lecture 3

lecture 3 - Chapter 6 Open Methods Open Methods 1 Simple...

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Unformatted text preview: Chapter 6 Open Methods Open Methods 1. Simple Fixed-Point Iteration 2. Newton-Raphson 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 Newton-Raphson method, Secant Method, Muller’s method, fixed-point iterations First one to consider is the fixed-point method x x sin x x sin 2 3 x x 3 x 2 x 2 2 Fixed-Point 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 Fixed-Point Iteration Two Alternative Graphical Methods f(x) = f 1 (x) – f 2 (x) = 0 f ( x ) = 0 f 1 ( x ) = f 2 ( x ) Fixed-Point 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 ) Fixed-point 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('Fixed-Point 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 root-finding methods Newton-Raphson 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 Newton-Raphson Method Newton-Raphson Method root x* 4 x 3 x x f 4 ) ( Newton-Raphson 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 Newton-Raphson 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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lecture 3 - Chapter 6 Open Methods Open Methods 1 Simple...

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