Page - 197 - in Programming for Computations – Python - A Gentle Introduction to Numerical Simulations with Python

6.3 TheSecantMethod 197 6.3 TheSecantMethod When finding the derivative f 0.x/ in Newton’s method is problematic, or when function evaluations take too long; wemay adjust themethod slightly. Instead of using tangent lines to thegraphwemayusesecants1. Theapproachis referred toas thesecantmethod, andthe idea is illustratedgraphically inFig.6.2forourexample problemx2 9D0. The idea of the secant method is to think as in Newton’s method, but instead of using f 0.xn/, we approximate this derivative by a finite difference or the se- cant, i.e., the slopeof the straight line that goes through thepoints .xn;f.xn//and .xn 1;f.xn 1//on thegraph,givenby the twomost recent approximationsxn and xn 1. This slope reads f.xn/ f.xn 1/ xn xn 1 : (6.2) Inserting this expression forf 0.xn/ inNewton’smethodsimplygivesus the secant method: xnC1 Dxn f.xn/f.xn/ f.xn 1/ xn xn 1 ; or xnC1 Dxn f.xn/ xn xn 1 f.xn/ f.xn 1/ : (6.3) Fig.6.2 Illustratestheuseofsecants inthesecantmethodwhensolvingx2 9D0;x2 Œ0;1000 . From twochosen startingvalues,x0 D 1000 andx1 D 700 the crossingx2 of the corresponding secantwith thex axis iscomputed, followedbya similarcomputationofx3 fromx1 andx2 1https://en.wikipedia.org/wiki/Secant_line