Page - 185 - in Programming for Computations – Python - A Gentle Introduction to Numerical Simulations with Python

6SolvingNonlinearAlgebraicEquations As a reader of this book you are probably well into mathematics and often “ac- cused” of being particularly good at “solving equations” (a typical comment at family dinners!). However, is it really true that you,with penandpaper, can solve many types of equations? Restricting our attention to algebraic equations in one unknownx, youcancertainlydo linearequations:axCbD0, andquadraticones: ax2CbxCcD0. Youmayalsoknowthat thereare formulas for the rootsof cu- bicandquarticequations too.Maybeyoucandothespecial trigonometricequation sinxCcosx D 1 aswell, but there it (probably) stops. Equations that are not re- ducible tooneof thementionedcannotbe solvedbygeneral analytical techniques, whichmeans thatmostalgebraicequationsarising inapplicationscannotbe treated withpenandpaper! Ifwe exchange the traditional idea of finding exact solutions to equationswith the ideaof ratherfindingapproximate solutions, awholenewworldofpossibilities opensup.Withsuchanapproach,wecan inprinciplesolveanyalgebraicequation. Letus start by introducingacommongeneric formforanyalgebraicequation: f.x/D0: Here,f.x/ is someprescribed formula involvingx. Forexample, theequation e x sinxD cosx 185©TheAuthor(s)2016 S.Linge,H.P.Langtangen,Programming for Computations –Python, Texts inComputational Science andEngineering15,DOI10.1007/978-3-319-32428-9_6