Numerical-Computing-slides-chapter5

Numerical-Computing-slides-chapter5 - Numerical Computing...

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Unformatted text preview: Numerical Computing chapter 5: Eigenproblems Dr. Nguyen V.M. Man Email: [email protected] Faculty of Computer Science and Engineering HCMUT Vietnam May 4, 2010 The structure of the chapter ——————————————————————————— Introduction . Eigenproblems arise in many areas of engineering as signal processing, stability of dynamic systems, control systems, circuit simulations ... We firstly review eigenvalues and eigenvectors, then will consider numerical methods to determine them. A/ Eigenvalues and characteristic polynomial B/ Eigenvectors and diagonalization Computation of eigenvalues two methods C/ Power method: be used to find the eigenvalue of largest magnitude D/ JACOBI METHOD: finds us all the eigenvalues of a real symmetric matrix. Why are Eigenvalue Problems important? ——————————————————————————— Practical problem in Petroleum Industry. PXP (Plains Exploration and Production) in Texas operates a number of chemical reaction plants, in which various complex reactions occur. Most of the reactions can be modeled by linear system of differential equations: x = d x dt = A x here x = [ x 1 ( t ) , x 2 ( t ) ,..., x n ( t )] ∈ R n is a vector of component concerntrations x i ( t ), and the matrix elements a i , j model the rate at which the constituent j is turned into product i . [currently the x i ( t ) are unknown differential functions over an interval J w.r.t. a variable t ] Why are Eigenvalue Problems important? ——————————————————————————— Example Consider one variable differential equation (DE) ( n = 1): x = 5 x x (0) =- 7; where x : R-→ R is differential. The equation has a unique solution x ( t ) =- 7 e 5 t ( t ∈ R ). —————————– In general case n ≥ 1, given an intial vector x (0), the simplest solution of such a system uses the matrix exponential x ( t ) = e At x (0) (see Appendix A) Solution of x ( t ) = e At x (0) ——————————————————————————— The matrix exponential then is easily defined in terms of the eigenvalues and eigenvectors of the matrix A : e At = V- 1 E ( t ) V where E ( t ) = e λ 1 t ... e λ 2 t ... . . . . . . . . . . . . ... e λ nt , the λ k are the eigenvalues of A , and V = [ v 1 v 2 ··· v n ] is the matrix whose columns are the corresponding eigenvectors. V is called the modal matrix . For instance, consider two variables differential equation ( n = 2): X = A X X (0) = c = [ c 1 , c 2 ]; where A =- 1 0 2 . This means x 1 =- x 1 x 2 = 2 x 2 ; where X = [ x 1 , x 2 ] : R 2-→ R 2 is differential....
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Numerical-Computing-slides-chapter5 - Numerical Computing...

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