数值分析课件session-9 Solution of Nonlinear Equations.ppt

Summaryoflastsession

-Chapter7.NumericalDifferentiationandintegration7.3.GaussianQuadrature7.4.RombergIntegration1

ChapterThree

SolutionofNonlinearEquations3.0Introduction3.1BisectionMethod3.2Newton’sMethod3.3SecantMethod3.4FixedPointsandFunctionalIteration3.5ComputingRootsofPolynomials3.6HomotopyandContinuationMethods2

3.0IntroductionOneofthebasicproblemsinscienceandengineeringistheroot-findingproblem.Thisprocessinvolvesfindingarootorsolution,ofanequationoftheform

f(x)=0

Theequationf(x)=0iscalledanalgebraicequationifitispurelyapolynomialinx.Forexample

x3-5x2-6x+3=0

isanalgebraicequation.

Itiscalledtranscendental(超越)equationiff(x)containstrigonometric(三角的),exponentialorlogarithmicfunctions.

Forexample

M=E–esinEandax2+log(x-3)+exsinx=0

aretranscendentalequations.3

3.0IntroductionTofindthesolutionofanequationf（x）=0，wefindthosevaluesofxforwhichf（x）=0issatisfied。Suchvaluesofxarecalledtherootsoff（x）=0.Examplesofnon-linearequationscanbefoundinmanyapplications.Generally,therootisobtainedinanumericalway,usingiterativemethod.Forexample:tofindthevalueofrootsquareof3isalternativelytofindthesolutionorrootofequation:x2–3=0(Continued)4

3.0IntroductionGivenainitialvaluex0(0),fromtheformula:xk=(xk-1+3/xk-1)/2wecangenerateasequence: