数值分析上机报告第一章、题目精确值为1（3-—-—）。22NN+11111）编制按从大到小的顺序SN二-+-+……+〒，计算SN的通用程序。N22—132—1N2-1N2）编制按从小到大的顺序SN二-+-+……+-^，计算SN的通用程NN2—1（N—1）2—122—1N序。3）按两种顺序分别计算S102,S104,S106，并指出有效位数。（编制程序时用单精度）4）通过本次上机题，你明白了什么？二、通用程序clearN=input('PleaseInputanN(N>1):');AccurateValue=single((0-1/(N+1)-1/N+3/2)/2);Sn1=single(0);fora=2:N;Sn1=Sn1+1/(aA2-1);endSn2=single(0);fora=2:N;Sn2=Sn2+1/((N-a+2)八2-1);endfprintf('ThevalueofSnusingdifferentalgorithms(N=%d)\n,N);disp('')fprintf('Accuratefprintf('Caculatefprintf('Caculatedisp('Calculationfromlargetosmallfromsmalltolarge%f\,AccurateValue);%f\n,Sn1);%f\n,Sn2);三、求解结果PleaseInputanN(N>1):10^2ThevalueofSnusingdifferentalgorithms(N=100)AccurateCalculation0.740049Caculatefromlargetosmall0.740049Caculatefromsmalltolarge0.740050PleaseInputanN(N>1):10A4ThevalueofSnusingdifferentalgorithms(N=10000)AccurateCalculation0.749900Caculatefromlargetosmall0.749852Caculatefromsmalltolarge0.749900PleaseInputanN(N>1):10A6ThevalueofSnusingdifferentalgorithms(N=1000000)AccurateCalculation0.749999Caculatefromlargetosmall0.749852Caculatefromsmalltolarge0.749999四、结果分析有效位数、\n顺序100100001000000从大到小633从小到大566可以得出，算法对误差的传播又一定的影响，在计算时选一种好的算法可以使结果更为精确。从以上的结果可以看到从大到小的顺序导致大数吃小数的现象，容易产生较大的误差，求和运算从小数到大数算所得到的结果才比较准确。第二章一、题目给定初值X0及容许误差£，编制牛顿法解方程f(x)=O的通用程序。给定方程f(x)—X33—x—0，易知其有三个根片*——3,=0,七*=、：3由牛顿方法的局部收敛性可知存在5〉0,当xoe(-6,+6)时，Newton迭代序列收敛于根x2*。试确定尽可能大的5。试取若干初始值，观察当x0e(s,-1),(-1,-5),(-5,+5),(5,1),(1,+8)时Newton序列的收敛性以及收敛于哪一个根。(3)通过本上机题，你明白了什么？二、通用程序文件fx.m%%定义函数f(x)functionFx=fx(x)Fx=xA3/3-x;文件dfx.m%%定义导函数df(x)functionFx=dfx(x)Fx=xA2-1;文件Newton.m%%Newton法求方程的根％%clear%%ef=10—6;%给定容许误差10—6k=0;x0=input('PleaseinputinitialvalueXo:');disp('kX]);fprintf('0%f\,X0);flag=1;whileflag==1&&k<=10入3x1=x0-fx(x0)/dfx(x0);ifabs(x1-x0)
=10A-6flag=0;endendfprintf('Themaximundeltais%f\n',delta);1.运行search.m文件结果为：Themaximumdeltais0.774597即得最大的5为0.774597,Newton迭代序列收敛于根x*=0的最大区间为(-0.774597,20.774597)。2.运行Newton.m文件在区间(s,-1),(-1,),(,+§),(§,1),(1,+8)上各输入若干个数，计算结果如下：区间(—8,-1)上取-1000,-100，-50，-30，-10，-8，-7，-5，-3，-1.5PleaseinputinitialvalueXo:-100000-100.00000013-1.732051kXk1-66.673334PleaseinputinitialvalueXo:-300-10000.0000002-44.458891kXk1-6666.6667333-29.6542630-30.0000002-4444.4445894-19.7920161-20.0222473-2962.9632095-13.2284472-13.3815444-1975.3090316-8.8696513-8.9711295-1316.8730257-5.9892314-6.0560006-877.9158568-4.1073245-4.1505037-585.2779979-2.9107556-2.9375248-390.18647010-2.2001897-2.2150469-260.12602211-1.8486878-1.85471410-173.41991112-1.7422359-1.74323611-115.61711813-1.73213910-1.73215812-77.08384514-1.73205111-1.73205113-51.39788015-1.73205112-1.73205114-34.278229PleaseinputinitialvalueXo:-50PleaseinputinitialvalueXo:-1015-22.871618kXkkXk16-15.2769490-50.0000000-10.00000017-10.2284591-33.3466721-6.73400718-6.8847802-22.2511252-4.59057019-4.6887723-14.8641053-3.21284020-3.2748074-9.9544584-2.37165321-2.4077145-6.7039605-1.92298122-1.9397506-4.5710136-1.75717523-1.7612597-3.2005207-1.73258024-1.7327628-2.3645158-1.73205125-1.7320519-1.9197039-1.73205126-1.73205110-1.756405PleaseinputinitialvalueXo:-10011-1.732548kXk12-1.732051PleaseinputinitialvalueXo:-8PleaseinputinitialvalueXo:-3kXkkXk0-8.0000000-3.0000001-5.4179891-2.2500002-3.7393792-1.8692313-2.6849343-1.7458104-2.0782464-1.7322125-1.8029285-1.7320516-1.7360236-1.7320517-1.732064PleaseinputinitialvalueXo:-1.58-1.732051kXk9-1.7320510-1.500000PleaseinputinitialvalueXo:-71-1.800000kXk2-1.7357140-7.0000003-1.7320621-4.7638894-1.7320512-3.3223185-1.7320513-2.4355334-1.9529155-1.7646306-1.7329317-1.7320518-1.732051PleaseinputinitialvalueXo:-5kXk0-5.0000001-3.4722222-2.5241803-1.9960684-1.7766185-1.7336746-1.7320537-1.7320518-1.732051结果显示’以上初值迭代序列均收敛于亠732051,即根x*。在区间(—1,-5)即区间(-1,-0.774597)上取-0.774598,-0.8,-0.85,-0.9,-0.99，计算结果如下：PleaseinputinitialvalueXo:-0.774598PleaseinputinitialvalueXo:0.85kXkkXk0-0.77459800.85000010.7746051-1.4753752-0.7746452-1.81944430.7748843-1.7379694-0.7763244-1.73208150.7850495-1.7320516-0.8406416-1.73205171.350187PleaseinputinitialvalueXo:-0.981.993830kXk91.7759630-0.900000101.73362812.557895111.73205322.012915121.73205131.781662131.73205141.734049PleaseinputinitialvalueXo:-0.851.732054kXk61.7320510-0.80000071.73205110.948148PleaseinputinitialvalueXo:-0.992-5.625370kXk3-3.8726250-0.9900004-2.766197132.5058295-2.121367221.6910816-1.818292314.4915217-1.73782249.7072388-1.73207956.5409069-1.73205164.46496610-1.73205173.13384082.32607591.902303101.752478111.732403121.732051131.732051计算结果显示，迭代序列局部收敛于亠732。51,即根x*，局部收敛于诈0251,即根x3。在区间(」,5)即区间(-0.774597,0.774597)上，由search.m的运行过程表明，在整个区间上均收敛于0,即根x*。2在区间(5,1)即区间(0.774597，1)上取0.774598，0.8，0.85，0.9，0.99，计算结果如下PleaseinputinitialvalueXo:0.774598PleaseinputinitialvalueXo:0.85kXkkXk00.77459800.8500001-0.7746051-1.47537520.7746452-1.8194443-0.7748843-1.73796940.7763244-1.7320815-0.7850495-1.73205160.8406416-1.7320517-1.350187PleaseinputinitialvalueXo:0.98-1.993830kXk9-1.77596300.90000010-1.7336281-2.55789511-1.7320532-2.01291512-1.7320513-1.78166213-1.7320514-1.734049PleaseinputinitialvalueXo:0.85-1.732054kXk6-1.73205100.8000007-1.7320511-0.948148PleaseinputinitialvalueXo:0.9925.625370kXk33.87262500.99000042.7661971-32.50582952.1213672-21.69108161.8182923-14.49152171.7378224-9.70723881.7320795-6.54090691.7320516-4.464966101.7320517-3.1338408-2.3260759-1.90230310-1.75247811-1.73240312-1.73205113-1.732051计算结果显示，迭代序列局部收敛于亠732。51,即根x*，局部收敛于诈。251，即根x3。区间(1,)上取100，60，20，10，7，6，4，3，1.5，计算结果如下:PleaseinputinitialvalueXo:10062.213605PleaseinputinitialvalueXo:4kXk71.854126kXk0100.00000081.74313604.000000166.67333491.73215612.844444244.458891101.73205122.163724329.654263111.73205131.834281419.792016PleaseinputinitialvalueXo:1041.740007513.228447kXk51.73210568.869651010.00000061.73205175.98923116.73400771.73205184.10732424.590570PleaseinputinitialvalueXo:392.91075533.212840kXk102.20018942.37165303.000000111.84868751.92298112.250000121.74223561.75717521.869231131.73213971.73258031.745810141.73205181.73205141.732212151.73205191.73205151.732051PleaseinputinitialvalueXo:60PleaseinputinitialvalueXo:761.732051kXkkXkPleaseinputinitialvalueXo:1.5060.00000007.000000kXk140.01111414.76388901.500000226.69074923.32231811.800000317.81884532.43553321.735714411.91676241.95291531.73206258.00084851.76463041.73205165.41854661.73293151.73205173.73973671.73205182.68515181.73205192.078360PleaseinputinitialvalueXo:6101.802967kXk111.73602706.00000012]1.73206414.114286131.73205122.915068141.73205132.202578PleaseinputinitialvalueXo:2041.849650kXk51.742392020.00000061.732142113.36675071.73205128.96132381.73205136.04954744.146328结果显示，以上初值迭代序列均收敛于1.732051，即根x*。3综上所述：(-°°,-1)区间收敛于-1.73205,(-1,6)区间局部收敛于1.73205,局部收敛于-1.73205,(-6,6)区间收敛于0,(6,1)区间类似于(-1,6)区间，(1产)收敛于1.73205。通过本上机题，明白了对于多根方程，Newton法求方程根时，迭代序列收敛于某一个根有一定的区间限制，在一个区间上，可能会局部收敛于不同的根。第三章一、题目列主元Gauss消去法对于某电路的分析，归结为求解线性方程组RI二V。其中‘31-13000-1000、0-1335-90-1100000-931-100000000-1079-30000-9R=000-3057-70-500000-747-300000000-3041000000-50027-2<000-9000-22229Vt=(—15,27,-23,0,-20,12,—7,7,105编制解n阶线性方程组Ax二b的列主元高斯消去法的通用程序；用所编程序线性方程组RI二V，并打印出解向量，保留5位有效数；二、通用程序%%列主元Gauss消去法求解线性方程组％%%%参数输入n=input('PleaseinputtheorderofmatrixA:n=');%输入线性方程组阶数nb=zeros(1,n);A=input('InputmatrixA(suchasa2ordermatrix:[12;3,4]):');b(1,:)=input('Inputthecolumnvectorb:');%输入行向量bb=b';C=[A,b];%得到增广矩阵%%列主元消去得上三角矩阵fori=1:n-1[maximum,index]=max(abs(C(i:n,i)));index=index+i-1;T=C(index,:);C(index,:)=C(i,:);C(i,:)=T;fork=i+1:n%%列主元消去ifC(k,i)~=0C(k,:)=C(k,:)-C(k,i)/C(i,i)*C(i,:);endendend%%回代求解%%x=zeros(n,1);x(n)=C(n,n+1)/C(n,n);fori=n-1:-1:1x(i)=(C(i,n+1)-C(i,i+1:n)*x(i+1:n,1))/C(i,i);endA=C(1:n,1:n);%消元后得到的上三角矩阵disp('Theupperteianguularmatrixis:')fork=1:nfprintf('%f',A(k,:));fprintf('\n');enddisp('Solutionoftheequations:');fprintf('%.5g\n',x);%以5位有效数字输出结果以教材第123页习题16验证通用程序的正确性。执行程序，输入系数矩阵A和列向量b,结果如下：PleaseinputtheorderofmatrixA:n=4InputmatrixA(suchasa2ordermatrix:[12;3,4])[1253-2-2-2351323]Inputthe21-2columnvectorb:[47-10]2.0000005.0000003.000000-2.0000000.0000003.0000006.0000003.0000000.0000000.0000000.500000-0.5000000.0000000.0000000.0000003.000000Solutionoftheequations:2-12-1结果与精确解完全一致。三、求解结果执行程序，输入矩阵A（即题中的矩阵R）和列向量b（即题中的V），得如下结果：PleaseinputtheorderofmatrixA:n=9InputmatrixA(suchasa2ordermatrix:[12;3,4]):[31-13000-10000-1335-90-1100000-931-100000000-1079-30000-9000-3057-70-500000-747-300000000-3041000000-50027-2000-9000-229]Inputthecolumnvectorb:[-1527-230-2012-7710]31.000000-13.0000000.0000000.0000000.000000-10.0000000.0000000.0000000.0000000.00000029.548387-9.0000000.000000-11.000000-4.1935480.0000000.0000000.0000000.0000000.00000028.258734-10.000000-3.350437-1.2772930.0000000.0000000.0000000.0000000.0000000.00000075.461271-31.185629-0.4519990.0000000.000000-9.0000000.0000000.0000000.0000000.00000044.602000-7.1796950.000000-5.000000-3.5779940.0000000.0000000.0000000.000000-0.00000045.873193-30.000000-0.784718-0.5615430.0000000.0000000.0000000.000000-0.000000-0.00000021.380698-0.513187-0.3672360.0000000.0000000.0000000.000000-0.000000-0.0000000.00000026.413085-2.4199960.0000000.0000000.0000000.000000-0.000000-0.0000000.0000000.00000027.389504Solutionoftheequations:-0.289230.34544-0.71281-0.22061-0.43040.15431-0.0578230.201050.29023由上述结果得：「-0.289230.34544-0.71281-0.22061V=-0.43040.15431-0.0578230.201050.29023第四章、题目IL)轎制弟第一型3疣样拎播值函數的通用釋序;(2)已知汽车门曲竣型值点的0I23452,5]3,504+044,705.225,5467S9105^85.405,57.5,705.80端点条件为从=。飞艸；=62/所醐觀序求牟门的$灰择告輛位晶教和甲出£(i斗0.5}{i二DJ,■,9)0二、通用程序黑更第四章上机题：求第一型3次样築插值函数的通用程序%%clearclc%%输几栢关参埶n=input(JInputn:n=J);n=n+l;sn=zeros{13n);yn=zerosii,n);zn(lj:〕=input(JInputk:J);yii(lj:)=input(JInputdyO=input(JInputthey：>\derivativeofytO)：J);弔输入.边畀条件dyn=input(JInputthederivativeofy(n):J);%%求d值d=zeros(n,1);h=zerosUjn-1);fl=zeros(ljn-1);f2=zeros(1,n-2):fori=1:n-1h(i)=Kn(i+l)-Kn(i);fl(i)=(yn(i+l)-yn(i))/h(i);黑一阶差商endfor1=2:n-1f2(i)二(f1(i)-f111-1))/'\n(i+l)-Kn(i-l))飞二阶差商d(i)=6*f2(i);endd(l)=6*(fl(l)-dyO)/h(l);d(n)=6*(dyn-f1(n-1))/h(n-1);%%我陋值A=zeros(nJ;miu=zeros(15n-2);lainda=zeros'13n-2);fori=l:n-2miu%%回代敢插値函埶symsk;fori=1:n-1=collect(yn1i)+'f1■i)-'M11)/3+M(i+l)/6)*h'i))*i玄-xn1i))...+M(i)/2*(k-kii(£}■)"2+(M(i+l)-M(i))/(6*h(i))*(x-sn(i))^3);SS(i)=VPA(Sk(i)j4);endS=zeros(ljn-1';fori=1:n-19S：求节总插眉K=Kn<1)+0.5;S(i)=yn(i)+(f1(i)-(M(i)/3+M'i+l)/6)*h(i))*(s-Kn(i))...+M(1)/2+(x-sn(1))2+(M(i+l)-M(1))/(6*h(i))*(x-Kn(i))^3;end%%输出结果dispCS(z)=J);fori=1:n-1fprintf(J%s(%d,%d.'n',char(Sx(i))Kn(i)r,sn(i+l))dispCJ)^nddispfS■i+0.5)J)dispCiK'i+0.5)S(i+0.5)J)fori=1:n-1fprintf(J%d%.4f%.4fnJjxn(i)+0.S(i))endi)=h'i)/17(h-i)+h'i+l));lamdaii)=l-miui);endA(l,2)=1;A(nJn-l)=l;fori=l:nA(iJi)=2;endfori=2:n-lA(i,;A(iji+l)=lajnda■i-l);endM=A\d;三、求解结果1、数据输入Inputn:n=10Inputx:[012345678910]Inputy:[2.513.304.044.705.225.545.785.405.575.705.80]Inputthederivativeofy(0):0.5Inputthederivativeofy(n):0.22、计算结果S仗)二0.5181畑"2-0.2281*x^3+0.+2.51(0,1)0.O5439*ZA3-0.329U^2+L.348*x+2.227(1,2)0.1134紘"2-0.01942松"d+0.4618*s+2.818(2,3)0.2689*k'，2-0.0367^3-0.00466*^+3.284(3,4)o.loea+K^-1.446^^2+6.855*k-5.862(4,5)4.17^2-0.2632^3-21.22*z+40.94(5,6)0.4265^^3-8.334*z2+53.-109.1(6,7)6.247*k-'2-0.2678*zA3-48.27*k+129.Q(7,8)0.05487*z-'3-1.498*x^2+13.69#k-36.18(8,9)0.05838*zA3-1.£93机"2+14.55#x-38.74(9,10)S(i+0.5)ik(i+0.5)S(i+0.-：5)10.50002.861021.50003.691232.50004.378143.50004.98915d.50005.383065.50005.723816.50005.594487.50005.4299g8.50005.6598109.50005.7323第五章、题目⑴總定积分/(/)=血)曲。取初始步枚h^k及祷度"应用罠忆嫌形公戎，采用遂炭二事步长的方济，并应用外推感瘪轎制计算/(/)的通用程序*计算至相邻两次近拟值之星的魁对值不越过E为止。严/3〔⑵列所撼釋序计算积分"C=(伽(异+y右旅川八取・=yX1O\JE警<]忆二、通用程序%Gettheapproximationofadoubleintegraloff(jc,y}%onareatangularregionR={(xTy)|a<=x<=btc<=y<=d^%withgivenstepm=(b-a}/handn=(d-a)/k%andgivenerrorepsilonfunction[bestresTres]=g匕七口oiibleIntegral(fra;b,c,d,in;nrepsilon)nes={];count=1;nes(countT1)=getT(fra,brdr2^co口11七吉131『2“口°1111七*11);count=2;res(countr1)=getT(fra,brdr2Aaount*m/2*kcjoun七*口)；res(count-1,2)=4/3*res(countfl}_l/3*res(oount-l,1};rest=res(1f1);res2=res(2f1);co=[4/3-1/3,16/15-1/15;64/63-1/63];whileabs(resl-res2)>epsiloncount=count+1;res(coun七『1)=getT(fa,b7a；2^cjount*in;2^count*n);%#okfox1=1:size(co,1)ifcount>iies{conn=co(1f1)*res(cjolui七一co(1f2)*i?es(aount-i,i);%#okendendi=sizeJco,l);whilesize(resr1)-1-1<0i=i-1;endresl=res(qch.iii七一res2=res{aoun七一)+丄#土)；bestres=res2;endend%Getapproximatiouofadoubleintegraloff(xy)%onareatangularregionR={(xTyJ}a<=x<=±>tc<=y<=d\%withgivenstepsm=^b-a)/hEindn=(d-o)/k%usingTheCompositeTrapezoidRulefunctionres=getT(fra,hra^djm^nJh=(b-a)/m;3c=(d-c}/n;x=a+(Q:m)'*11；y=a+(0:n}1*k;re呂=0;fori=1:mfoij=1:nr&s=res+h*k/4*(f(x(i)7y(j)>+f(x(i),y(j+1))+f(jc(1+1),y(j)}+f(k(1+1),y(j+l}));endendendfunationexp5|f=@myfun;a=D;b=pi/6;c=D;&=pi/3；m=1;n=1;epsilon=0,5e-5;[bestresfxes]=getDoubleIri七egral(ffaTb；c,d,mfn,epsilon);fprlntf{1===^==^======^===========\E)fprintf(1Resultis%D.7f7detail:\nJ^bestres)fprintf(1=========\n1)fprintf(1k\t\t\t2Ak\七\七\七工(2^k)\t\t\tS(2^k)\t\七\七匚(2Ak)\七\七\七R(2^k)\nfprintf(1=========\n1)fork=1:size{res,1)fprintfC1Qf1fkf2^k);sti?=1';for1=1:size(res,1)-k+1ifi<=size(i?esf2)str=sprin七壬(，慕呂\七\七%0,TfJfstr,res(kr1));endendfprin七：E(1%s\nrfstr);endfprintf{1=====^====^=====^=========^====^=====^==\口1)endfunotiani?总呂=myfun(x,y)res=tan(x*2+7*2};end三、运行结果Resultis0.3365205j,detail2AkT(2AkJS(2xk)C(2Ak)R(2Ak)2t)・51979650.3440324D.33739335240.38797340.3378083D.33659240.3365227380・3503495G.3366604D.3365238Q・33652054160.3400887Q.3365328D.3365205532t)・33742100.33652136640.3367464第六章、题目踽制RK,方法的通用程序*鋼制AB*方去的通用程序(由R百提供軻隹*编制A比-陋预测杜止方法通用程序(由R心撮供初值)；徧制葷配进的AB.-MA.喬测稅正亦法填旳程序(由提供捌值h对于初值问题ff=-//(0x1.5)ly<0)=3取册故A=41*应用(I】~(4)中的四科■方法进行计算•并将计界皓杲和棘易解y(*)■3/(1+.J)作比较*通过本上机些站论？二、通用程序1、rk4方法的通用程序%RK4方法的通用程序%f-函甦句栩%jc0=[xlrx2]-驶解范雨%yD-初值%h-计算步长functian[xTy]=RK4(ffx0ryOrh)x=[xO(1}:h:xO(2)]T;%#aky=zeros(leng七h(3t)f1);y(i)=yO;y(i)+l/2*h*kl);y(i)+l/2*h*k2);y£i)+h*k3);fori=1:length(x)-1kl=f(x(i);k2=f[「¥引=RK4(f“(口4J),yG.h)；y=zeros(length(x);y(l-4)=yO；for1=4:length(x)-1Y(i+1)=y(i)+■■■h/24*(55*f(x(i)-….59*f(x(i-l),y(i-l})+■…37*f(x(i-2),y(i-2})-….9*f(x(i-3);endend3、AB4-AB4预测校正方法的通用程序%AB4-AM4预测校正方法的通用方程%f-函数句辆%就1=住1’淞]-求解范围%yO-初值%h-计算步长function[x,y]=AB4AM4(fFxO,yOh)x=[xO£1):h:x0(2}]';%#ok=RK4(frx([l4])fyO.h);y=zeros(length(x)；1);y[-.yQl=RK4(f,x([l4J).yO.h);y=zeros(length(x)f1);y(l：4}=yOr-for1=4:length(x)-1yp=y(i)+■■h/24*(55*f(x(i)-…59*f7y(i-l})+…37*f(x(i-2)7y(i-2})一…9*f(x(i-3),y(i-3}});yc=y(i)+…h/24*(9*f(x(i+l)Typ)19*f(x(i}fy(i}).5*f(x(i-l)+..,f(x(i-2)fy(i-2))};y(i+l)=251/27a*yc+19/270*yp;endend三、结果比较表1各种方法运行结凰比较方法RK4AB4AR4-AM4改进AB4-AAI4■逞差盘范砌0.00003S0.0048090.0000200.000531按精度据序1432四、结论由图表可见RK4的粕度最髙，改进AB4-ANI斗得稍度ttAB4-AM4高.AB斗嚴卷
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