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

In the previous slide Rootfinding –multiplicity Bisection method –Intermediate Value Theorem –convergence measures False position –yet another simple enclosure method –advantage and disadvantage in comparison with bisection method 2

In this slide Fixed point iteration scheme –what is a fixed point? –iteration function –convergence Newtons method –tangent line approximation –convergence Secant method 3

Rootfinding Simple enclosure –Intermediate Value Theorem –guarantee to converge convergence rate is slow –bisection and false position Fixed point iteration –Mean Value Theorem –rapid convergence loss of guaranteed convergence 4

2.3 5 Fixed Point Iteration Schemes

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7 There is at least one point on the graph at which the tangent lines is parallel to the secant line

Mean Value Theorem 8

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Fixed points 10

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Number of fixed points According to the previous figure, a trivial question is –how many fixed points of a given function? 12

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Only sufficient conditions Namely, not necessary conditions –it is possible for a function to violate one or more of the hypotheses, yet still have a (possibly unique) fixed point 15

Fixed point iteration 16

Fixed point iteration 17

18 In action

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Any Questions? 21 About fixed point iteration

Relation to rootfinding Now we know what fixed point iteration is, but how to apply it on rootfinding? More precisely, given a rootfinding equation, f(x)=x 3 +x 2 -3x-3=0, what is its iteration function g(x) ? 22 hint

Iteration function 23

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25 In action

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Analysis 27

Convergence 28

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Order of convergence of fixed point iteration schemes 33

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Stopping condition 39

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Two steps 41

The first step 42

The second step 43

Any Questions? Fixed Point Iteration Schemes

Newtons Method

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Newtons Method Definition 48

49 In action

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In the previous example 51

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Any Questions? 53

Initial guess 54 example answer

Initial guess 55 answer

Initial guess 56

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Convergence analysis for Newtons method 58

59 The simplest plan is to apply the general fixed point iteration convergence theorem

Analysis strategy 60

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Any Questions? 65

Newtons Method Guaranteed to Converge? 66 hint answer

Newtons Method Guaranteed to Converge? 67 answer

Newtons Method Guaranteed to Converge? 68

69 Oh no! After these annoying analyses, the Newtons method is still not guaranteed to converge!?

Dont worry Actually, there is an intuitive method Combine Newtons method and bisection method –Newtons method first –if an approximation falls outside current interval, then apply bisection method to obtain a better guess (Can you write an algorithm for this method?) 70

Newtons Method Convergence analysis 71

72 Recall that

73 Is Newtons method always faster?

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75 In action

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Any Questions? Newtons Method

Secant Method

Secant method Because that Newtons method –2 function evaluations per iteration –requires the derivative Secant method is a variation on either false position or Newtons method –1 additional function evaluation per iteration –does not require the derivative Lets see the figure first 79 answer

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Secant method 81

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Any Questions? Secant Method