Numerical Analysis

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

2. 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

3. About the exercise?

4. 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

5. Rootfinding Given a function f, find a x such that f(x)=0

6. Is a rootfinding problem

10. Multiplicity

11. Definition

12. Multiplicity for polynomials • For polynomials, multiplicity can be determined by factoring the polynomial • That’s easy, but

13. For non-polynomials • What about this f(x)=0, where • Clearly, f(0)=0, so the f(x) has a root at x=0 • But what is the multiplicity of this root? • f(0)=f’(0)=f’’(0)=0, but f’’’(0)=-4 • the equation has a root of multiplicity 3 at x=0 answer

16. For non-polynomials • What about this f(x)=0, where • Clearly, f(0)=0, so the f(x) has a root at x=0 • But what is the multiplicity of this root? • f(0)=f’(0)=f’’(0)=0, but f’’’(0)=-4 • the equation has a root of multiplicity 3 at x=0

17. 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 • Newton’s method and secant method • fast, but require stronger conditions to guarantee convergence

19. A pathological example

20. 2.1 The Bisection Method

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

22. Drawing proof for Intermediate Value Theorem

23. 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

24. 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

25. Bisection method

26. In action f(x)=x3+2x2-3x-1, and (a1,b1)=(1,2)

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

31. An Algorithm Requires a stopping condition

32. Convergence of {pn}

34. Note • The bisection method converges to a root of f, not the root of f • what’s the difference? • f(a)f(b)<0 • guarantees the existence of a root, but not uniqueness, and the bisection method converge to one of these roots • The bisection method cannot locate roots ofeven multiplicity(the sign does not change on either side of such roots) • is common to all simple enclosure techniques

35. http://www.dianadepasquale.com/ThinkingMonkey.jpg Rate of convergence, O(1/2n)Order of convergence, α=1 and λ=1/2

36. We are now in position to select a stopping condition

37. Convergence measures • For any rootfinding technique, we have 3 convergence measures to construct the stopping condition • absolute error • relative error • test

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

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

41. Algorithm • Suppose that we decide to use the absolute error, but we don’t know the value of p • With the theorem, we can now construct an algorithm

43. Note • Performance measure • number of f evaluations rather than number of iterations (f could involve many floating point operations) • Underflow • both f(a) and f(p) will approaching zero • work with the signs rather than the sign of the product f(a)f(p)

44. 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)

45. 2.1 The Bisection Method

46. 2.2 The Method of False Position

47. False position • Very similar to bisection method • Only differ in selecting pn

49. Selecting pn • False position uses more information • values of f(an) and f(bn) • rather than just the signs

50. Which method is better?