Page - 180 - in Programming for Computations – Python - A Gentle Introduction to Numerical Simulations with Python 3.6, Volume Second Edition

180 7 SolvingNonlinearAlgebraicEquations # Return x and y values return [(x[i], y[i]) for i in minima], \ [(x[i], y[i]) for i in maxima] Themaxandmin functionsare standardPythonfunctionsforfinding themaximum andminimumelementofa listoranobject thatonecan iterateoverwitha for loop. Anapplication tof(x)= e−x2 cos(4x) looks like def demo(): from numpy import exp, cos minima, maxima = brute_force_optimizer( lambda x: exp(-x**2)*cos(4*x), 0, 4, 1001) print(’Minima:\n’, minima) print(’Maxima:\n’, maxima) Runningtheprogramgives Minima: [(0.70000000000000007, -0.5772302750838405), (2.1520000000000001, -0.0066704807422565023), (3.6600000000000001, -7.3338267339366542e-07)] Maxima: [(1.4159999999999999, 0.10965467991643564), (2.8999999999999999, 0.00012651823896373234), (0.0, 1.0)] 7.1.3 ModelProblemforAlgebraicEquations We shall consider the very simple problem of finding the square root of 9. That is, we want to solve x2 = 9, but will (for simplicity) seek only the positive solution. Knowingthesolutionbeforehand,allowsus toeasily investigatehowthenumerical method (and the implementation of it) performs in the search for a solution. The f(x) function thatcorrespondsto the equationx2 =9 is f(x)=x2−9 . Ourintervalof interest forsolutionswillbe [0,1000](theupperlimithereischosen somewhatarbitrarily). In the following,wewillpresent severalefficientandaccuratemethodsfor solv- ing nonlinear algebraic equations, both single equation and systems of equations. The methods all have in common that they search for approximate solutions. The methodsdiffer,however, in the way theyperformthesearch for solutions.The idea for the search influences the efficiency of the search and the reliability of actually findingasolution.Forexample,Newton’smethodisveryfast,butnotreliable,while thebisectionmethod is the slowest, butabsolutely reliable.Nomethod isbest at all problems,so weneeddifferentmethodsfordifferentproblems.