Page - 186 - in Programming for Computations – Python - A Gentle Introduction to Numerical Simulations with Python 3.6, Volume Second Edition

186 7 SolvingNonlinearAlgebraicEquations iteration_counter = 0 while abs(f_value) > eps and iteration_counter < 100: try: x = x - 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 = iteration_counter + 1 # Here, either a solution is found, or too many iterations if abs(f_value) > eps: iteration_counter = -1 return x, iteration_counter if __name__ == ’__main__’: def f(x): return x**2 - 9 def dfdx(x): return 2*x solution, no_iterations = Newton(f, dfdx, x=1000, eps=1.0e-6) if no_iterations > 0: # Solution found print(’Number of function calls: {:d}’.format(1+2*no_iterations)) print(’A solution is: {:f}’.format(solution)) else: print(’Solution not found!’) Handling of the potential division by zero is done by a try-except construc- tion.3 The division by zero will always be detected and the program will be stopped. The main purpose of our way of treating the division by zero is to give the user a more informativeerrormessageandstop theprogramina gentlerway. Calling sys.exitwith an argument different from zero (here 1) signifies that the program stopped because of an error. It is a good habit to supply the value 1, because tools in the operatingsystem can then be used by other programsto detect thatourprogramfailed. To prevent an infinite loop because of divergent iterations, we have introduced the integer variable iteration_counter to count the number of iterations in Newton’smethod.Withiteration_counterwecaneasilyextendtheconditionin thewhile loopsuchthatnomoreiterationstakeplacewhenthenumberofiterations reaches100.Wecouldeasily let this limitbeanargumentto thefunctionrather than afixedconstant. The Newton function returns the approximate solution and the number of iterations. The latter equals −1 if the convergence criterion |f(x)| < was not reachedwithin themaximumnumberof iterations. In thecallingcode,we printout 3 Professional programmers would avoid callingsys.exit inside a function. Instead, they would raise a new exception with an informative error message, and let the calling code have another try-exceptconstruction to stop the program.