數(shù)值分析課程設(shè)計比較各種迭代收斂速度模板

1、數(shù)值分析課程設(shè)計比較各種迭代收斂速度分別用雅可比迭代法(J)、高斯一塞德爾迭代法(G-S)、超松弛迭代X101X=82X10L 3-4 -1 0法(SOR)計算方程組A= -1 4 -1 0 -1 4并比較哪一種迭代方法收斂的速度更快方程真實值計算：A=4-10;-14-1;0-14;b=10810;jX=Ab得到結(jié)果：3.42863.71433.4286雅可比迭代：首先編寫jacdd.m的函數(shù)文件(見附錄一)調(diào)用程序，在命令窗口分別輸入如下語句：A=4-10;-14-1;0-14;b=10;8;10;X0=000;X=jacdd(A,b,X0,inf,0.00001,100)結(jié)果見表一高斯一

2、塞德爾迭代：首先編寫gsdddy.m的函數(shù)文件(見附錄二)調(diào)用程序，在命令窗口分別輸入如下語句：A=4-10;-14-1;0-14;b=10;8;10;X0=000;X=gsdddy(A,b,X0,inf,0.00001,100)結(jié)果見表一雅可比迭代誤差計算:x0=3.42863.71433.4286;%此為方程組的真實值x1=2.50003.00003.31253.37503.41413.42193.42683.42773.42833.42853.42853.4286;x2=2.00003.25003.50003.65633.68753.70703.71093.71343.71393.714

3、23.71423.7143;x3=2.50003.00003.31253.37503.41413.42193.42683.42773.42833.42853.42853.4286;formatlong%循環(huán)求二范數(shù)的平方fori=1:12t(i)=(x 1(i)-3.4286)人2+(x2 (i)-3.7143)人2+(x3 (i)-3.4286)人2;sqrt(t(i)end結(jié)果見表一高斯一塞德爾迭代誤差計算：x0=3.42863.71433.4286;%此為方程組的真實值x1=2.50003.15633.39453.42433.42803.42853.4286;x2=2.62503.578

4、13.69733.71223.71403.71433.7143;x3=3.15633.39453.42433.42803.42853.42863.4286;formatlong%循環(huán)求二范數(shù)的平方fori=1:6s(i)=(x 1(i)-3.4286)人2+(x 2(i)-3.7143)人2+(x3 (i)-3.4286)人2;sqrt(s(i)end結(jié)果見表一雅可比迭代高斯一塞德爾迭代次數(shù)X1X2X3誤差次數(shù)X1X2X3誤差12.50002.00002.50002.159495412.50002.62503.15631.457058623.00003.25003.00000.76352502

5、3.15633.57813.39450.306366733.31253.50003.31250.269968333.39453.69733.42430.038344543.37503.65633.37500.095445943.42433.71223.42800.004822953.41413.68753.41410.033745253.42803.71403.42850.000678263.42193.70703.42190.011961263.42853.71433.42860.000099973.42683.71093.42680.004247473.42863.71433.428608

6、3.42773.71343.42770.0015588893.42833.71393.42830.00058319103.42853.71423.42850.000173210113.42853.71423.42850.000173211123.42863.71433.4286012注：MATLAB計算的誤差為16位有效數(shù)字，由于表的位置有限, 這里只取8位有效數(shù)字畫圖比較：畫圖函數(shù)：k=1:12;x=2.15949540.76352500.26996830.09544590.03374520.01196120.00424740.00155880.00058310.00017320.00017

7、320;%J 的迭代誤差plot(k,x,b)holdony=1.45705860.30636670.03834450.00482290.00067820.0;%G-S 的迭代誤差plot(k,y,-.)legend(J迭代,G-S迭代)%畫出圖形，標(biāo)明各曲線的含義title(誤差圖);%加上標(biāo)題text(k(1),x (1) ,start)%注明起始和終止點text(k(11),x(11),end)xlabel(K迭代次數(shù));%標(biāo)注橫，縱坐標(biāo)ylabel(誤差);gridon%畫出網(wǎng)格誤差圖迭代日迭代B8K迭代彼數(shù)結(jié)論：從數(shù)據(jù)圖表可觀察到：雅可比的迭代次數(shù)明顯比高斯塞德爾 的迭代次數(shù)要多，因此高斯塞德爾比雅可比迭代的收斂速度快.G-S迭代與J迭代在本質(zhì)上沒有必然的聯(lián)系，求解方程組時，J 迭代的速度與G