Page - 175 - in Programming for Computations – Python - A Gentle Introduction to Numerical Simulations with Python 3.6, Volume Second Edition

7SolvingNonlinearAlgebraicEquations Asareaderof thisbook,youmightbewell intomathematicsandoften“accused”of beingparticularlygoodatsolvingequations(a typicalcommentat familydinners!). How true is it, however, that you can solve many types of equations with pen and paperalone?Restrictingourattentiontoalgebraicequationsinoneunknownx,you cancertainlydo linearequations:ax+b=0,andquadraticones:ax2+bx+c= 0. You may also know that there are formulas for the roots of cubic and quartic equationstoo.Maybeyoucandothespecial trigonometricequationsinx+cosx= 1 as well, but there it (probably?) stops. Equations that are not reducible to one of those mentioned, cannot be solved by general analytical techniques, which means thatmostalgebraicequationsarising in applicationscannotbe treatedwith penand paper! If we exchange the traditional idea of finding exact solutions to equations with the idea of ratherfindingapproximate solutions,a wholenewworldof possibilities opensup.Withsuchanapproach,wecan in principlesolveanyalgebraicequation. Letus start by introducinga commongeneric formforanyalgebraicequation: f(x)=0 . ©The Author(s) 2020 S.Linge, H.P. Langtangen, Programming forComputations -Python, Texts in Computational Science and Engineering 15, https://doi.org/10.1007/978-3-030-16877-3_7 175