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Lecture13

# Lecture13 - Roots of Equations The first real numerical...

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Roots of Equations The first real numerical method – Root finding Finding the value x where a function y = f(x) = 0 Very basic numerical procedure

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Two Basic Approaches Bracketing Methods Bisection False Position Open Methods Fixed-Point Iteration Newton-Raphson Secant Methods Roots of Polynomials
Chapter 5 Bracketing Methods

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5.1 Introduction and Background 5.2 Graphical Methods 5.3 Bracketing Methods and Initial Guesses 5.4 Bisection 5.5 False-Position Bracketing Methods
Graphical methods No root (same sign) Single root (change sign) Two roots (same sign) Three roots (change sign)

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Multiple Roots Discontinuity Special Cases
Graphical Method - Progressive Enlargement Two distinct roots

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Graphical Method Graphical method is useful for getting an idea of what’s going on in a problem, but depends on eyeball. Use bracketing methods to improve the accuracy: Bisection and false-position methods
Bracketing Methods Both bisection and false-position methods require the root to be bracketed by the endpoints. How to find the endpoints? * plotting the function * incremental search * trial and error

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Incremental Search
Incremental Search function xb=incsearch(func,xmin,xmax,ns) %incsearch: incremental search root locator % . .. see the rest of comments on page 120 of Chapara textbook if nargin<4,ns=50; end % if ns blank set to 50 % Incremental search x=linspace(xmin,xmax,ns); f=func(x); nb=0; xb=[]; %xb is null unless sign change detected for k=1:length(x)-1

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Lecture13 - Roots of Equations The first real numerical...

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