太原理工大学数值计算方法实验报告

1、实验名称：线性方程组的直接和迭代解法实验时间：2016.05.31实验目的和要求：1. 了解Gauss消元法、LU分解法、追赶法等线性方程组直接求解的基本方法、基本原理；2. 能够按照工程实际要求，选择适当的算法；3. 通过编写程序，进行算法设计和数值求解；4. 了解雅可比迭代法、高斯-赛德尔迭代法等线性方程组迭代求解的基本方法、基本原理；5. 能够按照工程实际要求，选择适当的算法;6. 通过编写程序，进行算法设计和数值求解。实验内容和原理:1合理利用主元素消元法、LU分解法、追赶法求解下列方程组:123_x114012x28241xL3J130.035.29111.2159.14-6.130

2、92rx159.17x46.782x13x21-4113-1511221_4215_x1-2_872101x2-74836x3-71261120x1-4-32使用雅可比迭代法或高斯-赛德尔迭代法对下列方程组进行求解。10x-x-2x=7.2123<x+10x2x=8.3123xx+5x=4.2v123主要仪器设备：计算机，vc2010x86。上机调试修改源程序:仁列主元素消元法#include"stdafx.h"#includestdlib.hvoidshuchu(doublea34)for(inti=0;i3;i+)for(intj=0;j4;j+)printf(&

3、quot;%lf",aij);printf("");printf("n");printf("n");int_tmain(intargc,_TCHAR*argv)doublea34,b4,x3,c;for(inti=0;i<3;i+)for(intj=0;j<4;j+)scanf("%lf",&aij);if(a10a00&&a10a20)for(inti=0;i4;i+)bi=a0i;a0i=a1i;a1i=bi;if(a20a00&&a20a10)fo

4、r(inti=0;i4;i+)bi=a0i;a0i=a2i;a2i=bi;shuchu(a);c=a10/a00;for(inti=0;i4;i+)a1i=a1i-c*a0i;c=a20/a00;for(inti=0;i4;i+)a2i=a2i-c*a0i;shuchu(a);c=a21/a11;for(inti=0;i4;i+)a2i=a2i-c*a1i;shuchu(a);x2=a23/a22;x1=(a13-a12*x2)/a11;x0=(a03-x2*a02-x1*a01)/a00;printf("x1=%lf",x0);printf("nx2=%lf&q

5、uot;,x1);printf("nx3=%lf",x2);printf("n");system("pause");return0;2.完全组元素消元法#include"stdafx.h"#includestdlib.h#include<math.hvoidshuchu(doublea55)for(inti=0;i5;i+)for(intj=0;j5;j+)if(abs(aij)1e-8)aij=0;printf("%lf",aij);printf("");printf

6、("n");printf("n");voidasd(intm,doublea55,doubleb)intc,d;doublee5,n=b;for(inti=m;i4;i+)for(intj=m;j<4;j+)if(abs(aij)abs(b)b=aij;c=i;d=j;if(b!=n)for(inti=0;i5;i+)ei=ami;ami=aci;aci=ei;for(inti=0;i5;i+)ei=aim;aim=aid;aid=ei;int_tmain(intargc,_TCHAR*argv)doublea55,b,x4;for(inti=0;

7、i5;i+)for(intj=0;j5;j+)scanf("%lf",&aij);asd(0,a,a00);shuchu(a);b=a10/a00;for(inti=0;i5;i+)a1i=a1i-b*a0i;b=a20/a00;for(inti=0;i5;i+)a2i=a2i-b*a0i;b=a30/a00;for(inti=0;i5;i+)a3i=a3i-b*a0i;shuchu(a);asd(1,a,a11);shuchu(a);b=a21/a11;for(inti=0;i5;i+)a2i=a2i-b*a1i;b=a31/a11;for(inti=0;i5;i

8、+)a3i=a3i-b*a1i;shuchu(a);asd(2,a,a22);shuchu(a);b=a32/a22;for(inti=0;i5;i+)a3i=a3i-b*a2i;shuchu(a)-xinr+(a4二3)=a3二二3-xinr+(a4二2)=(a2二4a2二3*xinr+(a4二3)、a2二2-xin?4二二二丄三一二4丁孑二二3丁in?42二盖二二二-xinr+(a4o)=(ao二4Xinr+(a4二3)*ao二3Xinr+(a4二2)*ao二2Xinr+(a4二二盖O二二)mso-for(inr+i=pi4-i+)if(a4二i=l)prinr+f(、x%d=%lf、in

9、r+xinr+(a4二i)八for(inr+i=pi4-i+)if(a4二i=2)prinr+f(、x%d=%lf、inr+xinr+八for(inr+i=pi4-i+)if(a4二i=3)prinr+f(、x%d=%lf、inr+xinr+八for(inr+i=pi4-i+)if(a4二i=4)prinr+f(、x%d=%lf、inr+xinr+八prinr+f(、n、)-sysr+em(、pause、)-rer+urnp一3LU第第ttinclude、sr+dafxh、ttincludesr+dlibhvoidshuchu(doublea45)for(inr+i=pi4-i+)for(in

10、r+j=pj4二+)prinr+f(、lf二j)-prinr+f(、)八一prinr+f(、n、)-一prinr+f(、n、)-一in<-+r+main(in<-+argc-TCHAR*argv口)doubleu4二514二5八for(inr+i=pi4-i+)for(inr+j=pj5二+)lij=O；scanf("%lf",&uij);for(inti=0;i4;i+)li0=ui0/u00;for(inti=1;i4;i+)for(intj=0;j<4;j+)uij=uij-liO*u0j;for(inti=1;i4;i+)li1=ui1/u

11、11;for(inti=2;i4;i+)for(intj=0;j<4;j+)uij=uij-li1*u1j;for(inti=2;i4;i+)li2=ui2/u22;for(inti=3;i4;i+)for(intj=0;j<4;j+)uij=uij-li2*u2j;l33=1;printf("L矩阵:n");shuchu(l);printf("U矩阵:n");shuchu(u);u14=u14-l10*u04;u24=u24-l20*u04-l21*u14;u34=u34-l30*u04-l31*u14-l32*u24;u34=u34/u3

12、3;u24=(u24-u23*u34)/u22;u14=(u14-u12*u24-u13*u34)/u11;u04=(u04-u01*u14-u02*u24-u03*u34)/u00;for(inti=0;i4;i+)printf("x%d=%lfn",i+1,ui4);system("pause");return0;4.高斯-赛德尔迭代法#include"stdafx.h"#includestdlib.h#include<math.hvoidshuchu(doublex3)for(inti=0;i3;i+)printf(%lf

13、,xi);printf("n");int_tmain(intargc,_TCHAR*argv)doublea34,x3=0,0,0,d3;for(inti=0;i3;i+)for(intj=0;j4;j+)scanf("%lf",&aij);shuchu(x);dox0=(a03-a01*x1-a02*x2)/a00;x1=(a13-a10*x0-a12*x2)/a11;x2=(a23-a20*x0-a21*x1)/a22;d0=abs(a03-a01*x1-a02*x2)/a00-x0);d1=abs(a13-a10*(a03-a01*x1-a

14、02*x2)/a00-a12*x2)/a11-x1);d2=abs(a23-a20*(a03-a01*x1-a02*x2)/a00-a21*(a13-a10*(a03-a01*x1-a02*x2)/a00-a12*x2)/a11)/a22-x2);shuchu(x);while(d00.5e-5&&d10.5e-5&&d20.5e-5);system("pause");return0;实验结果与分析1.列主元素消元法4_000000_000000_00000032X4.000000±.0000002.000000.000000_00

15、0000_00000013.0000008.000000:L4000000_000000_000000_0000004.0000001.0000000.00000010000002.000000250000013-000000a.000000*?5000004.0000001.0000000.0000001.0000002.0000002-500000±3.0000008.0000007.500000_000000=2-000000&c3=3-000000LiII_jI_II1_l2.完全组元素消元法0.S359.143T"59.175.291-6.130-1246

16、.7811.2952112112123400.0380003Mm59.170000-6.1300S05.291000-l.mm2.mm46.7800009.00000011.2000005.0000002.0000001.0000002.0000001.00000010000001.0000002.0000002.mml.mm3mm4.0000000.00000059.1400000.03Q0Q03.mm1.00000059.1700090.0000005.294110-0.6890432.103&5252.9131100.00000011.1954354.5434561.847819

17、-8.0045650.S0000O0.9989850.8985460.966182-0.0010152.mml.mm3mm4.00000059.1400000.0300003.0000001.00000059.1700000.000000i仁19E43E4.E434E6仁847819-8.0045658.S000005.294110-0.6890432.103&5252.9131108.S000000.9989850.8985460.966182-0.0010152.0000001.00000030000004.0000000.00000059.1400000.0360803.8000

18、061.00090859.1700090.60600011.1954354.5434561.847819-8.004565S.S0Q0000.000S0Q-2.8375581.22985456.6983170.0000000.0000000.4931260.8012?80.7132452.0000001.00000030000004.0000000.000000B9.1400000.03S0603MM1.00000059.1700000.00Q00011.1954354.5434561.847819-8.0045650.0000000.000000-2.8375581.22985456.698

19、31?0.4931260.8012980.7132452.mml.mm3.mm4.0000000.00000059.1400000.0300003.0000001.00000059.1700000.00000011.1954354.5434561.847819-8.0045650.6080008.000608-2.8375581.22985456.6983170.S0000O0.000S001.01502910.5665912.mm1.000Q0Q3mm4.0000000.000000xl=3.844784x2=1.607250x3=-15.469421x4=10.4101363LU分解法4215-287210-?4836-71261120-3L矩