1. Graphics Graphics Lab @ Korea University kucg.korea.ac.kr 2. Solving Equations of One Variable Korea University Computer Graphics Lab. Lee Seung Ho / Shin Seung Ho Roh Byeong Seok / Jeong So Hyeon

2. KUCG Graphics Lab @ Korea University kucg.korea.ac.kr Contents Bisection Method Regula Falsi and Secant Method Newton’s Method Muller’s Method Fixed-Point Iteration Matlab’s Method

3. Graphics Graphics Lab @ Korea University kucg.korea.ac.kr Bisection Method

4. KUCG Graphics Lab @ Korea University kucg.korea.ac.kr Bisection Method

5. KUCG Graphics Lab @ Korea University kucg.korea.ac.kr Finding the Square Root of 3 Using Bisection How can we get ?

6. KUCG Graphics Lab @ Korea University kucg.korea.ac.kr Approximating the Floating Depth for a Cork Ball by Bisection(1/2) Cork ball Radius : 1 Density : 0.25

7. KUCG Graphics Lab @ Korea University kucg.korea.ac.kr Approximating the Floating Depth for a Cork Ball by Bisection(2/2)

8. KUCG Graphics Lab @ Korea University kucg.korea.ac.kr Discussion of Bisection Method

9. KUCG Graphics Lab @ Korea University kucg.korea.ac.kr Fixed-Point Iteration Solution of equation Convergence Theorem of fixed-point iteration

10. KUCG Graphics Lab @ Korea University kucg.korea.ac.kr Fixed-Point Iteration to Find a Zero of a Cubic Function

11. KUCG Graphics Lab @ Korea University kucg.korea.ac.kr Matlab’s Methods(1/2) roots(p) p : vector Example EDU> r = roots(p); (p=[1 -7 14 -7]) r = 3.8019 2.445 0.75302

12. KUCG Graphics Lab @ Korea University kucg.korea.ac.kr Matlab’s Methods(2/2) fzero( ‘function name’,x0 ) function name: string x0 : initial estimate of the root Example function y = flat10(x) y = x.^10 – 0.5; z = fzero(‘flat10’,0.5) z = 0.93303

13. Graphics Graphics Lab @ Korea University kucg.korea.ac.kr Regular Falsi and Secant Methods 2005. 3. 23 Byungseok Roh

14. KUCG Graphics Lab @ Korea University kucg.korea.ac.kr Regula Falsi Method The regula falsi method start with two point, (a, f(a)) and (b,f(b)), satisfying the condition that f(a)f(b)<0. The straight line through the two points (a, f(a)), (b, f(b)) is The next approximation to the zero is the value of x where the straight line through the initial points crosses the x-axis.

15. KUCG Graphics Lab @ Korea University kucg.korea.ac.kr Regula Falsi Method (cont.) If there is a zero in the interval [a, c], we leave the value of a unchanged and set b = c. On the other hand, if there is no zero in [a, c], the zero must be in the interval [c, b]; so we set a = c and leave b unchanged. The stopping condition may test the size of y, the amount by which the approximate solution x has changed on the last iteration, or whether the process has continued too long. Typically, a combination of these conditions is used.

16. KUCG Graphics Lab @ Korea University kucg.korea.ac.kr Example Finding the Cube Root of 2 Using Regula Falsi Since f(1)= -1, f(2)=6, we take as our starting bounds on the zero a=1 and b=2. Our first approximation to the zero is We then find the value of the function: Since f(a) and y are both negative, but y and f(b) have opposite signs

17. KUCG Graphics Lab @ Korea University kucg.korea.ac.kr Example (cont.) Calculation of using regula falsi.

18. KUCG Graphics Lab @ Korea University kucg.korea.ac.kr Secant Method Instead of choosing the subinterval that must contain the zero, we form the next approximation from the two most recently generated points: At the k-th stage, the new approximation to the zero is The secant method, closely related to the regula falsi method, results from a slight modification of the latter. The secant method has converged with a tolerance of.

19. KUCG Graphics Lab @ Korea University kucg.korea.ac.kr Example Finding the Square Root of 3 by Secant Method To find a numerical approximation to, we seek the zero of. Since f(1)=-2 and f(2)=1, we take as our starting bounds on the zero and. Our first approximation to the zero is Calculation of using secant method.

20. Graphics Graphics Lab @ Korea University kucg.korea.ac.kr NEWTON’S METHOD

21. KUCG Graphics Lab @ Korea University kucg.korea.ac.kr Newton’s Method Newton’s method uses straight-line approximation which is the tangent to curve.. Intersection point

22. KUCG Graphics Lab @ Korea University kucg.korea.ac.kr Example Finding Square Root of ¾ approximate the zero of using the fact that. Continuing for one more step

23. KUCG Graphics Lab @ Korea University kucg.korea.ac.kr Finding Floating Depth for a Wooden Ball Volume of submerged segment of the Sphere To find depth at which the ball float, volume of submerged segment is time. Simplifies to

24. KUCG Graphics Lab @ Korea University kucg.korea.ac.kr Finding Floating Depth for a Wooden Ball (cont.) To find depth a ball, density is one-third of water float. Calculation f(x) using Newton’s Method

25. KUCG Graphics Lab @ Korea University kucg.korea.ac.kr Oscillations in Newton Method Newton’s method give Oscillatory result for some funtions & initial estimates. Ex)

26. Graphics Graphics Lab @ Korea University kucg.korea.ac.kr Muller’s Method

27. KUCG Graphics Lab @ Korea University kucg.korea.ac.kr Muller’s Method based on a quadratic approximation procedure 1. Decide the parabola passing through (x1,y1), (x2, y2) and (x3,y3) 2. Solve the zero(x4) that is closest to x3 3. Repeat 1,2 until x converge to predefined tolerance advantage Requires only function values; Derivative need not be calculated X can be an imaginary number.

28. KUCG Graphics Lab @ Korea University kucg.korea.ac.kr Muller’s Method (Cont’)

29. KUCG Graphics Lab @ Korea University kucg.korea.ac.kr Example Finding the sixth root of 2 using Muller’s method,,,

30. KUCG Graphics Lab @ Korea University kucg.korea.ac.kr Example (Cont’) ixy 10.5-1.9844 21.59.3906 31 41.0779-0.43172 51.117-0.05635 61.12550.00076162 71.1255-4.7432e-07 converge Calculation of using Muller’s method

31. KUCG Graphics Lab @ Korea University kucg.korea.ac.kr Another Challenging Problem stepxY 10-0.5 210.5 3 -0.49902 40.80875-0.38029 50.9081-0.11862 60.943250.057542 70.93269-0.0018478 80.93303-6.3021e-06 90.93303-3.1235e-10 Tolerance = 0.0001

32. KUCG Graphics Lab @ Korea University kucg.korea.ac.kr MATLAB function for Muller’s Method P.65~66 code