Page - 202 - in Programming for Computations – Python - A Gentle Introduction to Numerical Simulations with Python

202 6 SolvingNonlinearAlgebraicEquations 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 = ", 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 if return_x_list: return x_list, iteration_counter else: return x, iteration_counter The function is found in thefilenonlinear_solvers.py. Wecannowmakeacall x, iter = Newton(f, dfdx, x=1000, eps=1e-6, return_x_list=True) andget a listx returned. With knowledgeof the exact solutionx off.x/ D 0 wecancompute all the errorsen andall the associatedqn valueswith the compact function def rate(x, x_exact): e = [abs(x_ - x_exact) for x_ in x] q = [log(e[n+1]/e[n])/log(e[n]/e[n-1]) for n in range(1, len(e)-1, 1)] return q The errormodel (6.5)workswell forNewton’smethod and the secantmethod. For thebisectionmethod,however, itworkswell in thebeginning,butnotwhen the solution is approached. Wecancompute the ratesqn andprint themnicely, def print_rates(method, x, x_exact): q = [’%.2f’ % q_ for q_ in rate(x, x_exact)] print method + ’:’ for q_ in q: print q_, print The result forprint_rates(’Newton’, x, 3) is Newton: 1.01 1.02 1.03 1.07 1.14 1.27 1.51 1.80 1.97 2.00