Page - 186 - in Programming for Computations – Python - A Gentle Introduction to Numerical Simulations with Python

186 6 SolvingNonlinearAlgebraicEquations has f.x/D e x sinx cosx: Just move all terms to the left-hand side and then the formula to the left of the equality sign isf.x/. So, when dowe really need to solve algebraic equations beyond the simplest typeswecan treatwithpenandpaper?Thereare twomajorapplicationareas.One iswhenusing implicitnumericalmethodsforordinarydifferentialequations. These giverise tooneorasystemofalgebraicequations. Theothermajorapplicationtype isoptimization, i.e.,findingthemaximaorminimaofafunction.Thesemaximaand minimaarenormally foundbysolving thealgebraicequationF 0.x/D0 ifF.x/ is the function tobeoptimized.Differential equationsareverymuchused throughout science and engineering, and actuallymost engineeringproblemsareoptimization problems in the end, becauseonewants a design thatmaximizes performanceand minimizescost. Wefirstconsideronealgebraicequationinonevariable,withourusualemphasis on how to program the algorithms. Systems of nonlinear algebraic equationswith many variables arise from implicit methods for ordinary and partial differential equationsaswell as inmultivariateoptimization.Ourattentionwill be restricted to Newton’smethodfor suchsystemsofnonlinearalgebraicequations. Terminology When solving algebraic equationsf.x/ D 0, we often say that the solution x is a root of the equation. The solution process itself is thus often called root finding. 6.1 BruteForceMethods Therepresentationofamathematicalfunctionf.x/onacomputertakes twoforms. OneisaPythonfunctionreturningthefunctionvaluegiventheargument,while the other is a collection of points .x;f.x// along the functioncurve. The latter is the representationweuse for plotting, togetherwith an assumption of linear variation between the points. This representation is also very suited for equation solving and optimization: we simply go through all points and see if the function crosses thex axis, or for optimization, test for a localmaximumorminimumpoint. Be- cause there is a lot ofwork to examine a hugenumber of points, and also because the idea is extremely simple, such approaches are often referred to as brute force methods.However,wearenotembarrassedofexplaining themethods indetail and implementing them.