數(shù)值分析實(shí)驗(yàn)報(bào)告1

數(shù)學(xué)與信息工程學(xué)院實(shí)驗(yàn)報(bào)告課程名稱(chēng)：計(jì)算方法實(shí)驗(yàn)室：5206實(shí)驗(yàn)臺(tái)號(hào)：班級(jí)：姓名：實(shí)驗(yàn)日期：2014年2月26日實(shí)驗(yàn)名稱(chēng)插值實(shí)驗(yàn)?zāi)康暮鸵?.插值函數(shù)及其在定點(diǎn)的函數(shù)值計(jì)算2.完成實(shí)驗(yàn)內(nèi)容3..準(zhǔn)備實(shí)驗(yàn)報(bào)告實(shí)驗(yàn)內(nèi)容和步驟：實(shí)驗(yàn)內(nèi)容：1.熟悉與插值有關(guān)的Matlab函數(shù)2.設(shè)計(jì)插值程序3.數(shù)據(jù)實(shí)驗(yàn)LagrangeInterpolation實(shí)驗(yàn)題目：(1)、FindtheLagrangeinterpolatingpolynomialrespectivelyforthefollowingdataset,andgiveapproximatevaluetof(0).D={(0,5),(1,-2),(2,3),(3,4),(4,-1),(5,7),(6,5),(7,2)}(2)、WhataretheLagrangefromoftheinterpolatingpolynomialforthedatainthefollowingtable,andgiveapproximatevaluetof(7)x5-7-60y1-23-54-954求Lagrange多項(xiàng)式的程序：functionyy=lagrintp(x,y,xx)n=length(x);m=length(y);ifm~=nerror('向量x與y的長(zhǎng)度不一致')endS=0;fork=1:np=1;forj=1:nifj~=kp=conv(p,poly(x(j)))/(x(k)-x(j));endendS=S+p*y(k);endyy=poly2sym(S)y0=subs(yy,'x',xx)3、實(shí)驗(yàn)結(jié)果：（1）、>>lagrintp(0:7,[5,-2,3,4,-1,7,5,2],0)yy=23/560*x^7-691/720*x^6+2089/240*x^5-5537/144*x^4+1689/20*x^3-7343/90*x^2+4369/210*x+5y0=5ans=23/560*x^7-691/720*x^6+2089/240*x^5-5537/144*x^4+1689/20*x^3-7343/90*x^2+4369/210*x+5（2）、>>lagrintp([5,-7,-6,0],[1,-23,-54,-954],7)yy=4*x^3+35*x^2-84*x-954y0=1545ans=4*x^3+35*x^2-84*x-954二、Newtoninterpolating1、實(shí)驗(yàn)題目：(1)、Findthecubicinterpolatingpolynomialrespectivelyforthefollowingdataset,andgiveapproximatevaluetof(0).D={(0,5),(1,-2),(2,3),(3,4),(4,-1),(5,7),(6,5),(7,2)}(2)、Findthequadraticinterpolatingpolynomialforthedatainthefollowingtable,andgiveapproximatevaluetof(7)x5-7-60y1-23-54-954求牛頓多項(xiàng)式的程序建立差商表：functionD=divdiff(x,y)n=length(y);iflength(x)~=nfprintf('出錯(cuò)信息：節(jié)點(diǎn)向量與函數(shù)值向量的維數(shù)不相等.');endD=zeros(n,n);D(:,1)=y(:);forj=2:n;fori=j:nD(i,j)=(D(i,j-1)-D(i-1,j-1))/(x(i)-x(i-j+1));end求m次牛頓差值多項(xiàng)式function[D,yy]=newton(x,y,xx,m)n=length(y);iflength(x)~=nerror('xandyarenotcompatible');endifnargin<4m=n-1;endD=divdiff(x,y);symsts=D(1,1);w=1;fori=1:mw=w*(t-x(i));s=D(i+1,i+1)*w+s;endfprintf('所求的%d次牛頓插值多項(xiàng)式為:\nN(t)',m);display(collect(s,'t'))ifnargin<3return;endyy=eval(subs(s,'xx','t'));fprintf('\n且N(%d)=%10.4d\n',xx,yy);end3、實(shí)驗(yàn)結(jié)果：（1）、>>newton(0:7,[5,-2,3,4,-1,7,5,2],0,3)所求的3次牛頓插值多項(xiàng)式為:N(t)=5-8/3*t^3+14*t^2-55/3*t且N(0)=0005ans=Columns1through75.0000000000-2.0000-7.0000000003.00005.00006.000000004.00001.0000-2.0000-2.6667000-1.0000-5.0000-3.0000-0.33330.5833007.00008.00006.50003.16670.87500.058305.0000-2.0000-5.0000-3.8333-1.7500-0.5250-0.09722.0000-3.0000-0.50001.50001.33330.61670.1903Column800000000.0411（2）、>>newton([5,-7,-6,0],[1,-23,-54,-954],7,2)所求的2次牛頓插值多項(xiàng)式為:N(t)=-114+3*t^2+8*t且N(7)=0089ans=1000-23200-54-3130-954-150-174三、Cubicsplineinterpolating1、實(shí)驗(yàn)題目：(1)、Givetablex-1.5012y0.125-119SolveacubicsplinefunctionS(x)ontheinterval[-1.5,2]whichsatisfiesS’(-1.5)=0.75,S’(2)=14,andgiveapproximatevaluetoS(1.5)(2)、Givetablex0.250.320.400.460.55y0.52000.54780.62500.68080.7288SolveacubicsplinefunctionS(x),whichsatisfiesS’(0.125)=1,S’(0.55)=0.68,andgiveapproximatevaluetoS(0.45)求三次樣條插值多項(xiàng)式的程序function[]=spline3(X,Y,dY,x0,m)N=size(X,2);s0=dY(1);sN=dY(2);interval=0.025;disp('x0為插值點(diǎn)')x0h=zeros(1,N-1);fori=1:N-1h(1,i)=X(i+1)-X(i);endd(1,1)=6*((Y(1,2)-Y(1,1))/h(1,1)-s0)/h(1,1);d(N,1)=6*(sN-(Y(1,N)-Y(1,N-1))/h(1,N-1))/h(1,N-1);fori=2:N-1d(i,1)=6*((Y(1,i+1)-Y(1,i))/h(1,i)-(Y(1,i)-Y(1,i-1))/h(1,i-1))/(h(1,i)+h(1,i-1));endmu=zeros(1,N-1);md=zeros(1,N-1);md(1,N-1)=1;mu(1,1)=1;fori=1:N-2u=h(1,i+1)/(h(1,i)+h(1,i+1));mu(1,i+1)=u;md(1,i)=1-u;endp(1,1)=2;q(1,1)=mu(1,1)/2;fori=2:N-1p(1,i)=2-md(1,i-1)*q(1,i-1);q(1,i)=mu(1,i)/p(1,i);endp(1,N)=2-md(1,N-1)*q(1,N-1);y=zeros(1,N);y(1,1)=d(1)/2;fori=2:Ny(1,i)=(d(i)-md(1,i-1)*y(1,i-1))/p(1,i);endx=zeros(1,N);x(1,N)=y(1,N);fori=N-1:-1:1x(1,i)=y(1,i)-q(1,i)*x(1,i+1);endfprintf('M為三對(duì)角方程的解\n');M=x;fprintf('\n');symst;digits(m);fori=1:N-1pp(i)=M(i)*(X(i+1)-t)^3/(6*h(i))+M(i+1)*(t-X(i))^3/(6*h(i))+(Y(i)-M(i)*h(i)^2/6)*(X(i+1)-t)/h(i)+(Y(i+1)-M(i+1)*h(i)^2/6)*(t-X(i))/h(i);pp(i)=simplify(pp(i));coeff=sym2poly(pp(i));iflength(coeff)~=4tt=coeff(1:3);coeff(1:4)=0;coeff(2:4)=tt;endifx0>X(i)&x0<X(i+1)L=i;y0=coeff(1)*x0^3+coeff(2)*x0^2+coeff(3)*x0+coeff(4);endval=X(i):interval:X(i+1);fork=1:length(val)fval(k)=coeff(1)*val(k)^3+coeff(2)*val(k)^2+coeff(3)*val(k)+coeff(4);endifmod(i,2)==1plot(val,fval,'r+')elseplot(val,fval,'b.')endholdonclearvalfvalans=sym(coeff,'d');ans=poly2sym(ans,'t');fprintf('在區(qū)間[%f,%f]內(nèi)\n',X(i),X(i+1));fprintf('三次樣條函數(shù)S(%d)=',i);pretty(ans)endfprintf('x0所在區(qū)間為[%f,%f]\n',X(L),X(L+1));fprintf('函數(shù)在插值點(diǎn)x0=%f的值為\n',x0);y03、實(shí)驗(yàn)結(jié)果：（1）、>>X=[-1.5012];>>Y=[0.125-119];>>x0=1.5;>>m=5;dY=[0.75,14];>>spline3(X,Y,dY,x0,m)x0為插值點(diǎn) x0=1.5000M為三對(duì)角方程的解在區(qū)間[-1.500000,0.000000]內(nèi)三次樣條函數(shù)S(1)=321.t+2.t-1.在區(qū)間[0.000000,1.000000]內(nèi)三次樣條函數(shù)S(2)=22.t-1.在區(qū)間[1.000000,2.000000]內(nèi)三次樣條函數(shù)S(3)=322.t-4.t+6.t-3.x0所在區(qū)間為[1.000000,2.000000]函數(shù)在插值點(diǎn)x0=1.500000的值為y0=3.7500（2）、>>X=[0.250.320.400.460.55];>>Y=[0.52000.54780.62500.68080.7288];>>x0=0.45;>>m=6;dY=[1,0.68];>>spline3(X,Y,dY,x0,m)x0為插值點(diǎn)x0=0.4500M為三對(duì)角方程的解在區(qū)間[0.250000,0.320000]內(nèi)三次樣條函數(shù)S(1)=32136.809t-120.795t+35.7460t-3.00444在區(qū)間[0.320000,0.400000]內(nèi)三次樣條函數(shù)S(2)=32-53.5821t+61.9798t-22.7420t