Graphics Graphics 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

KUCG Graphics 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

Graphics Graphics Korea University kucg.korea.ac.kr Bisection Method

KUCG Graphics Korea University kucg.korea.ac.kr Bisection Method

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

KUCG Graphics 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

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

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

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

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

KUCG Graphics Korea University kucg.korea.ac.kr Matlab’s Methods(1/2) roots(p) p : vector Example EDU> r = roots(p); (p=[ ]) r =

KUCG Graphics 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 =

Graphics Graphics Korea University kucg.korea.ac.kr Regular Falsi and Secant Methods Byungseok Roh

KUCG Graphics 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.

KUCG Graphics 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.

KUCG Graphics 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

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

KUCG Graphics 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.

KUCG Graphics 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.

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

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

KUCG Graphics 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

KUCG Graphics 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

KUCG Graphics 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

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

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

KUCG Graphics 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.

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

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

KUCG Graphics Korea University kucg.korea.ac.kr Example (Cont’) ixy e-07 converge Calculation of using Muller’s method

KUCG Graphics Korea University kucg.korea.ac.kr Another Challenging Problem stepxY e e-10 Tolerance =

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