ECOR 2606 - Lecture 10 (golden search)

Iterations x 0 1 n

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Unformatted text preview: ternal point is not about to become one of XL and XU. The error formula assumes that XL and XU have yet to be updated (i.e. start of iteration values are assumed) but the Matlab code updates them before applying the formula (i.e. end of iteration values are used). This is a mistake. As usual the text uses relative error (the error is divided by the magnitude of the solution). The Matlab code in the text completely negates the advantages of the Golden Section search. Both interior points are recalculated on every iteration and the function is evaluated twice per iteration. If one is going to do this it would make more sense to locate the interior points just on either side of the interval midpoint (and so get rid of nearly half of the interval on every iteration). 4 2/5/2010 function x = golden (f, xL, xU, Edes, display) % GOLDEN Finds a minimum by performing a golden section search. % Inputs: f = a function of one variable % xL = lower bound of region containing minimum % xU = upper bound of region containing minimum % Edes = function stops when x within Edes of minimum % display = display option (0 = no display (default), 1 = display) % Outputs: x ‐ estimate of minimum if nargin < 5; display = 0; end p1 = ((1 + sqrt(5)) / 2) ‐ 1; % golden ratio ‐ 1 if display fprintf ... (' k xL end x2 x1 xU Emax\n'); % set up for first iteration x1 = xL + p1 * (xU ‐ xL); fx1 = f(x1); x2 = xU ‐ p1 * (xU ‐ xL); fx2 = f(x2); for k = 1 : 1000 Emax = (xU ‐ xL) / 2; if display fprintf ('%5d %12.6f %12.6f %12.6f %12.6f %12.6f\n', k, xL, x2, x1, xU, Emax); end if Emax <= Edes x = (xL + xU) / 2; return; end if fx2 < fx1 xU = x1; x1 = x2; fx1 = fx2; % old x2 becomes new x1 x2 = xU ‐ p1 * (xU ‐ xL); fx2 = f(x2); % brand new x2 required else xL = x2; x2 = x1; fx2 = fx1; % old x1 becomes new x2 x1 = xL + p1 * (xU ‐ xL); fx1 = f(x1); % brand new x1 required end end error ('Golden section search has not converged.'); end 5 2/5/2010 Example Golden Section Search: >> f = @(x) -5 * x^3 + 115.3 * x^2 - 700 * x + 757.5; >> minx = golden (f, 2, 8, 0.001, 1) k xL x2 x1 1 2.000000 4.291796 5.708204 2 2.000000 3.416408 4.291796 3 3.416408 4.291796 4.832816 4 3.416408 3.957428 4.291796 5 3.957428 4.291796 4.498447 6 3.957428 4.164079 4.291796 7 3.957428 4.085145 4.164079 8 4.085145 4.164079 4.212862 9 4.085145 4.133929 4.164079 10 4.133929 4.164079 4.182712 11 4.133929 4.152562 4.164079 12 4.152562 4.164079 4.171196 13 4.152562 4.159680 4.164079 14 4.159680 4.164079 4.166797 15 4.159680 4.162398 4.164079 16 4.159680 4.161360 4.162398 17 4.161360 4.162398 4.163040 18 4.162398 4.163040 4.163437 minx = 4.1632 xU 8.000000 5.708204 5.708204 4.832816 4.832816 4.498447 4.291796 4.291796 4.212862 4.212862 4.182712 4.182712 4.171196 4.171196 4.166797 4.164079 4.164079 4.164079 Emax 3.000000 1.854102 1.145898 0.708204 0.437694 0.270510 0.167184 0.103326 0.063859 0.039467 0.024392 0.015075 0.009317 0.005758 0.003559 0.002199 0.001359 0.000840 fminbnd Details: Options may be specified (details as for fzero). fminbnd combines a golden section search and parabolic interpolation (see text) >> opts = optimset('display', 'iter'); >> bestL = fminbnd (Tfunc, L1, L3, opts) % example from last lecture Func‐count x f(x) Procedure 1 502.564 14802.2 initial 2 503.036 14802.1 golden 3 503.328 14802.1 golden 4 503.095 14802.1 parabolic 5 503.095 14802.1 parabolic 6 503.095 14802.1 parabolic 7 503.095 14802.1 parabolic Optimization terminated: the current x satisfies the termination criteria using OPTIONS.TolX of 1.000000e‐004 bestL = 503.0946 6...
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This note was uploaded on 09/13/2013 for the course ECOR 2606 taught by Professor Goheen during the Fall '10 term at Carleton CA.

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