5 - Incremental Search(incsearch.m function brackets =...

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1/25/2010 1 function [ brackets ] = incsearch ( f, min, max, points ) % INCSEARCH: locates roots by incremental search % Inputs: f = a function of one variable (need not be able to deal with vectors) % min = lower bound of range to be searched % max = upper bound of range to be searched % points = number of search steps % O b k (i 1) l b d f i h b k Incremental Search (incsearch.m) % Outputs: brackets(i, 1) = lower bound for ith bracket % brackets(i, 2) = upper bound for ith bracket % **** if no brackets found, brackets = [] **** nb = 0; brackets = []; % brackets is initially 0 by 0 x = linspace (min, max, points); flo = f(x(1)); for i = 2: points fhi = f(x(i)); if sign(flo) ~= sign(fhi) nb = nb + 1; brackets(nb, 1) = x(i 1); brackets(nb, 2) = x(i); end flo = fhi; end end f = @(x) 50 * sin(0.5 * x) x.^2 17 * x + 60; % from last problem brackets = incsearch ( f, 0, 20, 100 ); [n m] = size(brackets); % note vector on left hand side The script below finds and outputs all roots of f between 0 and 20. if n == 0 fprintf ('No roots found.\n'); else for k = 1 : n x = fzero(f, brackets(k, :)); % select whole row fprintf ('There is a root at x = %f\n', x); end end Output: There is a root at x = 1.583780 There is a root at x = 6.636039 There is a root at x = 12.825489 There is a root at x = 16.266248
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1/25/2010 2 Bisection Search (basic idea) start with x LOW (less than root) and x HIGH (greater than root) while true pick x half way between x and x ROOT LOW HIGH if termination conditions satisfied stop endif if f( x MID ) has same sign as f(x LOW ) x LOW = x ROOT else x HIGH = x ROOT endif endwhile Possible Termination Conditions: 1/. The error in x ROOT has become acceptably small (i.e. we have got close enough to the actual root).
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