Page - 193 - in Programming for Computations – Python - A Gentle Introduction to Numerical Simulations with Python 3.6, Volume Second Edition

7.5 RateofConvergence 193 A single q in (7.5) is defined in the limit n → ∞. For finiten, and especially smallern, q will vary withn. To estimateq, we can compute all the errors en and set up(7.5) for threeconsecutiveexperimentsn−1,n, andn+1: en=Ceqn−1, en+1 =Ceqn . Dividing, e.g., the latter equation by the former, and solving with respect to q, we get that q= ln(en+1/en) ln(en/en−1) . Since thisq will vary somewhat withn, we call itqn. Asngrows, we expectqn to approacha limit (qn→q). ModifyingOurFunctions toReturnAll Approximations To computeall theqn values,we needall thexn approximations.However,ourprevious implementations of Newton’smethod, the secant method,and thebisection method returned just the finalapproximation. Therefore, we have modified our solvers5 accordingly, and placed them in nonlinear_solvers.py.A user can choose whether the final value or the whole history of solutions is to be returned. Each of the extended implementations now takesanextraparameterreturn_x_list.Thisparameter is aboolean,set toTrue if the function is supposed to return all the root approximations, or False, if the functionshouldonly return thefinalapproximation. Asanexample, letus takea closer lookatNewton: def Newton(f, dfdx, x, eps, return_x_list=False): f_value = f(x) iteration_counter = 0 if return_x_list: x_list = [] while abs(f_value) > eps and iteration_counter < 100: try: x = x - float(f_value)/dfdx(x) except ZeroDivisionError: print(’Error! - derivative zero for x = {:g}’.format(x)) sys.exit(1) # Abort with error f_value = f(x) iteration_counter += 1 if return_x_list: x_list.append(x) # Here, either a solution is found, or too many iterations if abs(f_value) > eps: iteration_counter = -1 # i.e., lack of convergence 5 An implemented numerical solution algorithm isoften called a solver.