Page - 184 - in Programming for Computations – Python - A Gentle Introduction to Numerical Simulations with Python 3.6, Volume Second Edition

184 7 SolvingNonlinearAlgebraicEquations WhyNotUsean Arrayfor thexApproximations? Newton’smethod isnormally formulatedwithan iteration indexn, xn+1 =xn− f(xn) f ′(xn) . Seeingsuchan index,manywould implement this as x[n+1] = x[n] - f(x[n])/dfdx(x[n]) Such an array is fine, but requires storage of all the approximations. In large industrial applications, where Newton’s method solves millions of equations at once, one cannot afford to store all the intermediate approximations in memory, so then it is important to understand that the algorithm in Newton’s method has no more need forxn whenxn+1 is computed. Therefore, we can workwithonevariablexandoverwrite thepreviousvalue: x = x - f(x)/dfdx(x) Runningnaive_Newton(f, dfdx, 1000, eps=0.001)resultsintheapprox- imate solution 3.000027639.A smaller value ofepswill produce a more accurate solution. Unfortunately, the plain naive_Newton function does not return how many iterations it used, nor does it print out all the approximationsx0,x1,x2,.. ., whichwould indeedbeanice feature. Ifwe insert suchaprintout(print(x) in the while loop), a rerunresults in 500.0045 250.011249919 125.02362415 62.5478052723 31.3458476066 15.816483488 8.1927550496 4.64564330569 3.2914711388 3.01290538807 3.00002763928 We clearly see that the iterations approach the solution quickly. This speed of the search for the solution is the primary strength of Newton’s method compared to othermethods. 7.2.2 MakingaMoreEfficientandRobustImplementation Thenaive_Newton function works fine for the example we are considering here. However, for more general use, there are some pitfalls that should be fixed in an improvedversionof thecode.An examplemay illustratewhat theproblemis. Let us use naive_Newton to solve tanh(x) = 0, which has solution x = 0 (interactively,youmaydefinef(x)= tanh(x)andf ′(x)=1−tanh2(x)asPython