计算方法C语言编程

计算方法C语言编程第二章2已知方程在区间[1，2]内有一根，试问用二分法求根，使其具有5位有效数字至少应二分多少次？【程序设计】#include<math.h>main(){intn=0;floatx1=1.0,x2=2.0,x=1.0,x0;do{x0=x;x=(x1+x2)/2;n++;if(x*x*x+x-4>0)x2=x;elsex1=x;}while(fabs(x-x0)>0.00005);printf("N=%d\n",n);}〖运行结果〗N=154用迭代法求的正根，要求准确到小数点后第5位。【程序设计】#include<math.h>main(){floatx0,x=1.5,y;do{y=(log(x+0.2))/5;x0=x;x=exp(y);}while(fabs(x-x0)>0.000005);printf("X=%f\n",x);}-〖运行结果〗x=1.0447639用牛顿法求方程在x0=2附近的根，要求准确到小数点后第3位。【程序设计】#include<math.h>main(){floatx=2.0,x0;do{x0=x;x=x-(x*x*x-3*x-1)/(3*x*x-3);}while(fabs(x-x0)>0.0005);printf("X=%f\n",x);}〖运行结果〗x=1.87938511.分别用单点和双点弦截法求方程-x-1=0在[1,1.5]内的根。要求|xn+1-xn|<=0.000005【程序设计】#include<math.h>floatf(floatx){floatf;f=x*x*x-x-1;returnf;}floatg(floatx1,floatx2){floatg;g=(x1*f(x2)-x2*f(x1))/(f(x2)-f(x1));returng;}main(){floatx1=1.0,x2=1.5,x,x0;x=x1;do{x0=x;x=g(x1,x2);if(f(x)>0)x2=x;elsex1=x;}while(fabs(x-x0)>0.000005);printf("X=%f\n",x);}〖运行结果〗x=1.324717第三章1.分别用列主元素消去法求解下列方程组.(计算取4位小数).【程序设计】#definen4main(){floata[n][n]={1.1161,0.1254,0.1397,0.1490,0.1582,1.1675,0.1768,0.1871,0.1968,0.2071,1.2168,0.2271,0.2368,0.2471,0.2568,1.2671},b[n]={1.5471,1.6471,1.7471,1.8471},k,x[n+1],y[n+1];inti,j,t;y[n]=0.0;for(i=0;i<n;i++)for(j=i+1;j<n;j++){k=(-1)*a[j][i]/a[i][i];a[j][i]=0;for(t=i+1;t<n;t++)a[j][t]=a[i][t]*k+a[j][t];b[j]=b[i]*k+b[j];}for(i=n-1;i>=0;i--){for(j=n-2;j>=0;j--){if(a[i][j]!=0)b[i]=b[i]-y[j+1]*a[i][j+1];elsebreak;}y[i]=b[i]/a[i][j+1];x[i+1]=y[i];}for(i=1;i<=n;i++)printf("X%d=%f\n",i,x[i]);}〖运行结果〗x1=1.040584x2=0.986957x3=0.935052x4=0.8812979设有方程组取初始向量,分别用雅可比迭代法与赛德尔迭代法求解,要求时迭代终止.【程序设计】#include<math.h>main()/*赛德尔*/{floatx1[100],x2[100],x3[100],y;intk=0;x1[0]=-3;x2[0]=1;x3[0]=1;do{x1[k+1]=((-2)*x2[k]-x3[k]-12)/5；x2[k+1]=(x1[k+1]-2*x3[k]+20)/4;x3[k+1]=((-2)*x1[k+1]+3*x2[k+1]+3)/10;if(fabs(x3[k+1]-x3[k])>fabs(x2[k+1]-x2[k]))y=x3[k+1]-x3[k];elseif(fabs(x2[k+1]-x2[k])>fabs(x1[k+1]-x1[k]))y=x2[k+1]-x2[k];elsey=x1[k+1]-x1[k];k++;}while(fabs(y)>0.001);printf("x1=%f\nx2=%f\nx3=%f\n",x1[k],x2[k],x3[k]);printf("K=%d\n",k);}〖运行结果〗x1=-3.999974x2=3.000043x3=2.000008k=7#include<math.h>main()/*雅可比*/{floatx1[100],x2[100],x3[100],y;intk=0;x1[0]=-3;x2[0]=1;x3[0]=1;do{x1[k+1]=((-2)*x2[k]-x3[k]-12)/5;x2[k+1]=(x1[k+1]-2*x3[k]+20)/4; x3[k+1]=((-2)*x1[k+1]+3*x2[k+1]+3)/10;if(fabs(x3[k+1]-x3[k])>fabs(x2[k+1]-x2[k]))y=x3[k+1]-x3[k];elseif(fabs(x2[k+1]-x2[k])>fabs(x1[k+1]-x1[k]))y=x2[k+1]-x2[k];elsey=x1[k+1]-x1[k];k++;}while(fabs(y)>0.001);printf("x1=%f\nx2=%f\nx3=%f\n",x1[k],x2[k],x3[k]);printf("K=%d\n",k);}〖运行结果〗x1=-4.000152x2=2.999648x3=2.000160k=1310设有方程组证明解此方程组的雅可比迭代法收敛,而相应的赛德尔迭代法发散.取初始向量,用雅可比迭代法求解,要求迭代三次.【程序设计】main(){floatx1[10],x2[10],x3[10];intk;x1[0]=0;x2[0]=0;x3[0]=0;for(k=0;k<=2;k++){x1[k+1]=1-2*x2[k]+2*x3[k];x2[k+1]=3-x1[k]-x3[k];x3[k+1]=5-2*x1[k]-2*x2[k];}printf("x1=%f\nx2=%f\nx3=%f\n",x1[k],x2[k],x3[k]);}〖运行结果〗x1=1.000000x2=1.000000x3=1.00000011设有方程组其等价形式为证明解等价方程组的简单迭代法发散,而赛德尔法收敛取初始向量,用赛德尔迭代法求解,要求迭代四次【程序设计】main(){floatx1[10],x2[10],x3[10];intk;x1[0]=0;x2[0]=0;x3[0]=0;for(k=0;k<=3;k++){x1[k+1]=(x2[k]+x3[k])/2;x2[k+1]=(x1[k+1]+1)/2;x3[k+1]=x1[k+1];}printf("x1=%f\nx2=%f\nx3=%f\n",x1[k],x2[k],x3[k]);}〖运行结果〗x1=0.578125x2=0.789062x3=0.578215第五章x1.12751.15031.17351.1972Y=f(x)0.11910.139540.159320.179031.已知函数表：应用拉格朗日插值公式计算f(1.1300)的近似值。（计算取4位小数）【程序设计】floatfun(float,float,float,int);floatx[4]={1.1275,1.1503,1.1735,1.1972};floaty[4]={0.1191,0.13954,0.15932,0.17903};voidmain(){intk=0;floatsum=0;for(k=0;k<=3;k++)sum+=fun(x[k],y[k],1.1300,k);printf("%5.4f",sum);}floatfun(floata,floatb,floatm,inti){floatshang=1.0,chu=1.0,shu;intj;loop:for(j=0;j<=3;j++)if(i==j)continue;else{shang*=(m-x[j]);chu*=(a-x[j]);}shu=shang*b/chu;returnshu;}〖运行结果〗0.1214X1.6151.6341.7021.8281.921Y=f(x)2.414502.464592.652713.030353.340667.利用函数表造出差商表，并利用牛顿插值公式计算f(x)在x=1.682,1.813处的近似值.（计算取5位小数）.【程序设计】doublef(doublex[5],doubley[25],double(c){doublez;inti,j,k;for(i=1;i<5;i++) {for(j=4;i>=I;j--)y[j]=(y[j]-y[j-1]/(x[j]-x[j-i]);z=y[4];for(k=3;k>=0;k--)z=z*(c-x[k])+y[k];returnz;}}main(){doublem[5]={1.615,1.634,1.072,1.828,1.921},n[5]={2.41450,2.46459,2.65271,3.03035,3.34066}a=1.682,b=1.813,c,d;c=f(m,n,a);d=f(m,n,b);printf(“c=%805f\nd=%8.5f\n,c,d);}〖运行结果〗c=2.059592d=2.98376第七章xk1.82.02.22.42.6F(xk)3.120144.426596.042418.0301410.466752已知函数f(x)的下列数据:用柯特斯公式计算积分.(计算取5位小数)【程序设计】main(){doubley[5]={3.12014,4.42659,6.04241,8.03014,10.46675},m,n,a=1.8,b=2.6;n=b-a;m=(b-a)/90*(7*y[0]+32*y[1]+12*y[2]+32*y[3]+7*y[4]);printf(“m=%7.5f”.m);}〖运行结果〗m=5.03279xk00.10.20.30.40.5f(xk)11.0049711.0195361.0426681.0727071.107432xk0.60.70.80.91.0f(xk)1.1441571.7598591.2113071.2352111.2483757已知函数f(x)的下列数据:用复化梯形公式和复化辛浦生公式计算积分.(计算取6位小数)注意:本题f(x)=cosx+sin²x,,积分真值I=1.11414677.【程序设计】main(){doubley[11]={1,1.004971,1.019536,1.042668,1.072707,1.107432,1.144157,1.759859,1.211307,1.235211,1.248375};doubleI,h,m=0,n=10.0,a=0,b=1.0,c=0;intI;h=(b-a)/n;for(i=1;i<=9;i++){m=m+y[2i-1];c=c+y2i};}I=h/6*(y[0]+4*m+2*c+y[10]);Printf(“I=%8.6f,”I);}〖运行结果〗I=1.11414511用龙贝格法计算积分I=,要求误差不超过(计算取6位小数).【程序设计】#include<math.h>doublef1(doublex)/*龙贝格:*/{doubley;y=exp(-x);returny;}doublef2(doublea,doubleb,doublec){intk=0;doubleh1,h2,t1,t2,s1=0,s2=0,c1=0,c2,r1=0,r2=0,max;doubles,x;h1=b-a;t1=h1*(f(a)+f(b))/2;while(1){h2=h1/2;x=a+h2;s=0;k++;}do{s=s+f(x);x=x+h1;}while(x<b)t2=1/2*t1+h1/2*s;if(k>=1)s2=(4*t2-t1)/3;if(k>=2)c2=(16*s2-s1)/15;if(k>=3)r2=(64*c2-c1)/63;if(k>=4){max=fabs(r2-r1);if(max<c)break;}h1=h2;t1=t2;s1=s2;c1=c2;r1=r2;}returnr2}main(){doublem,I,n=0,d=1.0,q=0.00001;m=srp(3.14);I=2/m*f2(n,d,q);Printf(“I=%9.6f“,I);}〖运行结果〗I=0.713271第八章2用梯形法与欧拉预估-校正法求解初值问题:取步长h=0.1,并与准确解y=-x-1+2相比较.(计算取6位小数).【程序设计】#include<math.h>doublef1(doublen,doublem){doublet;t=n+m；returnt;}voidf2(doublea,doubleb,doublex0,doubley0,doubleh){intk,N;doublex,y,Yp,Yc;N=(int)/((b-a)/h+o.5);for(k=0;k<N;k++){x=x0+h;Yp=y0+h*f1(x0,y0);Yc=y0+h*f1(x,Yp);y=(Yp+Yc)/2;printf(“y=%4.6\n”,y)；x0=x;y0=y;}}voiddoublef3(doublet1){doubled;d=-t1-1+2*exp(t1);printf(“d