Page - 189 - in Programming for Computations – Python - A Gentle Introduction to Numerical Simulations with Python

6.1 BruteForceMethods 189 Note thatif rootsevaluates toTrue ifroots is non-empty. This is a general test inPython:if Xevaluates toTrue ifX isnon-emptyorhasanonzerovalue. 6.1.2 BruteForceOptimization Numerical algorithm We realize that xi corresponds to a maximum point if yi 1 < yi > yiC1. Similarly,xi corresponds to aminimumifyi 1 > yi < yiC1. Wecando this test for all “inner” points i D 1;:: :;n 1 tofindall localminima andmaxima. In addition, we need to add an end point, i D 0 or i D n, if the correspondingyi is aglobalmaximumorminimum. Implementation The algorithm above can be translated to the following Python function (filebrute_force_optimizer.py): def brute_force_optimizer(f, a, b, n): from numpy import linspace x = linspace(a, b, n) y = f(x) # Let maxima and minima hold the indices corresponding # to (local) maxima and minima points minima = [] maxima = [] for i in range(n-1): if y[i-1] < y[i] > y[i+1]: maxima.append(i) if y[i-1] > y[i] < y[i+1]: minima.append(i) # What about the end points? y_max_inner = max([y[i] for i in maxima]) y_min_inner = min([y[i] for i in minima]) if y[0] > y_max_inner: maxima.append(0) if y[len(x)-1] > y_max_inner: maxima.append(len(x)-1) if y[0] < y_min_inner: minima.append(0) if y[len(x)-1] < y_min_inner: minima.append(len(x)-1) # Return x and y values return [(x[i], y[i]) for i in minima], \ [(x[i], y[i]) for i in maxima] Themax andmin functions are standardPython functions forfinding themaxi- mumandminimumelementofa listoranobject thatonecan iterateoverwithafor loop.