Numerical Analysis

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

2. 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

3. In this slide • Fixed point iteration scheme • what is a fixed point? • iteration function • convergence • Newton’s method • tangent line approximation • convergence • Secant method

4. 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

5. 2.3 Fixed Point Iteration Schemes

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

8. Mean Value Theorem • We use a slightly different formulation • An example of using this theorem • proof the inequality

10. Fixed points • Consider the function • thought of as moving the input value of to the output value • the sine function maps to • the sine function fixes the location of • is said to be a fixed point of the function

12. Number of fixed points • According to the previous figure, a trivial question is • how many fixed points of agiven function?

15. 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

16. Fixed point iteration

17. Fixed point iteration • If it is known that a function has a fixed point, one way to approximate the value of that fixed point is fixed point iteration scheme • These can be defined as follows:

18. http://thomashawk.com/hello/209/1017/1024/Jackson%20Running.jpghttp://thomashawk.com/hello/209/1017/1024/Jackson%20Running.jpg In action

21. About fixed point iteration

22. 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)=x3+x2-3x-3=0, what is its iteration function g(x)? hint

23. Iteration function • Algebraically transform to the form • … • Every rootfinding problem can be transformed into any number of fixed point problems • (fortunately or unfortunately?)

25. http://thomashawk.com/hello/209/1017/1024/Jackson%20Running.jpghttp://thomashawk.com/hello/209/1017/1024/Jackson%20Running.jpg In action

27. Analysis • #1 iteration function converges • but to a fixed point outside the interval • #2 fails to converge • despite attaining values quite close to #1 • #3 and #5 converge rapidly • #3 add one correct decimal every iteration • #5 doubles correct decimals every iteration • #4 converges, but very slow

28. Convergence • This analysis suggests a trivial question • the fixed point of is justified in our previous theorem

32. demonstrates the importance of the parameter • when , rapid • when , dramatically slow • when , roughly the same as the bisection method

33. Order of convergence of fixed point iteration schemes All about the derivatives,

39. Stopping condition

41. Two steps

42. The first step

43. The second step

44. 2.3 Fixed Point Iteration Schemes

45. 2.4 Newton’s Method

48. Newton’s MethodDefinition

49. http://thomashawk.com/hello/209/1017/1024/Jackson%20Running.jpghttp://thomashawk.com/hello/209/1017/1024/Jackson%20Running.jpg In action