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

In the previous slide Fixed point iteration scheme –what is a fixed point? –iteration function –convergence Newton’s method –tangent line approximation –convergence Secant method 2

In this slide Accelerating convergence –linearly convergent –Newton’s method on a root of multiplicity >1 –(exercises) Proceed to systems of equations –linear algebra review –pivoting strategies 3

2.6 4 Accelerating Convergence

Accelerating convergence Having spent so much time discussing convergence –is it possible to accelerate the convergence? How to speed up the convergence of a linearly convergent sequence? How to restore quadratic convergence to Newton’s method? –on a root of multiplicity > 1 5

Accelerating convergence Linearly convergence Thus far, the only truly linearly convergent sequence –false position –fixed point iteration Bisection method is not according to the definition 6

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Aitken’s Δ 2 -method Substituting Eq. (2) into Eq. (1) Substituting Eq. (4) into Eq. (3) The above formulation should be a better approximation to p than p n 8

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Aitken’s Δ 2 -method Accelerated? 10 which implies super- linearly convergence later answer

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Any Questions? 12 About Aitken’s Δ 2 -method

Accelerating convergence Anything to further enhance? 13

14 Why not use p-head instead of p ?

Steffensen’s method 15

16 Restoring quadratic convergence to Newton’s method

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Any Questions? 19

Two disadvantages Both the first and the second derivatives of f are needed Each iteration requires one more function evaluations 20 answer

Any Questions? 21 Chapter 2 Rootfinding (2.7 is skipped)

Exercise /4/21 9:00am to or hand over in class. You may arbitrarily pick one problem among the first three, which means this exercise contains only five

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27 (Programming)

Chapter 3 28 Systems of Equations

Systems of Equations Definition 29

Linear Algebra Review (vectors and matrices)

Matrix Definitions 31

Any Questions? 32 m, n, m, i, j, E QUAL, S UM, S CALAR M ULTIPLICATION, P RODUCT …

The Inverse Matrix 33 (cannot be skipped)

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Any questions? 35 answerquestion

The Determinant 36 (cannot be skipped, too)

37 cofactor

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Link the concepts –All these theorems will be extremely important throughout this chapter Nonsingular matrices Determinants Solutions of linear systems of equations 39

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41 (Hard to prove)

Any Questions? Linear Algebra Review

Gaussian Elimination (I suppose you have already known it)

An application problem 44

I 1 -I 2 -I 3 =0 I 2 -I 4 -I 5 =0 I 3 +I 4 -I 6 =0 2I 3 +I 6 =7 I 2 +2I 5 =13 -I 2 +2I 3 -3I 4 =0 45

Following Gaussian elimination 46

Any Questions? 47 Gaussian elimination

Gaussian elimination Operation Counts 48

Operation Counts Comparison Gaussian elimination –forward elimination –back substitution Gauss-Jordan elimination Compute the inverse matrix 49

Pivoting Strategy

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53 Compare to x 1 =1, x 2 =7, x 3 =1

Pivoting strategy To avoid small pivot elements A scheme for interchanging the rows (interchanging the pivot element) Partial pivoting 54

55 In action

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57 Compare to x 1 =1, x 2 =7, x 3 =1

Any Questions? 58

From the algorithm view How to implement the interchanging operation? –change implicitly Introduce a row vector r –each time a row interchange is required, we need only swap the corresponding elements of the vector –number of operations from 3n to 3 59 hint answer

60 In action

61 Without pivoting

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x = [1.000, , , ] T –exact solution x = [1,-1,1,-1] T –no r x = [1.131, , , ] T 64

Scaled Partial Pivoting 65

Scaled partial pivoting An example 66

Any Questions? 67

Scaled partial pivoting A blind spot of partial pivoting 68 answer

Scaled partial pivoting 69

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71 In action

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x = [1.000, , 1.000, ] T –exact solution x = [1,-1,1,-1] T –no s x = [1.000, , , ] T –no r x = [1.131, , , ] T 74

Any Questions? Pivoting Strategy