Seite - 195 - in Programming for Computations – Python - A Gentle Introduction to Numerical Simulations with Python

6.2 Newton’sMethod 195 except ZeroDivisionError: print "Error! - derivative zero for x = ", x sys.exit(1) # Abort with error f_value = f(x) 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 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" % (1 + 2*no_iterations) print "A solution is: %f" % (solution) else: print "Solution not found!" Handlingofthepotentialdivisionbyzeroisdonebyatry-exceptconstruction. Python tries to run thecode in thetryblock. Ifanythinggoeswronghere,ormore precisely, if Python raises an exception caused by a problem (such as division by zero,arrayindexoutofbounds,useofundefinedvariable,etc.), theexecutionjumps immediately to theexceptblock. Here, the programmer can take appropriate ac- tions. In thepresent case,we simply stop theprogram. (Professional programmers would avoid callingsys.exit inside a function. Instead, theywould raise a new exceptionwith an informativeerrormessage, and let the callingcodehaveanother try-exceptconstruction tostop theprogram.) The division by zerowill always be detected and the programwill be stopped. Themain purpose of our way of treating the division by zero is to give the user amore informativeerrormessageandstop theprograminagentlerway. Calling sys.exitwith an argument different from zero (here 1) signifies that the programstopped because of an error. It is a goodhabit to supply the value1, because tools in theoperatingsystemcan thenbeusedbyotherprograms todetect thatourprogramfailed. To prevent an infinite loop because of divergent iterations, we have introduced theintegervariableiteration_countertocountthenumberof iterations inNew- ton’smethod.Withiteration_counterwecaneasilyextendthecondition in the whilesuchthatnomoreiterations takeplacewhenthenumberof iterationsreaches 100.Wecouldeasily let this limitbeanargumentto thefunctionrather thanafixed constant. TheNewton function returns the approximate solutionand thenumberof itera- tions. The latter equals 1 if theconvergencecriterion jf.x/j< wasnot reached within themaximum number of iterations. In the calling code, we print out the