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1、2008-2009 年（1）福建农林大学机电工程学院研究生数值分析课程试卷姓名：郑金凤专业年级：08级机械设计及理论学号：1081302 成绩: 注：以下试题编程语言任选。要求上机调试通过，提供原程序清单、计算结果。1、分别用列主元法和 Gauss-Seidel迭代法(要求 XnXn 110 4)求解方程组。10x! 2x2 x332x110x2 3x315x1 2x2 5x310(1 )列主元法#in clude#in cludeGaussdieda i(n, a,x)int n;double *a,*x;int i,j,k,r;double m;for(k=0;k n-1;k+)r=k;m
2、=fabs(ak*( n+1)+k);for(i=k+1;i m)r=i;m=fabs(ai*( n+1)+k);if(rk)for(j=k;j n +1;j+)m=ak*( n+1)+j;ak*( n+1)+j=ar*( n+1)+j;ar*(n +1)+j=m;for(i=k+1;i n ;i+)m=ai*( n+1)+k/ak*( n+1)+k;for(j=k+1;j =0;i-)m=ai*( n+1)+n;for( j=i+1;j(2) Gauss-Seidel 迭代法#i ncludestdio.h#i ncludemath.h#i ncludemalloc.hGaussSeidel
3、( n, a,b,x)int n;double *a,*b,*x;int i,j;double t,u,m=0,eps;while(1)eps=0;for(i=0;i n;i+)t=xi;for(j=0;jeps)eps=fabs(u_t);if(eps0.000001)x0=x1;x1=cos(x0);printf(%sn,解得);prin tf(x=%fn,x1);prin tf(n);X-0 押阴Press any key to Gontinue3、已知函数表如下，用二次插值求y(0.85)的近似值。Xi0.500.700.800.90Yi0.100.120.130.14#in clud
4、e math.hdouble n Ig3(double x,double y,i nt n double t) i nt i,j,k,m;double z,s;z=0.0;if(*1) return(z);if(n=1) z=yO;return(z);if(n=2)z=(y0*(t-x1)-y1*(t-x0)/(x0-x1); return(z);if (t=x n-2) k= n-3;m=n-1;elsek=1;m=n;while (m-k !=1)i=(k+m)/2;if (txi-1) m=i;else k=i;k=k-1;m=m-1;if(fabs(t-xk)fabs(t-xm) k=
5、k-1;else m=m+1;z=0.0;for(i=k;i=m;i+)s=1.0;for(j=k；j=m;j+)if(j!=i) s=s*(t-xj)/(xi-xj);z=z+s*yi;return(z); main () double t,z;static double x4=0.50,0.70,0.80,0.90;static double y4=0.10,0.12,0.13,0.14;prin tf(n);t=0.85;z=nl g3(x,y,5,t);prin tf(x=%f,f(x)=%f n,t,z);.850J-f=0.135000Press any key to contin
6、ue4、已知数据表如下，试求其最小二乘二次拟合多项式。Xi2345678910Yi654211234#in elude math.h#i nclude stdio.hvoid spir(double x,double y,i nt n, double a,i nt m,double dt)int i,j,k;double z,p,c,g,q,d1,d2,s20,t20,b20;for(i=0;in) m=n;if(m20) m=20;z=0.0;for(i=0;i=n-1;i+) z=z+xi/(1.0*n);bO=1.O;d1=1.O* n;p=0.0;c=0.0; for(i=0;i1)t
7、1=1.0;t0=-p; d2=0.0;c=0.0;g=0.0;for(i=0;i=n _1;i+)q=xi-z-p;d2=d2+q*q;c=c+yi*q;g=g+(xi-z)*q*q;C=c/d2;p=g/d2;q=d2/d1;d1=d2;a1=c*t1;a0=c*t0+a0;for( j=2;j=3)for(k=j-2;k=1;k-)sk=-p*tk+tk-1-q*bk;sO=-p*tO-q*bO;d2=0.0;c=0.0;g=0.0;for(i=0;i=0;k-)q=q*(xi-z)+sk; d2=d2+q*q;c=c+yi*q;g=g+(xi-z)*q*q;C=c/d2;p=g/d2;
8、q=d2/d1;d1=d2;aj=c*sj;tj=sj;for(k=j-1;k=0;k-)ak=c*sk+ak;bk=tk;tk=sk;main ()double x9=2,3,4,5,6,7,8,9,10,y9=6,5,4,2,1,1,2,3,4,a3,dt3;int i;spir(x,y,9,a,3,dt);for (i=0; i-1.545455e*00013.16666?e-0012=2.34S4S5e-001Press any key to continue5.用Romberg方法计算积分I e dx的近似值要求误差不超过10 -4。0#i nclude stdio.h#in clu
9、de math.h double romb(double a,double b,double eps,double(*f)()int m,n,i,k;double y10,h,ep,p,x,s,q;h=b-a;y0=h*(*f) (a)+(*f) (b)/2.0; m=1; n=1; ep=eps+1.0; while (ep=eps )&(m =9) p=0.0;for (i=0;i=n-1; i+)x=a+(i+0.5)*h;p=p+(*f) (x);p=(y0+h*p)/2.0;s=1.0;for (k=1; k=m; k+)s=4.0*s;q=(s*p-yk-1)/(s-1.0);yk
10、-1=p；p=q;ep=fabs(q-ym-1);m=m+1;ym-1=q;n=n+n;h=h/2.0;return(q);main ()double a,b,eps,t,rombf(double);a=0.0; b=1.0; eps=0.0001;t=romb(a,b,eps,rombf);prin tf(%sn,解得);prin tf(I=%f n ,t);prin tf(n);double rombf(double x)double y;y=exp(_x*x);return(y);FFWI=0.74&834mPresskey to continn6、用RK方法求初值问题的数值解。取步长
11、h=0.1 ,计算结果取8位小数。2xy1 x20x2y(0) 0#in clude #in clude double f(double x,double y) double rus;rus=x-(2*x*y)/(1+x*x);return(rus);void mai n()double x0=0; /范围上限double x1=2;/ 范围下限double h=0.1;/ 步长double n;/计算出的点的个数double k1,k2,k3,k4;double y0=0; /用于存放计算出的常微分方程数值解int i=0;int j;n=(x1-x0)/h);printf(%sn,结果如下
12、);for( j=O;jc0 =H _0干x:0=0 ZL , x0 0 _ 2,c0 =0 - 9 .V=0044039X0 =0 _ 5 -1,-2,-3,-4-5,-y=0 _11250029 _ S&?68ir=0 - -4?i_5X-4V=0-25756X3?sr-0 _343V8S5 =0 _ 3 7*5 000-40y=0-433?H23V-0 -5075-4X305/=0 _579SG359ir=0 - &5554B92y =0 _ 7355 W29SJ=0 _ 89?564if=S _ 90823300V=.001038B8ii-X . 09 8 27040V=A.20000032r v Re m t o coot zLrujie
福建农林大学2008数值分析报告考卷及问题详解
