1、安徽建筑大學(xué)數(shù)值分析 設(shè)計(jì)報(bào)告書題 目 松弛迭代法中松弛因子 院 系 數(shù)理系 專 業(yè) 信息與計(jì)算科學(xué) 班 級(jí) 信息班 姓 名 穆海山 時(shí) 間 2013-12-102013-12-23 指導(dǎo)教師 劉華勇 題目：選用Jacobi迭代法、Gauss-Seidel迭代法和超松弛迭代法求解下面的方程組（考慮等于150）=考慮初值的變化和松弛因子的變化收斂效果的影響；對(duì)上述方程組還可以采用哪些方法求解？選擇其中一些方法編程上機(jī)求解上述方程組，說明最適合的是什么方法；將計(jì)算結(jié)果進(jìn)行比較分析，談?wù)勀銓?duì)這些方法的看法。一、摘要本課程設(shè)計(jì)用matlab就線性方程組數(shù)值方法，Jacobi迭代法，Gauss-Seid

2、el迭代法，超松弛法對(duì)所設(shè)計(jì)的問題進(jìn)行求解，并編寫程序在Matlab中實(shí)現(xiàn)，在文章中對(duì)各種迭代法進(jìn)行了收斂性分析。接著用幾種不同方法對(duì)線性方程組進(jìn)行求解及結(jié)果分析，最后對(duì)此次課程設(shè)計(jì)進(jìn)行了總結(jié)。關(guān)鍵詞：線性方程組，迭代，Matlab，結(jié)果分析二、設(shè)計(jì)目的用熟悉的計(jì)算機(jī)語言編程上機(jī)求解線性方程組。三、理論基礎(chǔ)對(duì)方程組 做等價(jià)變換 如：令 ，則 則，我們可以構(gòu)造序列 若 同時(shí)：所以，序列收斂，與初值的選取無關(guān)則轉(zhuǎn)化為矩陣形式 (1)令 或者 故迭代過程(1)化為 (2) 等價(jià)線性方程組為 稱(2)式為解線性方程組(1)的Jacobi迭代法(J法)迭代矩陣 考慮迭代式(2) 即 將上式改為 (3)

3、(4)超松弛迭代記 則 可以看作在前一步上加一個(gè)修正量。若在修正量前乘以一個(gè)因子有 對(duì)GaussSeidel迭代格式 迭代矩陣四、 程序代碼及運(yùn)算結(jié)果1.利用Jacobi迭代法求解：編制名為majacobifun.m的文件，內(nèi)容如下：function x,n=jacobifun(A,b,x0,eps)%jacobifun為編寫在雅可比迭代函數(shù)%A為線性方程組的系數(shù)矩陣%b為線性方程組的常數(shù)向量%x0為迭代初始向量%eps為解的精度%x為線性方程組的解%n為求出所需精度的解實(shí)際的迭代步數(shù)D=diag(diag(A);B=D(D-A);f=Db;x=B*x0+f;n=0;if max(abs(ei

4、g(B)>=1 disp('Warning:不收斂！'); return;endwhile norm(x-x0)>=eps x0=x; x=B*x0+f; n=n+1;end2.利用Gauss-Seidel迭代法求解：編制名為G_Seidelifun.m的文件，內(nèi)容如下：nction x,n=G_Seidelifun(A,b,x0,eps)%G_Seidelifun為編寫在高斯-賽德爾迭代函數(shù)%A為線性方程組的系數(shù)矩陣%b為線性方程組的常數(shù)向量%x0為迭代初始向量%eps為解的精度%x為線性方程組的解%n為求出所需精度的解實(shí)際的迭代步數(shù)D=diag(diag(A);

5、L=-tril(A,-1);U=-triu(A,1);G=(D-L)U;f=(D-L)b;x=G*x0+f;n=0;if max(abs(eig(G)>=1 disp('Warning:不收斂！'); return;endwhile norm(x-x0)>=eps x0=x; x=G*x0+f; n=n+1;end3.利用SOR迭代法求解：編制名為SORfun.m的文件，內(nèi)容如下：function x,n=SORfun(A,b,x0,w,eps)%SORfun為編寫在松弛迭代函數(shù)%A為線性方程組的系數(shù)矩陣%b為線性方程組的常數(shù)向量%x0為迭代初始向量%w為松弛因子%

6、eps為解的精度%x為線性方程組的解%n為求出所需精度的解實(shí)際的迭代步數(shù)if(w<=0 |w>=2) error return;endD=diag(diag(A);L=-tril(A,-1);U=-triu(A,1);S=(D-w*L)(1-w)*D+w*U);f=w*(D-w*L)b);x=S*x0+f;n=0;if max(abs(eig(S)>=1 disp('Warning:不收斂！'); return;end while norm(x-x0)>=eps x0=x; x=S*x0+f; n=n+1; end在Matlab命令窗口輸入：>&g

7、t; clearw=1.5;eps=1e-6;n=150;A=diag(8*ones(1,n-1),-1)+diag(6*ones(1,n)+diag(ones(1,n-1),1);b(1)=10.5;b(n)=21;for i=2:n-1 b(i)=22.5;endb=b'x0=zeros(n,1);x1,n=jacobifun(A,b,x0,eps);x2,n=G_Seidelifun(A,b,x0,eps);x3,n=SORfun(A,b,x0,w,eps);plot(x1);hold onplot(x2,'r');hold onplot(x3,'g*&#

8、39;)表1 Jacobi、Guass_seidel、SOR迭代結(jié)果表x0=zeros(n,1)x0=ones(n,1)w=1.5w=1.1w=1.5w=1.1雅可比高斯-賽德爾超松弛雅可比高斯-賽德爾超松弛不收斂！不收斂！n =n =n =n =n =n =n =n =084355663083353655x =x =x =x =x =x =x =x =1.0e+028 *1.0e+029 *1.0e+028 *1.0e+029 *1.7501.501.583301.503.7501.502.2501.503.7501.502.2501.503.7501.502.2501.503.7501.5

9、02.2501.503.7501.502.2501.503.7501.502.2501.503.7501.502.2501.503.7501.502.2501.503.7501.502.2501.503.7501.502.2501.503.7501.502.2501.503.7501.502.2501.503.7501.502.2501.503.7501.502.2501.503.7501.502.2501.503.7501.502.2501.503.7501.502.2501.503.7501.502.2501.503.7501.502.2501.503.7501.502.2501.503.

10、7501.502.2501.503.7501.502.2501.503.7501.502.2501.503.7501.502.2501.503.7501.502.2501.503.7501.502.2501.503.7501.502.2501.503.7501.502.2501.503.7501.502.2501.503.7501.502.2501.503.7501.502.2501.503.7501.502.2501.503.7501.502.2501.503.7501.502.2501.503.7501.502.2501.503.7501.502.2501.503.7501.502.250

11、1.503.7501.502.2501.503.7501.502.2501.503.7501.502.2501.503.7501.502.2501.503.7501.502.2501.503.7501.502.2501.503.7501.502.2501.503.7501.502.2501.503.7501.502.2501.503.7501.502.2501.503.7501.502.2501.503.7501.502.2501.503.7501.37502.2501.37503.750202.250203.7500.502.2500.503.750302.250303.750002.250

12、003.750402.250403.750002.250003.750002.250003.750002.250003.750002.250003.750002.250003.750002.250003.750002.250003.750002.250003.750002.250003.750002.250003.750002.250003.750002.250003.750002.250003.750002.250003.750002.250003.750002.250003.750002.250003.750002.250003.750002.250003.750002.250003.75

13、0002.250003.750002.250003.750002.250003.750002.250003.750002.250003.750002.250003.750002.250003.750002.250003.750002.250003.750002.250003.750002.250003.750002.250003.750002.250003.750002.250003.750002.250003.750002.250003.750002.250003.750002.250003.750002.250003.750002.250003.750002.250003.750002.2

14、50003.750002.250003.750002.250003.750002.250003.750002.250003.750002.250003.750002.250003.750002.250003.750002.250003.750002.250003.750002.250003.750002.250003.750002.250003.750002.250003.750002.250003.750002.250003.750002.250003.750002.250003.750002.250003.750002.250003.750002.250003.750002.250003.

15、750002.250003.750002.250003.750002.250003.750002.250003.750002.250003.750002.250003.750002.250003.750002.250003.750002.250003.750002.250003.750002.250003.750002.250003.750002.250003.75-0.0001002.25-0.0001003.750.0002002.250.0002003.75-0.000300.00012.25-0.000300.00013.750.00060-0.00022.250.00060-0.00

16、023.75-0.001200.00042.25-0.001200.00043.750.00240-0.00072.250.00240-0.00073.75-0.004900.00142.25-0.004900.00143.750.00970-0.00282.250.00970-0.00283.75-0.019500.00572.25-0.019500.00573.750.03890-0.01132.250.03890-0.01133.75-0.077600.02262.25-0.077600.02263.750.15460-0.0452.250.15460-0.0453.75-0.30680

17、0.08932.25-0.306800.08933.750.60390-0.17572.250.60390-0.17573.75-1.168700.342.25-1.168700.343.752.18170-0.63472.252.18170-0.63473.75-3.7401.0882.25-3.7401.0883.54.98660-1.45072.16674.98660-1.4507以下為方程組的解的圖：圖1 初值x0=0；.；0 松弛因子=1.5時(shí)圖圖2 初值x0=1；.；1 松弛因子=1.5時(shí)圖圖3 初值x0=1；.；1 松弛因子=1.1時(shí)圖五、結(jié)果分析 通過以上數(shù)據(jù)測(cè)試可以分析出以下幾點(diǎn)：1. 系數(shù)矩陣為150階時(shí)，經(jīng)判斷Jacobi迭代是發(fā)散的、Gauss-Seidel、SOR兩種迭代是收斂的，當(dāng)初值一定時(shí)，Gauss-Seidel、SOR對(duì)應(yīng)的迭代次數(shù)是逐漸減少的，也就是說SOR迭代比Gauss-Seidel迭代收斂更快。2. 當(dāng)初值變大時(shí)，Gauss-Seidel、SOR對(duì)應(yīng)的迭代次數(shù)是逐漸減少的，當(dāng)初值相同，松弛因子 變小時(shí)，SOR對(duì)應(yīng)