05-Roots, Bracketing Methods

05-Roots, Bracketing Methods - 5: Roots of Equations...

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5: Roots of Equations 5: Roots of Equations Bracketing Methods Bracketing Methods
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Oct 10, 2006 14:15 2 Objectives understand what roots problems are. determine a root graphically incremental search and its shortcomings bisection method error estimates for bisection false position method
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Oct 10, 2006 14:15 3 Problem Bungee jumper: Question : what value of m will result in v(t) = 36 m/s after t = 4 s, using c d = 0.25 kg/m? Root finding: find values of x such that f(x) = 0 v t  = g m c d tanh g c d m t (5.1) 36 = 9.81 × m 0.25 tanh 9.81 × 0.5 m × 4 9.81 × m 0.25 tanh 9.81 × 0.5 m × 4 36 = 0 f m = g m c d tanh g c d m t v t (5.2)
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Oct 10, 2006 14:15 4 Graphical Methods (1) Plot, visually estimate a root: cd = 0.25; g = 9.81; v = 36; t = 4; mp = linspace(50,300); fp = sqrt(g*mp/cd).*tanh(sqrt(g*cd./mp)*t)-v; plot(mp,fp), grid root ≈ 140 kg
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Oct 10, 2006 14:15 5 Graphical Methods (2) not very precise impossible to automate useful to determine starting values for better methods can be used to determine when problems might arise
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Oct 10, 2006 14:15 6 Types of Roots Fig 5.1 Fig 5.2
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Oct 10, 2006 14:15 7 Basic methods Bracketing methods : initial guesses on each side of a root. Open methods : one or more initial guesses, not necessarily a bracket. Bracketing methods are more “robust”. But brackets may be difficult to find.
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Oct 10, 2006 14:15 8 5.3.1: Incremental Search (1) find intervals over which f(x) changes sign there will be at least one real root (actually, an odd number of real roots) choice of increment length is important, difficult function xb = incsearch(func,xmin,xmax,ns) % xb = incsearch(func,xmin,xmax,ns): % finds brackets of x that contain sign changes of % a function on an interval % input: % func = name of function % xmin, xmax = endpoints of interval % ns = (optional) number of subintervals along x % used to search for brackets % output: % xb(k,1) is the lower bound of the kth sign change % xb(k,2) is the upper bound of the kth sign change % If no brackets found, xb = []. if nargin < 4, ns = 50; end %if ns blank set to 50
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Oct 10, 2006 14:15 9 Incremental Search (2) % Incremental search x = linspace(xmin,xmax,ns); f = feval(func,x); nb = 0; xb = []; %xb is null unless sign change detected for k = 1:length(x)-1 if sign(f(k)) ~= sign(f(k+1)) %check for sign change nb = nb + 1; xb(nb,1) = x(k); xb(nb,2) = x(k+1); end end if isempty(xb) %display that no brackets were found
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05-Roots, Bracketing Methods - 5: Roots of Equations...

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