高斯-赛德尔迭代法解线性方程组

1、数值分析实验五班级：10信计二班 学号：59 姓名：王志桃 分数： .实验名称高斯-赛德尔迭代法解线性方程组实验目的1. 学会利用高斯赛德尔方法解线性方程组2. 明白迭代法的原理3. 对于大型稀疏矩阵方程组适用于迭代法比较简单三. 实验内容利用Gauss-Seidel迭代法求解下列方程组8x1 -3x2 2x3 = 20« 4x1 +11 x2 -1 x3 = 33 , 其中取 x(°)= 0。6x1 +3x2 +12x3 =36四、算法描述由Jacobi迭代法中，每一次的迭代只用到前一次的迭代值，若每一次迭代充分利用当前最新的迭代值，即在计算第i个分量为*刊时，用最新分量

2、xk*), x2(k扣xi-1(z)代替Gauss-Seidel 迭代法。(初始向量)，旧分量x1(k) , x2(k) xik),就得到所谓解方程组的其迭代格式为(0),(0)(0)(0)、Tx =(x ，*2 ，' ' '，xn )(i 1)1( k 1)l(k).xi=(bi£ aijxj Z aijxj ) (k=0,1,2,；i = 0,1,2,n)aiij 1j =t 1或者写为x：k *)= x(k) + 皈 (k = k = 0,1,2,；i = 0,1,2, n)(i +)1寸(k+)寸(k)x i =( bi a § X j a

3、§ X j )aiij ztjd=+五、编码#include#include<stdio.h><stdlib.h>#include#include<conio.h><math.h>#define MAX_n 100#define PRECISION0.0000001#define MAX_Number1000void VectorInput(float x,int n) /输入初始向量int i;for(i=1;i<=n;+i)printf("x%d=",i);scanf("%f",&

4、;xi);void MatrixInput(float AMAX_n,int m,int n) /输入增广矩阵int i, j;printf("n=Begin input Matrix elements=n");for(i=1;i<=m;+i)printf("Input_Line %d : ",i);for(j=1;j<=n;+j)scanf("%f",&Aij);void VectorOutput(float x,int n) /输出向量int i;for(i=1;i<=n;+i)printf("

5、nx%d=%f",i,xi);判断是否在规定精度内 int IsSatisfyPricision(float x1,float x2,int n) /int i;for(i=1;i<=n;+i)if(fabs(x1i-x2i)>PRECISION) return 1;return 0;int Jacobi_(float AMAX_n,float x,int n) /具体计算(float x_formerMAX_n;int i,j,k;printf("nInput vector x0:n");VectorInput(x,n);k=0;do(for(i=1

6、;i<=n;+i)(printf("nx%d=%f",i,xi);x_formeri=xi;printf("n");for(i=1;i<=n;+i)(xi=Ain+1;for(j=1;j<=n;+j)if(j!=i) xi-=Aij*xj;if(fabs(Aii)>PRECISION)xi/=Aii;elsereturn 1;+k;while(IsSatisfyPricision(x,x_former,n) && k<MAX_Number);if(k>=MAX_Number)return 1;else(

7、printf("nG-S %d times!",k);return 0;int main() / 主函数(int n;float AMAX_nMAX_n,xMAX_n;printf("nInput n=");scanf("%d”,&n);if(n>=MAX_n-1)(printf("n007n must <%d!”,MAX_n);exit(0);MatrixInput(A,n,n+1);if(Jacobi_(A,x,n)printf("nG-S Failed!");else(printf(&qu

8、ot;nOutput Solution:");VectorOutput(x,n);printf("nn007Press any key to quit!n"); getch();F:窝用文档Debugshiyanwu.exe"Input n=3=Begin input Matrix elenents=Input_Line 1：8-3220InputJLine 2：411-133Input_Line 3：631236Input uector x0：xU=0x2=0x3J=0xl=0.000000x2)-0.000000x31=0.000000xtl=2.5

9、00000x2J=2.090909x31=1.227273xtl=2.977273x2=2.028925x31=1.004132xU=3.009814x(21-1.996807xC31-0.995891xU=2.999830x21=1.999689x3=l.000163xl=2.999842x2J=2.000072x31=1.000061xU =3.000012x2=2.000001x31=0.999994xtl=3.000002-:用文尚* Debug shiyansu. exeX【11=3.000002H2 J=L999999b<(3 J=0-959999:(1 J-3 .00000B U23=2.000000 t<3 1=1.000000Mil "3.000000 21=2.000000 k【3 】=1 - 000000G-S 10 t imes t Output SoIu.t