计算方法试验三解线性方程组的迭代法

1、实验三 解线性方程组的迭代法(1)雅可比迭代法1、实验程序实现雅可比迭代法的MATLAB数文件agui_jacobi.m2 34 -5 - 6 -7 - 8 - 9 -10 - 11 -12 - 13 -14 - 15 -16 - 17 - IS - 19 - 20 - 21 -22 -23function z = agui_jacobi (a, b)为系数矩阵, b为右端向量,区口为初始向量(默认为零向量)加为精度默认为154)，W为最大迭代次数(默认】。口)1笈为返回解向量 n=lengthCb) tN=1DO;e=1e-4;xO=zeros (n3 1) hx=xO;x0=x+2*e;k

2、=0;d= di ag (diag (a);l=-tril(a,-10 hu二一triu(a, 1);while norm (xO_Xj inf) >e4k<Nk=k+l;xO=x;x=inv(d) * (1+u) *x+inv(d) *b;kdisp )endif k=N WARNING C已达最大迭代次数;end24在MATLAB令窗口输入及实验结果和操作界面J Command WindowTo get startedj select MATLAB H.lp or Dem口理 from the Help menu.The element type "naneA mus

3、t be terminated by the matching end-tag *</name”.Could not parse the file: c:natlab7toolboxccslinkccslinkinfo.xml» a=4 -1 0 -1 0 0；-l 4 -1 0 -1 。;0 -1 4 -1 0 -1,-1 0 -1 4 -1 0.0 -1 0 -1 4 -1;D 0 -1 0 -1 44-10-10-14-10-10T4-10-10-14-10T0-1400-10-10070-14» b=0;5；-2；5-2；6b =05-25-26»

4、K=agui_jacobiC&b)k -1L 2500-0,5000 L500001,2500-0.50000. 62501.00000.50001.00000.50001. 25000.50001.65630.31251.65630.31251.7500k =40.8281k =50. 7656k =61.53130.76561. 83980. 67971. 53130.76561.65631. 83980.67971. 88280.89061.92530.85061.92530.85061.9453k =80.9626k =90. 94901. 89790.94901. 89790

5、.94901.92531. 96510.93031. 96510.93031.974510U. 982611l.952qU.9,621.9524U.9,621.9651120.99191.97780.98891.97780.98891.9837k =130.98891.99240. 98481. 99240.98481.9944140. 9962150. 9948161.98961.99650. 99480.99291.98961.99650. 99480.99291.99241. 9974170. 99761. 99830.99671. 99830.99671. 9988180.99921.

6、99770.99891. 99770.99891. 9983190.99891.99920.99851.99920.99851.9994200.99961.99890.99951.99890.99951.999221220.99981.99950.99981. 99950.99981.9996k =230. 99981.99980.99971. 99980.99971. 9999240. 99991.99980.99991.99980.99991. 9998250.99991.99990.99981.99990.99981.999926O+9S9S 2.00000.99992.00000. 9

7、9992* 口叩口0.9999 2. 0000 0.99992. 0000 0. 9999 2. 0000(2)高斯-赛德尔迭代法1、实验程序实现高斯-赛德尔迭代法的MATLA函数文件agui_GS.mRft FiM IN J Tpmr;Qarinjir-qa2 M由寄吐*距b由E*向*- xl力M由何I监i匕电代表酸d力延回即4 - h=«h£rhft). 3 - R=IW.4 - «=1I: F - Riarxglb.3 " X"iD.X13Bz+2"!i.Tlfirnlfi2"un-r-iL 如la IK*EHljE【

8、F Ulf XU<H k=k+L.Nt=E.K=T北(4-411 STft+n，. ff t IcridxiplM，H! k«H /Uai!*/EE：士华亡MV ) «il23456789101112131415161718192021222324function x = agui_GS(a,b)为系数矩阵，b为右端向量, 乂口为初始向量(默认为零向量)灰为精度(默认为1总7)，N为最大迭代次数默认为1口。)为返回解向量- n=length(b);- N=100,- e=le-4;- xO=zeros (n3 1);- x=x0;- x0=x+2*e；- k=0;-

9、al=tril(a);- a2=inv(al);- while norm(kO-Zj inf)>eftk<N- k=k+1；- xO=z;- x-a2*(a-al) *x0+a2*b;- forjuatlong- k- disp(£)- end- if k=N WARNING C已达最大迭代次数Q- end在MATLAB令窗口输入及实验结果和操作界面» a=4-1 0 -10 0;-l4-10-10;0 -14-10 -1;-1 0-14-1 0;0 -10-14 -l;0 0-10-1 4a =4-10-100-14-10-100-14-10-1-10-14-

10、100-10-14-100-10-14» b=0;5;-2;5;-2; 6b =05-25-26» x=agui_GS (a, b)k =101.25000000000000 -0.187500000000001.203125000000000.113281250000001.481445312500000.61328125000000k =30.73643493652344k =40.87300521135330k =50.93943285173737k =60.970875459909621.384765625000001.715141296386721.8638888

11、89551161.934219151618891.968361701603950.517333984375000.764286994934080.884101506322620.944355651925430.973321806562521.560974121093751.776879549026491.893842517398301.949282688019591.975603240263230.606796264648440.818263351917270.913342248415570.958215694580760.979901944623431.781032562255861.895

12、637586712841.949360938684551.975642836626551.988305937796490.98599123546679k =80.99326789987023k =90.99676495834023k =100.99844528982311k =110.999252825423461.984803746663191.992697628842581.996490403508261.998313328403621.999189418716950.987178231180730.993836533883830.997037942987460.9985765225339

13、10.999315901092071.988267852817741.994362204518351.997290755784171.998697973290231.999374263356490.990344384319350.995360121808990.997770080803910.998928326910420.999484973608631.994380653875021.997299163923211.998702005947841.999376212361081.99970021867517k =12File Edit Debug Desktop Wndcw Help0.99

14、9640920518361.999610448804760.999671232709111.933599281709020.999762487297241.99985533000159130. 99982M32628451.999812788158701.993855479973250. 99988104953338L999930732375180.999917067032991.99991002913342D.999924067870461.999930546109210.999942834404451.99995572556873x -0.999�.999924067870461.999930546109210.999942834404451.99996672556873结果分析:从上面的雅可比迭代法和高斯一赛德尔迭代法这两种方法所得的实验结果可知，对于同样的矩阵：a =4-10-100-14-10-100-14-10-1