数值分析上机题目

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1、数值分析上机编程题目1631110XXXX材料科学与工程学院一第2章插值法L2.7给定数据表215用Newton插值公式计算3次插值多项式N3(x)表215X11.502f(x)1.252.501.005.50a. Matlab代码如下,two.m,%第二章，P45,练习題2第七題clear();x=1,1.5,0,2;y(:J)=1.25,2.50,1.00,5.50*;%已知点集合x和ysymstw;w(1)=1；%计算基函数序列w和差商表y,以及函数序列的权数diag(y),计算的牛顿三次多项式表述为t的函数forj=2:length(x)fori=j:length(x)y(U)=(y(

2、lj-i)-y(i-iJ-i)/(x(i)-x(H+i)；i=i+1;endw(j)=prod(t-x(1:j-1);j=j+1;enddisp(三次牛顿插值多项式为J;disp(collect(w*diag(y);plot(x,y(J)J);holdon;fplot(collect(w*diag(y)/-0.5,2.5);legend(己知点集T三次牛顿插值多项式函数locationJnorthwestJfontsize,14);xlabelCx/fontsize,16);ylabel(y/fontsize16);holdoff;b. 计算结果如下：京令ft口two三次牛顿插值多项式为t*3

3、-tM2+t/4+1AX二第3章函数ii近与数据拟合L3.2设f(x)=|x|,xe-1Jo求在P=span1Jx2x4上的最佳平方逼近。a.Matlab代码fthree.m,%第三章函数逼近与数据拟合，P68练习題，第2题clear();symsx;%所使用的非找性基函数序列，用符号表示y=abs(x);%被逼近函数f=1rxA2,xA4;%求解法方程的系数矩阵a*Gn=b,其中a和b均为行向虽Gn=ones(length(f)Jength(f);fori=1:length(f)forj=1:length(f)Gn(iJ)=int(f(i)f(j),-1,1);j=j+1；endb(i)=i

4、nt(f(i)-y,-1J)；匸汁1;enda二b/Gn;%最佳平方逼近的系数行向凤dis逼近函数表达式)；disp(vpa(f*a);dis最佳函数逼近得平方误差)；disp(vpa(int(yA2r-1,1)-a*b);fplot(y,-1,1);holdon;fplot(aT-UJ)；legend(被逼近函数丁逼近函数LocationJnorth/OrientationJhorizontarjFontSize*,16,FontWeight7bold);xlabelCx/FontSizeO/FontWeight/bold);ylabeKy/FontSizeO/FontWeight/bold

5、);holdoff;b.运行结果如下：three逼近函數裘达式0.820312評/4+1.64062员2+0.1171875最佳函数逼近得平方误差0.00260416666666666666666666666666670.9-被逼近函数逼近函数0.80.70.60.403-0.8-0.6-0.4-0.20020.40.60.81三第4章数值积分与数值微分例49用Romberg求积法计算定积分警dx,使计算值的误差不超过0.5*10A-6%T(1,col)对应的计算的是多少步的值cokcoln关系col=col+1;%此时求得是第n+1次均分后的结果,使用的是第n次的结果,注意在矩阵%计算的第n

6、斜列是第n-1次均分的结果forj=1:colifj=1h=(t-a)/2A(col-2);%使用n+1之前的第n次结果s=0;fori=1:2A(col-2)s=s+f(a+(i-1/2)*h);endT(colJ)=1/2*T(col-1J)+h/2*s;%以上为第一列，也就是半分区间得到的复化梯形公式求积elseT(col+HJ)=(4A(j-1)T(col+2T*1H(col+1j.j-1)/(4八(H)-1);%以上为第2到col列,也就是第1到第col-1次外推公式endendenddis”从第2列开始是外推得到的结果，第1列是使用复化梯形公式得到的结果);disp(T);b.Ma

7、tlab函数文件,f.mfunctiony=f(x)ifx=0y二1;elsey=sin(x)/x;c. end运行结果如下:four从第2列开始是外推得到的结果，第1列是使用复化梯形公式得到的结果0.9207354924039480.9397932848061770.9461458822735870.9460830040636740.9460830703872230.9460869339517940.9460830693509170.9445135216653900.9460833108884720.945690863582701第5章线性方程姐的直接解法L5.6用Gauss-Jordan消

8、去法解方程组a.Matlab代码,flve.m%矩阵的三角分解P142练习五第6題clear();formatlong;a=2340;1192;12-61;forj=1:length(a)-1c=max(abs(a(j:length(a)-1,j);x,y=find(abs(aG：length(a)-1J)=c);ifx=1d=a(j,：)；a(j,:)=a(x+j-1/);a(x+j-1/)=d;endl=a(:,j)/a0J);%l=l(j+1:end);fori=1:length(a)-1ifi=ja(i,:)=a(i,:)-a(j/)*l(i);end匸汁1;endenddis”经过转

9、化后的系数矩阵J;disp(a);%进行转化后的系数矩阵%回代过程X=zeros(1Jength(a)-1);fori=length(a)-1:-1:1X(i)=(a(ijength(a)-X*a(i,1:length(a)-1)*)/a(ij);enddispC求解的结果工disp(X);b.运行结果如下：命令仙口five经过转化后的系数矩阵1.0e+02*1.500000000000000-0.2300000000000000.0300000000000000.0200000000000000000.005000000000000000-0.010000000000000求解的结果75-4

10、6-3五第6章线性方程组的迭代解法例6.2用J法和GS法分别求解方程组。/1O31/xi/14I2-103112I=I-SI1310/3/14/a.Matlab代码,slx.m%第六章,P149例6.2线性方程组的迭代解法formatlong;clear();A=10-2-2;-2-23;b=10.51;D=diag(diag(A);L=zeros(length(A);U=zeros(length(A);forcol=1:length(A)-1forrow=col+1:length(A)L(rowrcol)=-A(row/col);row=row+1;endcol=col+1;endU=-A-

11、L+D;%Jacobi迭代法的使用!%x(J)=Wforcol=2:3forrow=1:length(A)ifrow=1x(row,col)=1/D(rowrrow)*(b(row)+U(row,row+1:length(A)*x(row+1:length(A),col-1);elseifrow=length(A)x(row,col)=1/D(row,row)*(b(row)+L(rowJ:row-1)*x(1:row-1,001-1);elsex(rowrcol)=1/D(row,row)*(b(row)+L(row,1:row-1)*x(1:row-1,col-1)+U(row/row+1

12、:length(A)*x(row+1:length(A),coH);endendendwhilenorm(x(:icol-x(:col-1)jnf)10A-5col=col+1;forrow=1:length(A)ifrow=1x(row,col)=1/D(rowrrow)*(b(row)+U(row,row+1:length(A)*x(row+1:length(A),col-1);elseifrow=length(A)x(row,col)=1/D(row,row)*(b(row)+L(rowJ:row-1)*x(1:row-1,001-1);elsex(rowrcol)=1/D(row,ro

13、w)*(b(row)+L(row,1:row-1)*x(1:row-1,col-1)+U(row/row+1:length(A)*x(row+1:length(A),coH);endendenddispCJocobi迭代使得结果向虽的无穷范数不超过第一行为初始值0,0,0)；dispX);%Gauss-Seidel迭代法的使用!%X(:J)=0,0,0;forcol=2:3forrow=1:length(A)ifrow=1X(row,col)=1/D(row/row)*(b(row)+U(row,row+1:length(A)*X(row+1:length(A)/col-1);elseifro

14、w=length(A)X(row,col)=1/D(row/row)*(b(row)+L(row,1:row-1)*X(1:row-1,col);elseX(row/col)=1/D(rowfrow)*(b(row)+L(rowJ:row-1)*X(1:row-1rcol)+U(row/row+1:length(A)*X(row+1:length(A)/co1-1)；endendendwhilenorm(X(:rcol)-X(:/col-1)Jnf)10A-5col=col+1;forrow=1:length(A)ifrow=1X(row,col)=1/D(row/row)*(b(row)+U

15、(row/row+1:length(A)*X(row+1:length(A)/coM);elseifrow=length(A)X(row,col)=1/D(row/row)*(b(row)+L(row,1:row-1)*X(1:row-1,col);elseX(row/col)=1/D(rowfrow)*(b(row)+L(rowf1:row-1)*X(1:row-1rcol)+U(row/row+1:length(A)*X(row+1:length(A),coendendenddispCGauss-Seidel迭代使得结果向且的无穷范数不超过10A-5,第一行为初始值0Q0);disp(X)

16、;b.运行结果如下：2SIXJocobL迭代使得结杲向里的无穷范数不超过，第一行为初始值山0,000.1000000000000000.1766666666666670.2006666666666670.2172888888888890.2240044444444440.2276856296296300.2294037333333330.2302656572839510.2306861680987650.2308928383736630.2309944577132510.2310442975880600.2310688006160860.2310808268390390.23108673535

17、97240.23108963667263600.0500000000000000.1033333333333330.1253333333333330.1362444444444440.141S355555555560.1444601481481480.1457929777777780.1464342775308640.1467527877530860.1469083740312760.1469849591282300.1470225444238570.1470410053832690.1470500730044100.1470545257472290.14705671283354000.333

18、3333333333330.4000000000000000.4611111111111110.4837777777777780.4965925925925930.5025585185185190.5055353086419750.5069965629629630.5077114041152260.5080639145349790.5082365288120710.5083214586565710.5083631288119250.5083836037942080.5083936576159530.508398595618060Seidel迭代使得结果向里的无穷范数不超过第一行为初始值0,0G

19、auss-00.1000000000000000.1966666666666670.2233333333333330.2293111111111110.2306844444444440.2309989629629630.2310710222222220.2310875308641980.23109131298765400.0700000000000000.1306666666666670.1432666666666670.1461911111111110.1468600000000000.1470132740740740.1470483881481480.1470564327901230.14

20、705827581234600.4133333333333330.4860000000000000.5032888888888890.5072311111111110.5081348148148150.5083418370370370.5083892661728400.5084001321481480.508402621537449六第7章非线性方程的数值解法例7.9用Steffensen迭代法求方程f(x)=x3-x-1=0的实根。a.Matlab代码,seven.m%第7章.非线性方程的迭代解法，P180,例7.9,Steffensen迭代法clear();f=(x)xA3-1;%方程f(

21、x)=xA3-x-1=0的迭代公式可以选择为x=x3-1,虽然是发散的F=(x)x-(f(x)-x)A2/(f(f(x)-2*f(x)+x);X九5;x(2)=F(x(1);i=2；whileabs(x(i)-x(i-1)10A-7i=i+1;x(i)=F(x(i-1);enddisp(Steffensen迭代法，在误差小于10A-7时的迭代结果J;disp(x);b.运行结果如下：sevenSteffenseng代法，在淒差小于时的送代结果1.5000000000000001.4162929745889391.3556504414766441.3289487772840111.3248044

22、890410441.3247179939688151.324717957244753Al七第9章矩阵特征值问题的数值计算例93用寡法求矩阵(110.5A二110.250.50.252的主特征值和主特征向虽。a.Matlab代码,nlne.m%第九章，P212,例9.3,使用幕法求主特征值和主特征向虽clear();formatlong;A=110.5;110.25;0.50.252;r0=2.5365258;%主特征值的准确值(保留八位有效数字)v(:J)=111*；%v向虽中包含主特征值的序列，其最大值收敛到主特征值u(:J)=111：%u为主特征向虽的序列1=1;whileabs(max(

23、v(:J)-r0)10A-5v(:j+1)=A*u(:ri);u(:，汁1)=v(:J+1)/max(v(:J+1);i=i+1;enddis特征值收敛序列J;disp(transpose(max(v);disp(特征向虽收敛序列)；disp(u);b.运行结果如下：nine特征值收皴序列1.0000000000000002.7500000000000002.6590909090909092.6047008547008552.5752666119770302.5587919088954372.5494055538697352.5440034899273562.5408764046190672.

24、5390601567190542.5380032047956452.5373874231842652.5370284314858842.5368190641496382.5366969317923142.5366256777836372.5365841038638952.5365598460331562.5365456915433272.5365374322468362.536532612816242特征向重收敘序列1.0000000006000000.9090909090909090.8376068376068380.7990155865463500.7774149876562870.765

25、1081907518500.7580253450823990.7539253079726680.7515439582311710.7501581495045290.7493507754775210.7488800881951660.7486055789434050.7484454466962930.7483520229323110.7482975138297650.7482657084914290.7482471500167330.7482363209478800.7482300020143260.7482263147972921.6000000000000000.81818181818181

26、80.7435897435897440.7030352748154230.6803376602691730.6674058339752420.6599632695446230.6556550025309330.6531527104138730.6516965201735220.6508481417820170.6503535495532020.6500650987117420.6498968337766710.6497986652699270.6497413877960500.6497079671497630.6496884661398410.6496770870915830.64967044

27、72363180.6496655727554061.0060000000000001.0000000000000001.0000000000000001.0000000000000001.0000000000000001.0000000000000001.0060000000000001.0000000000000001.0000000000000001.0000000000000001.0000000000000001.0000000000000001.0060000000000001.0000000000000001.ooooooooooopooo1.0000000000000001.00

28、00000000000001.0000000000000001.0000000000000001.0000000000000001.000000000000000A.第10章常微分方程的数值解法L10.3用改进的Euler方法解初值问题(y=x2+x-yky(o)=oa.Matlab代码，ten.m%第10章，练习10第3题，使用Euler方法解初值问題clear();f=(szt)sA2+s弋h=0.1;x=0:h:0.5;Yn(1)=0;%Yx是显式Euler方法求得的函数值作为改进Euler方法的初值Yt(1)=Yn(1);%改进的Euler方法的初值fori=2:length(x)Yn

29、(i)=Yt(i-1)+h*f(x(i-1)，丫t();Yt(i)=Yt(i-1)+h/2(f(x(i-1),Yt(i-1)+f(x(i)”n(i);i=i+1;enddisp(改进的Euler方法计算得到的数值)；disp(transpose(丫t);disp(改进的Euler方法计算得到的y(0.5)；disp(Yt(end);ft=(t)tA2-t+1-exp(-t);disp(y=xA2-x+1-exp(-x)在x=0.5时的准确值);disp(ft(0.5);b.运行结果如下：ten改进的Euler方法计算得到的数值00.0055000000000000-0219275000000000-0501443875000000.0909306706875000.144992256972188改进的EuXr方法计算得到的ygd)0-144992256972188y=:2-：+l-e?:p(-：)在m=05时的准确值0.143469340287367