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

In the previous slide 2

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

3.7 4 Special matrices

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

Strictly diagonally dominant 6

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Symmetric positive definite 8

Symmetric positive definite Theorems for verification 9

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

Cholesky decomposition 12

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Tridiagonal 15

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Any Questions? Special matrices

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

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

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

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

Iterative techniques for linear systems

Iterative techniques 23

Iterative techniques Basic idea 24

Iteration matrix Immediate questions 25

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(in section 2.3 with proof) 29 Recall that

30 Recall that

Iteration matrix For these questions 31 question hint answer

Iteration matrix For these questions 32 hint answer

Iteration matrix For these questions 33 answer

Iteration matrix For these questions 34

Splitting methods 35

Splitting methods 36

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Gauss-Seidel method 39

Gauss-Seidel method Iteration matrix 40

41 The SOR method (successive overrelaxatoin)

Any Questions? 42 Iterative techniques for linear systems

3.9 Conjugate gradient method 43

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

Conjugate gradient method Quadratic functional 45

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Minimizing quadratic functional 48

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50 Global optimization problem

Any Questions? 51 Conjugate gradient method

Nonlinear systems of equations

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Generalization of root-finding 54

Generalization Newton’s method 55

Generalization of Newton’s method Jacobian matrix 56

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58 A lots of equations bypassed…

59 And this is a friendly textbook :)

Any Questions? 60 Nonlinear systems of equations

Exercise /5/2 2:00pm to or hand over in class. Note that the fourth problem is a programming

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Implement LU decomposition 65

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