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

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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

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

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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

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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

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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

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Fixed point iteration 16

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Fixed point iteration 17

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18 In action

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Any Questions? 21 About fixed point iteration

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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

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Iteration function 23

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25 In action

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Analysis 27

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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

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The first step 42

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The second step 43

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Any Questions? Fixed Point Iteration Schemes

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Newtons Method

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Newtons Method Definition 48

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49 In action

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In the previous example 51

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Any Questions? 53

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Initial guess 54 example answer

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Initial guess 55 answer

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Initial guess 56

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Convergence analysis for Newtons method 58

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

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Analysis strategy 60

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Any Questions? 65

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Newtons Method Guaranteed to Converge? 66 hint answer

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Newtons Method Guaranteed to Converge? 67 answer

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Newtons Method Guaranteed to Converge? 68

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

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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

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Newtons Method Convergence analysis 71

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72 Recall that

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73 Is Newtons method always faster?

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75 In action

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Any Questions? Newtons Method

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Secant Method

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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

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