Page - 183 - in Programming for Computations – Python - A Gentle Introduction to Numerical Simulations with Python 3.6, Volume Second Edition

7.2 Newton’sMethod 183 Withx0 =1000,wegetx1 ≈500,whichis inaccordancewiththegraphinFig.7.1. Repeating theprocess,weget x2 =x1− f(x1) f ′(x1) ≈250 . Thegeneral scheme2 ofNewton’smethodmaybewrittenas xn+1 =xn− f(xn) f ′(xn) , n=0,1,2,.. . (7.1) The computation in (7.1) is repeated until f (xn) is close enough to zero. More precisely,we test if |f(xn)|< , with beinga smallnumber. We moved from 1000 to 250 in two iterations, so it is exciting to see how fast we can approach the solution x = 3. A computer program can automate the calculations. Our first try at implementing Newton’s method is in a function naive_Newton(foundinnaive_Newton.py): def naive_Newton(f, dfdx, x, eps): while abs(f(x)) > eps: x = x - (f(x))/dfdx(x) return x The argument x is the starting value, called x0 in our previous mathematical description. Tosolve theproblemx2 =9we alsoneed to implement def f(x): return x**2 - 9 def dfdx(x): return 2*x print(naive_Newton(f, dfdx, 1000, 0.001)) whichinnaive_Newton.pyis includedbyuseofanextrafunctionandatestblock. 2 The term scheme isoften used as a synonym for method or computational recipe.