Page - 194 - in Programming for Computations – Python - A Gentle Introduction to Numerical Simulations with Python

194 6 SolvingNonlinearAlgebraicEquations -1.05895313436 0.989404207298 -0.784566773086 0.36399816111 -0.0330146961372 2.3995252668e-05 Adjustingx0 slightly to 1.09 gives division by zero! The approximations com- putedbyNewton’smethodbecome -1.09331618202 1.10490354324 -1.14615550788 1.30303261823 -2.06492300238 13.4731428006 -1.26055913647e+11 Thedivisionbyzeroiscausedbyx7 D 1:26055913647 1011,becausetanh.x7/ is 1.0 tomachine precision, and thenf 0.x/ D 1 tanh.x/2 becomes zero in the denominator inNewton’smethod. The underlying problem, leading to the division by zero in the above example, is that Newton’s method diverges: the approximations move further and further away from x D 0. If it had not been for the division by zero, the condition in thewhile loopwould always be true and the loopwould run forever. Divergence ofNewton’smethodoccasionally happens, and the remedy is to abort themethod whenamaximumnumberof iterations is reached. Another disadvantage of the naive_Newton function is that it calls the f.x/ function twice asmany times asnecessary. This extrawork is of noconcernwhen f.x/ isfast toevaluate,butin large-scaleindustrialsoftware,onecall tof.x/might takehoursordays, and then removingunnecessarycalls is important. Thesolution inour function is to store thecallf(x) inavariable (f_value)and reuse thevalue insteadofmakinganewcallf(x). To summarize,wewant towrite an improved function for implementingNew- ton’smethodwherewe avoiddivisionbyzero allowamaximumnumberof iterations avoid theextraevaluationoff.x/ Amore robustandefficientversionof the function, inserted inacompleteprogram Newtons_method.pyfor solvingx2 9D0, is listedbelow. def Newton(f, dfdx, x, eps): f_value = f(x) iteration_counter = 0 while abs(f_value) > eps and iteration_counter < 100: try: x = x - float(f_value)/dfdx(x)