Page - 203 - in Programming for Computations – Python - A Gentle Introduction to Numerical Simulations with Python

6.6 SolvingMultipleNonlinearAlgebraicEquations 203 indicating that q D 2 is the rate forNewton’smethod. A similar computation using thesecantmethod,gives the rates secant: 1.26 0.93 1.05 1.01 1.04 1.05 1.08 1.13 1.20 1.30 1.43 1.54 1.60 1.62 1.62 Here it seems thatq 1:6 is the limit. Remark Ifwe in the bisectionmethod think of the length of the current interval containing the solution as the error en, then (6.5) works perfectly since enC1 D 1 2 en, i.e., q D 1 andC D 12, but if en is the true error jx xnj, it is easily seen from a sketch that this error can oscillate between the current interval length and apotentiallyverysmallvalueasweapproachtheexactsolution. Thecorresponding ratesqnfluctuatewidelyandareofno interest. 6.6 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, onlyNewton’smethod is suitable forextension to systemsofnonlinearequations. 6.6.1 AbstractNotation Supposewehavennonlinearequations,written in the followingabstract form: F0.x0;x1;:: :;xn/D0; (6.6) F1.x0;x1;:: :;xn/D0; (6.7) :::D ::: (6.8) Fn.x0;x1;:: :;xn/D0: (6.9) (6.10) Itwill beconvenient to introduceavectornotation F D .F0;:: :;F1/; x D .x0;:: :;xn/: ThesystemcannowbewrittenasF.x/D0. Asa specificexampleon thenotationabove, thesystem x2 Dy xcos. x/ (6.11) yxCe y Dx 1 (6.12)