrices. Vector norms and matrix norms may be computed differently.

16 Vector Norms For a vector {X} of size n, the p-norm is: X p n i1 x i p Important examples of vector p-norms include: 1/ p n p 1:sum of the absolute values X 1 x i i1 n i1 p 2 :Euclidian norm (length) X 2 X e x i 2 p :maximum magnitude X max x i 1in

17 Matrix Norms Common matrix norms for a matrix [A] include: Note - max is the largest eigenvalue of [A] T [A]. column - sum norm A 1 Frobenius norm A f a ij 2 row - sum norm spectral norm (2 norm) A n max a ij 1 jn i1 n n i1 j1 n max a ij 1in j1 1/2 A 2 max

18 Matrix Condition Number The matrix condition number Cond[A] is obtained by calculating Cond[A]= A A -1 In can be shown that: X X Cond A A A The relative error of the norm of the computed solution can be as large as the relative error of the norm of the coefficients of [A] multiplied by the condition number. If the coefficients of [A] are known to t digit precision, the solution [X] may be valid to only t-log 10 (Cond[A]) digits.

19 MATLAB Commands MATLAB has built-in functions to compute both norms and condition numbers: norm(x,p) Compute the p norm of vector X, where p can be any number, inf, or fro (for the Euclidean norm) norm(a,p) Compute a norm of matrix A, where p can be 1, 2, inf, or fro (for the Frobenius norm) cond(x,p) or cond(a,p) Calculate the condition number of vector X or matrix A using the norm specified by p.

20 Resumen Introducción. Matrices y vectores Eliminación de Gauss Pivoteo y eliminación de Gauss. Problemas - solución numérica. Descomposición LU. Solución numérica Factorización Cholesky Inversión de una matriz.