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Numerical Analysis 1 EE, NCKU Tien-Hao Chang (Darby Chang)

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In the previous slide 2

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In this slide Special matrices –strictly diagonally dominant matrix –symmetric positive definite matrix Cholesky decomposition –tridiagonal matrix Iterative techniques –Jacobi, Gauss-Seidel and SOR methods –conjugate gradient method Nonlinear systems of equations (Exercise 3) 3

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3.7 4 Special matrices

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Linear systems –which arise in practice and/or in numerical methods –the coefficient matrices often have special properties or structure Strictly diagonally dominant matrix Symmetric positive definite matrix Tridiagonal matrix 5

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Strictly diagonally dominant 6

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Symmetric positive definite 8

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Symmetric positive definite Theorems for verification 9

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Symmetric positive definite Relations to Eigenvalues Leading principal sub-matrix 11

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Cholesky decomposition 12

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Tridiagonal 15

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Any Questions? 17 3.7 Special matrices

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Before entering 3.8 So far, we have learnt three methods algorithms in Chapter 3 –Gaussian elimination –LU decomposition –direct factorization Are they algorithms? What’s the differences to those algorithms in Chapter 2? –they report exact solutions rather than approximate solutions 18 question further question answer

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Before entering 3.8 So far, we have learnt three methods algorithms in Chapter 3 –Gaussian elimination –LU decomposition –direct factorization Are they algorithms? What’s the differences to those algorithms in Chapter 2? –they report exact solutions rather than approximate solutions 19 further question answer

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Before entering 3.8 So far, we have learnt three methods algorithms in Chapter 3 –Gaussian elimination –LU decomposition –direct factorization Are they algorithms? What’s the differences to those algorithms in Chapter 2? –they report exact solutions rather than approximate solutions 20 answer

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Before entering 3.8 So far, we have learnt three methods algorithms in Chapter 3 –Gaussian elimination –LU decomposition –direct factorization Are they algorithms? What’s the differences to those algorithms in Chapter 2? –they report exact solutions rather than approximate solutions 21

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3.8 22 Iterative techniques for linear systems

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Iterative techniques 23

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Iterative techniques Basic idea 24

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Iteration matrix Immediate questions 25

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(in section 2.3 with proof) 29 http://www.dianadepasquale.com/ThinkingMonkey.jpg Recall that

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30 http://www.dianadepasquale.com/ThinkingMonkey.jpg Recall that

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Iteration matrix For these questions 31 question hint answer

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Iteration matrix For these questions 32 hint answer

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Iteration matrix For these questions 33 answer

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Iteration matrix For these questions 34

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Splitting methods 35

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Splitting methods 36

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Gauss-Seidel method 39

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Gauss-Seidel method Iteration matrix 40

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41 The SOR method (successive overrelaxatoin)

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Any Questions? 42 Iterative techniques for linear systems

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3.9 Conjugate gradient method 43

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Conjugate gradient method Not all iterative methods are based on the splitting concept The minimization of an associated quadratic functional 44

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Conjugate gradient method Quadratic functional 45

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46 http://fuzzy.cs.uni-magdeburg.de/~borgelt/doc/somd/parabola.gif

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Minimizing quadratic functional 48

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50 Global optimization problem http://www.mathworks.com/cmsimages/op_main_wl_3250.jpg

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Any Questions? 51 Conjugate gradient method

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3.10 52 Nonlinear systems of equations

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Generalization of root-finding 54

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Generalization Newton’s method 55

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Generalization of Newton’s method Jacobian matrix 56

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58 A lots of equations bypassed… http://www.math.ucdavis.edu/~tuffley/sammy/LinAlgDEs1.jpg

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59 And this is a friendly textbook :)

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Any Questions? 60 Nonlinear systems of equations

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Exercise 3 61 2011/5/2 2:00pm Email to darby@ee.ncku.edu.tw or hand over in class. Note that the fourth problem is a programming work.darby@ee.ncku.edu.tw

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Implement LU decomposition 65

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Slide 1Fig 26-CO, p.795. Slide 2Fig 26-1, p.796 Slide 3Fig 26-2, p.797.

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