Page - 195 - in Programming for Computations – Python - A Gentle Introduction to Numerical Simulations with Python 3.6, Volume Second Edition

7.6 SolvingMultipleNonlinearAlgebraicEquations 195 7.6 SolvingMultipleNonlinearAlgebraicEquations So far in this chapter, we have considered a single nonlinear algebraic equation. However, systems of such equations arise in a number of applications, foremost nonlinear ordinary and partial differential equations. Of the previous algorithms, only Newton’s method is suitable for extension to systems of nonlinear equa- tions. 7.6.1 AbstractNotation Supposewehavennonlinearequations,written in the followingabstract form: F0(x0,x1,.. .,xn)=0, (7.6) F1(x0,x1,.. .,xn)=0, (7.7) ...= ... (7.8) Fn(x0,x1,.. .,xn)=0 . (7.9) Itwill beconvenient to introducea vectornotation F = (F0,.. .,F1), x= (x0,.. .,xn). Thesystem cannowbe writtenasF(x)=0. Asa specificexampleon thenotationabove, thesystem x2 =y−xcos(πx) (7.10) yx+e−y =x−1 (7.11) canbewritten inourabstract formbyintroducingx0 =x andx1 =y.Then F0(x0,x1)=x2−y+xcos(πx)=0, F1(x0,x1)=yx+e−y−x−1 =0 . 7.6.2 TaylorExpansionsforMulti-VariableFunctions We follow the ideas of Newton’s method for one equation in one variable: approximate the nonlinearf by a linear functionand find the root of that function. Whennvariables are involved,we need to approximatea vector functionF(x)by somelinear function F˜ =Jx+c,whereJ is ann×nmatrixandc is somevector of lengthn. The technique for approximatingF by a linear function is to use the first two terms in a Taylor series expansion. Given the value ofF and its partial derivatives withrespect tox atsomepointxi,wecanapproximatethevalueatsomepointxi+1