Page - 182 - in Programming for Computations – Python - A Gentle Introduction to Numerical Simulations with Python 3.6, Volume Second Edition

182 7 SolvingNonlinearAlgebraicEquations Fig. 7.1 Illustrates the idea of Newton’s method with f(x) = x2 − 9, repeatedly solving for crossing of tangent lineswith thex axis How do we compute the tangent of a functionf(x) at a pointx0? The tangent function,herecalled f˜(x), is linearandhas twoproperties: 1. theslopeequals tof ′(x0) 2. the tangent touches thef(x)curveatx0 So, if we write the tangent function as f˜(x)= ax+b, we must require f˜ ′(x0)= f ′(x0)and f˜(x0)=f(x0), resulting in f˜(x)=f(x0)+f ′(x0)(x−x0). Thekeystep in Newton’smethod is tofindwhere the tangentcrosses thex axis, whichmeanssolving f˜(x)=0: f˜(x)=0 ⇒ x=x0− f(x0) f ′(x0) . This isournewcandidatepoint,whichwe callx1: x1 =x0− f(x0) f ′(x0) .