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lecture20 - 1.00 Lecture 20 More on root finding Numerical...

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1.00 Lecture 20 October 27, 2005 More on root finding Numerical Integration Newton’s Method For small increment and smooth function, st 1 st derivative is small, etc. ... 2 / ) ( ) ( ' ) ( ) ( 2 + + + + δ δ δ x f 0 ) ( = + δ ) ( ' / ) ( x f = δ Based on Taylor series expansion: higher order derivatives are small and implies If high order derivatives are large or 1 derivative is small, Newton can fail miserably Converges quickly if assumptions met Has generalization to N dimensions that is one of the few available See Numerical Recipes for ‘safe’ Newton- Raphson method, which uses bisection when ' ' x f x f x f x f x f 1
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Newton’s Method ) f(x) f’(x Initial guess of root Newton’s Method Pathology f’(x) f(x) Initial guess of root 2
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Newton’s Method public class Newton { // NumRec, p. 365 public static double newt(MathFunction2 func, double a, double b, double epsilon) { double guess= 0.5*(a + b); for (int j= 0; j < JMAX; j++) { double fval= func.fn(guess); double fder= func.fd(guess); double dx= fval/fder; guess -= dx; System.out.println(guess); if ((a - guess)*(guess - b) < 0.0) { System.out.println("Error: out of bracket"); return ERR_VAL; // Experiment with this } // It’s conservative if (Math.abs(dx) < epsilon) return guess; } System.out.println("Maximum iterations exceeded"); return guess; } Newton’s Method, p.2 public static int JMAX= 50; public static double ERR_VAL= -10E10; public static void main(String[] args) { double root= Newton.newt(new FuncB(), -0.0, 8.0, 0.0001); System.out.println("Root: " + root); System.exit(0); } } class FuncB implements MathFunction2 { public double fn(double x) { return x*x - 2; } public double fd(double x) { return 2*x; } } public interface MathFunction2 { public double fn(double x); // Function value public double fd(double x); } // 1st derivative value 3
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Examples f(x)= x 2 + 1 No real roots, Newton generates ‘random’ guesses f(x)= sin(5x) + x 2 – 3 Root= -0.36667 Bracket between –1 and 2, for example Bracket between 0 and 2 will fail with conservative Newton (outside bracket) f(x)= ln(x 2 –0.8x + 1) Roots= 0, 0.8 Bracket between 0 and 1.2 works Bracket between 0.0 and 8.0 fails 8 Use Roots.java from lab to experiment!
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