Page - 179 - in Programming for Computations – Python - A Gentle Introduction to Numerical Simulations with Python 3.6, Volume Second Edition

7.1 BruteForceMethods 179 def demo(): from numpy import exp, cos roots = brute_force_root_finder( lambda x: exp(-x**2)*cos(4*x), 0, 4, 1001) if roots: print(roots) else: print(’Could not find any roots’) Note thatif rootsevaluates toTrue ifroots isnon-empty.This is ageneral test inPython:if Xevaluates toTrue ifX isnon-emptyorhasa nonzerovalue. Running theprogramgives theoutput [0.39270091800495166, 1.1781066425246509, 1.9635022750438742, 2.7489089483136029, 3.534319340895673] 7.1.2 BruteForceOptimization Numerical Algorithm We realize that xi corresponds to a maximum point if yi−1 <yi > yi+1. Similarly,xi corresponds to a minimum if yi−1 > yi < yi+1. We can do this test for all “inner” points i = 1,.. .,n−1 to find all local minima and maxima. In addition, we need to add an end point, i = 0 or i = n, if the correspondingyi is a globalmaximumorminimum. 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(1, 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)