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

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In the previous slide Error (motivation) Floating point number system –difference to real number system –problem of roundoff Introduced/propagated error Focus on numerical methods –three bugs 2

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Any Questions? 3 About the exercise

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In this 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 4

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Rootfinding 5

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6 Is a rootfinding problem

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Multiplicity 10

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Definition 11

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Multiplicity for polynomials For polynomials, multiplicity can be determined by factoring the polynomial Thats easy, but 12

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For non-polynomials 13 answer

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For non-polynomials 16

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Rootfinding methods 2 categories –simple enclosure methods –fixed point iteration schemes Simple enclosure –bisection and false position –guaranteed to converge to a root, but slow Fixed point iteration –Newtons method and secant method –fast, but require stronger conditions to guarantee convergence 17

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A pathological example 19

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The Bisection Method

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Bisection method The most basic simple enclosure method All simple enclosure methods are based on Intermediate Value Theorem 21

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22 Drawing proof for Intermediate Value Theorem

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In Plain English Find an interval of that the endpoints are opposite sign Since one endpoint value is positive and the other negative, zero is somewhere between the values, that is, at least one root on that interval 23

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Bisection method The objective is to systematically shrink the size of that root enclosing interval The simplest and most natural way is to cut the interval in half Next is to determine which half contains a root –Intermediate Value Theorem, again Repeat the process on that half 24

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Bisection method 25

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In action 26

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Any Questions? 29

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30 You know what the bisection method is, but so far it is not an algorithm, why?

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31 An algorithm requires a stopping condition

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32 Convergence of {p n }

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Note 34

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36 We are now in position to select a stopping condition

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Convergence measures 37

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Which is the Best? 38 No one is always better than another answer

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Which is the Best? 40 No one is always better than another

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Algorithm 41

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Note 43

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Summary of bisection method Advantage –straightforward –inexpensive (1 evaluation per iteration) –guarantee to converge Disadvantage –error estimation can be overly pessimistic –(drawing for a extreme case of bisection method) 44

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Any Questions? The Bisection Method

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The Method of False Position

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False position 47

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Which method is better? 50

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Which method is better From another aspect to only the convergence rate –bisection method provides a theoretical bound of error, but no error estimate –false position provides computable error estimate –(the only one advantage of false position) Thus, we can have a more appropriate stopping condition for false position –(we will use this advantage in Section 2.6) 51

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52 Since false position has no theoretical bound of error, it requires effort to prove the convergence

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Convergence analysis One observation to proceed the convergence analysis –one of the endpoints remains fixed –the other endpoint is just the previous approximation Namely – a n =a n-1, b n =p n-1 or – b n =b n-1, a n =p n-1 55 observation

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56 The first problem

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57 The second problem

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58 The third problem

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Convergence analysis 60

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Go back to the equation (4) 61

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Any Questions? 63

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Guarantee to convergence 64 answer

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Guarantee to convergence 65

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The first condition 66 The remaining three conditions can be proved in a similar fashion

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67 Now its time to select a stopping condition

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Stopping condition 68

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70 The first problem

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71 The second problem

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72 The third problem

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Any Questions? The Method of False Position

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