数值实验指导书.doc

数值实验指导书.doc救值实验描导吊数值实验一实验名称:非线性方程求根(SolutionofNon-linearEquation)实验目的:拿握二分法、不动点迭代、牛顿迭代法等常川的非线性方程迭代算法;加深对不同算法收敛速度、对初值的依赖性等的认识。基本要求:应川C语言或Fortran语言及Matlab编程,并上机调试通过;2学时。算法描述:计算/(%)=0的二分法(bisectionMethod):PURPOSE:TofindasolutiontoF(x)=0giventhecontinuousfunctionFontheinterval[A,B1,whereF(A)andF(B)haveoppositesignsINPUT: endpoints:A,B,tolerance:TOL>,maximumnumberofiterationsNOUTPUT:approximationsolutionPormessagethatthealgorithmfailsSet1=1FA=F(A);FB=F(B)WhileI<NdoSteps3-6Step3SetP=A+(B-A)/2;FP=F(P)・Step4IFFP=0or(B-A)/2<T0LTHENOUTPUT(P)(essfully)STOPENDIFStep5 Set1=1+1Step6 IFFAxFP〉0THENSetA=P;FA=FPELSESetB=P;FB=FPENDIFStep7OUTPUT(MethodfailedafterNiteration)=g(x)的不动点迭代(Fixed-PointIteration):PURPOSE:TofindasolutiontoX=g(x)givenaninitialapproximationpoINPUT: initialapproximationg:toleranceTOL;maximumnumberofiterationsNOUTPUT:approximationsolutionpormessageoffailsStep1Set1=1Step2WhileI<NdoSteps3-6Step3SetP=G(P())(computePi)Step4IFP-Pq\<TOLTHENOUTPOT(P)(essful.)STOPENDIFStep5Step6Set1=1+1SetP{)=P(Update£))Step7OUTPUT(MethodfailedafterNiteration)/(x)=0的牛顿法(MewtonMethod):PURPOSE:Tofindasolutiontof(x)=0givenaninitialapproximationp(}:INPUT: initialapproximation;toleranceTOL;:approximatesolutionpormessageoffailure・Step1Set/=1Step2Whilei<NdoSteps3-6Step3Set /(“))(*Computep.*)/Vo)Step4Ifp-p(>|<TOLthenOutput(p)StopStep5Setp()=pStep6Setz=z+lStep7Output(ThemethodfailedafterNiterations)(*essful*)实验步骤与注意事项:选择C、Fortran语言或Matlab编写以上三种算法的通用程序。用同一初值,用不同的迭代式求方程/(x)=?+^2-3x-3=,比较收敛速度。对同一迭代式,取不同的初值,观察算法对初值的敏感性。4•总结你的实验结果。5•进一步思考:如何比较迭代法收敛的快慢?何为收敛阶数?如何加速迭代序列的收敛速度?埃特金加速法的处理思想是什么?数值实验二实验名称:多项式插值(PolynomialInterpolation)实验目的:掌握多项式逼近的思想,熟悉Lagrange插值算法,分段低次插值(piecewisePolynomialApproximation),三次样条(CubicSpline)插值,体会它们不同的特征。基本要求:应用C语言或Fortran语言及Matlab编程,并上机调试通过;2学时。算法描述:二次样条插值(CubicSplineInterpolation)PURPOSE:TofindapiecewisecubicsplinefunctionS(x)=-^(x,.-x)3+矿(1耳$+(护-〒勺心+ g)・<x<xf.,/=i,2,—,/?-l).whereSff(x.)=M.(/=0,1,-andS\x)=m + xe[x.,x/+]],INPUT: i