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POMatlab: poor accuracy of optimizers/solvers
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copied!<p>I am having difficulty achieving sufficient accuracy in a root-finding problem on Matlab. I have a function, <code>Lik(k)</code>, and want to find the value of <code>k</code> where <code>Lik(k)=L0</code>. Basically, the problem is that various built-in Matlab solvers (<code>fzero</code>, <code>fminbnd</code>, <code>fmincon</code>) are not getting as close to the solution as I would like or expect.</p> <p><code>Lik()</code> is a user-defined function which involves extensive coding to compute a numerical inverse Laplace transform, etc., and I therefore do not include the full code. However, I have used this function extensively and it appears to work properly. <code>Lik()</code> actually takes several input parameters, but for the current step, all of these are fixed except <code>k</code>. So it is really a one-dimensional root-finding problem.</p> <p>I want to find the value of <code>k >= 165.95</code> for which <code>Lik(k)-L0 = 0</code>. <code>Lik(165.95)</code> is less than <code>L0</code> and I expect <code>Lik(k)</code> to increase monotonically from here. In fact, I can evaluate <code>Lik(k)-L0</code> in the range of interest and it appears to smoothly cross zero: e.g. <code>Lik(165.95)-L0 = -0.7465, ..., Lik(170.5)-L0 = -0.1594, Lik(171)-L0 = -0.0344, Lik(171.5)-L0 = 0.1015, ... Lik(173)-L0 = 0.5730, ..., Lik(200)-L0 = 19.80</code>. So it appears that the function is behaving nicely.</p> <p>However, I have tried to "automatically" find the root with several different methods and the accuracy is not as good as I would expect...</p> <p>Using <code>fzero(@(k) Lik(k)-L0)</code>: If constrained to the interval <code>(165.95,173)</code>, <code>fzero</code> returns <code>k=170.96</code> with <code>Lik(k)-L0=-0.045</code>. Okay, although not great. And for practical purposes, I would not know such a precise upper bound without a lot of manual trial and error. If I use the interval <code>(165.95,200)</code>, <code>fzero</code> returns <code>k=167.19</code> where <code>Lik(k)-L0 = -0.65</code>, which is rather poor. I have been running these tests with <em>Display</em> set to <em>iter</em> so I can see what's going on, and it appears that <code>fzero</code> hits <code>167.19</code> on the 4th iteration and then stays there on the 5th iteration, meaning that the change in <code>k</code> from one iteration to the next is less than <code>TolX</code> (set to 0.001) and thus the procedure ends. The exit flag indicates that it successfully converged to a solution.</p> <p>I also tried minimizing <code>abs(Lik(k)-L0)</code> using <code>fminbnd</code> (giving upper and lower bounds on <code>k</code>) and <code>fmincon</code> (giving a starting point for <code>k</code>) and ran into similar accuracy issues. In particular, with <code>fmincon</code> one can set both <code>TolX</code> and <code>TolFun</code>, but playing around with these (down to 10^-6, much higher precision than I need) did not make any difference. Confusingly, sometimes the optimizer even finds a k-value on an earlier iteration that is closer to making the objective function zero than the final k-value it returns.</p> <p>So, it appears that the algorithm is iterating to a certain point, then failing to take any further step of sufficient size to find a better solution. Does anyone know why the algorithm does not take another, larger step? Is there anything I can adjust to change this? (I have looked at the list under <em>optimset</em> but did not come up with anything useful.)</p> <p>Thanks a lot!</p>

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