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Unformatted text preview: Chapter 6 Open Methods Open Methods 1. Simple FixedPoint Iteration 2. NewtonRaphson Method 3. Secant Methods 4. MATLAB function: fzero 5. Polynomials Open Methods What if we don ' t have two endpoints bracketing the root? There exist open methods which do not require bracketed intervals NewtonRaphson method, Secant Method, Muller’s method, fixedpoint iterations First one to consider is the fixedpoint method x x sin x x sin 2 3 x x 3 x 2 x 2 2 FixedPoint Iteration First open method is fixed point iteration Idea: rewrite original equation f(x) = 0 into another form x = g(x). Use iteration x i+1 = g ( x i ) to find a value that reaches convergence Example: Simple FixedPoint Iteration Two Alternative Graphical Methods f(x) = f 1 (x) – f 2 (x) = 0 f ( x ) = 0 f 1 ( x ) = f 2 ( x ) FixedPoint Iteration Convergent Divergent Convergence We decide where to stop based on either: the relative change in x = or the difference between x and g(x) 1 i i 1 i x x x Need to rearrange equation in order to find r 2 r 1 r 3 r 2 x i+1 = g(x i ) Fixedpoint iteration doesn’t always work Basically, if g’(x) < 1 near the intersection with the x line, it will work 1)/3 (x (x) g x 1) (3x (x) g x 1 3x x f(x) 3 2 1/3 1 3 Convergence Criterion Give two roots only Third root >> x=2:0.02:2; y=my_func1(x); z=x*0; >> H=plot(x,y,'r',x,x,'b'); hold on; plot(x,z,'k'); >> set(H,'LineWidth',2.0); >> axis([2 2 2 2]); xlabel('x'); ylabel('g(x) = (x^3 + 1)/3'); >> Title('FixedPoint Iteration : f(x) = x^3  3x + 1 = 0'); 1 ) x ( ' g 1 ) x ( ' g 1 ) x ( ' g x = g(x) 1 ) x ( ' g divergent 2 i 1 i i 1 i i i 1 i x x ! 2 f x x x f x f x f Newton’s Method King of the rootfinding methods NewtonRaphson method Based on Taylor series expansion Truncate the Taylor series to get i 1 i i i 1 i x x x f x f x f At the root, f ( x i+1 ) = 0 , so i 1 i i i x x x f x f i i i 1 i x f x f x x NewtonRaphson Method NewtonRaphson Method root x* 4 x 3 x x f 4 ) ( NewtonRaphson Method False position  secant line Newton’s method  tangent line x i x i+1 Newton’s Method Note that an evaluation of the derivative (slope) is required You may have to do this numerically Open Method – Convergence depends on the initial guess (not guaranteed) However, Newton method can converge very quickly (quadratic convergence) Bungee Jumper Problem NewtonRaphson method Need to evaluate the function and its derivative t m gc h t m 2 g t m gc mc g 2 1...
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 Spring '08
 Rottman
 Secant method, Rootfinding algorithm

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