Download
2 solving equations of one variable n.
Skip this Video
Loading SlideShow in 5 Seconds..
2. Solving Equations of One Variable PowerPoint Presentation
Download Presentation
2. Solving Equations of One Variable

2. Solving Equations of One Variable

239 Views Download Presentation
Download Presentation

2. Solving Equations of One Variable

- - - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript

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

  2. Contents • Bisection Method • Regula Falsi and Secant Method • Newton’s Method • Muller’s Method • Fixed-Point Iteration • Matlab’s Method kucg.korea.ac.kr

  3. Bisection Method kucg.korea.ac.kr

  4. Bisection Method kucg.korea.ac.kr

  5. Finding the Square Root of 3 Using Bisection How can we get ? kucg.korea.ac.kr

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

  7. Approximating the Floating Depth for a Cork Ball by Bisection(2/2) kucg.korea.ac.kr

  8. Discussion of Bisection Method kucg.korea.ac.kr

  9. Fixed-Point Iteration • Solution of equation • Convergence Theorem of fixed-point iteration kucg.korea.ac.kr

  10. Fixed-Point Iteration to Find a Zero of a Cubic Function kucg.korea.ac.kr

  11. 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 kucg.korea.ac.kr

  12. 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 kucg.korea.ac.kr

  13. Regular Falsi and Secant Methods 2005. 3. 23 Byungseok Roh kucg.korea.ac.kr

  14. 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. kucg.korea.ac.kr

  15. 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. Regula Falsi Method (cont.) kucg.korea.ac.kr

  16. 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 Example kucg.korea.ac.kr

  17. Example (cont.) • Calculation of using regula falsi. kucg.korea.ac.kr

  18. 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. Secant Method • The secant method has converged with a tolerance of . kucg.korea.ac.kr

  19. 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. kucg.korea.ac.kr

  20. NEWTON’S METHOD kucg.korea.ac.kr

  21. Newton’s Method • Newton’s method uses straight-line approximation which is the tangent to curve. • . • Intersection point kucg.korea.ac.kr

  22. Example • Finding Square Root of ¾ • approximate the zero of using the fact that . • Continuing for one more step kucg.korea.ac.kr

  23. 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 kucg.korea.ac.kr

  24. 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 kucg.korea.ac.kr

  25. Oscillations in Newton Method • Newton’s method give Oscillatory result for some funtions & initial estimates. Ex) kucg.korea.ac.kr

  26. Muller’s Method kucg.korea.ac.kr

  27. Muller’s Method • based on a quadratic approximation • procedure • Decide the parabola passing through (x1,y1), (x2, y2) and (x3,y3) • Solve the zero(x4) that is closest to x3 • 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. kucg.korea.ac.kr

  28. Muller’s Method (Cont’) kucg.korea.ac.kr

  29. Example • Finding the sixth root of 2 using Muller’s method • , , , kucg.korea.ac.kr

  30. Example (Cont’) converge Calculation of using Muller’s method kucg.korea.ac.kr

  31. Another Challenging Problem Tolerance = 0.0001 kucg.korea.ac.kr

  32. MATLAB function for Muller’s Method • P.65~66 code kucg.korea.ac.kr