北航研究生数值分析A作业三

《数值分析（A）》计算实习题目三一、题目关于x，y，t，u，v，w的下列方程组0.5cost+u+v+w-x=2.67t+0.5sinu+v+w-y=1.070.5t+u+cosv+w-x=3.74t+0.5u+v+sinw-y=0.79以及关于z，t，u的下列二维数表确定了一个二元函数z=f(x，y)。

1．试用数值方法求出f(x，y)在区域D={(x,y)︱0≤x≤0.8,0.5≤y≤1.5}上的一个近似表达式,0(,)kr srs r s p x y c x y==∑要求p(x,y)一最小的k 值达到以下的精度102027((,)(,))10iji j i j f x yp x y σ-===-≤∑∑其中x i =0.08i ，y j =0.5+0.05j 。

2．计算f(x i *,y j *),p(x i *,y j *)(i=1,2,…,8;j=1,2,…,5)的值，以观察p(x,y)逼近f(x,y)的效果，其中x i *=0.1i,y j *=0.5+0.2j 。

二、算法设计方案1.将0.08(0,1,,10)i x i i *== 和0.50.05(0,1,,20)j y j j *=+= 代入非线性方程组中，用牛顿法解出i t 和j u ；2.以采取分片二次插值，选择（m ，n ）满足,2322m i m h h t t t m -<≤+≤≤ ,2322n j n u u u n ττ-<≤+≤≤如果12ih t t ≤+或42ih t t >-，则m=1或4；如果12ju u τ≤+或42ju u τ>-，则n=1或n=4。

选择(,)(1,,1;1,,1)k r t u km m m r n n n =-+=-+为插值节点，相应的Lagrange形式的插值多项式为),()(~)(),(111122r k r m m k n n r k u t f u l t l u t p ∑∑+-=+-==其中11()m w k w m k ww kt t l t t t +=-≠-=-∏(k=m-1, m, m+1)∏+≠-=--=11)(~n rw n w wr w r y y y y u l (r=n-1, n, n+1)并将i t 和j u 代入22(,)p t u ，便得到了数表,,(,)i j i j x y f x y 。

3.进行曲面拟和系数矩阵[]rs c *=C ，11()()TTT--=C B B B U G G G其中0011101011[()]1kk r i k x x x x x x x ϕ⎡⎤⎢⎥⎢⎥==⎢⎥⎢⎥⎢⎥⎣⎦B，0011101011[()]1kk s j k y y y y G y y y ψ⎡⎤⎢⎥⎢⎥==⎢⎥⎢⎥⎢⎥⎣⎦[(,)]i j f x y =Uk 从0逐渐增大，直到710σ-≤，便得到了要求精度的系数rs c 。

4.由前面得到的函数关系，根据重新取值的x ，y 可以分别得到新的数表，比较两组数据观察逼近效果三、全部源程#include "stdio.h"#include "stdafx.h"#include "math.h"//高斯列主元法解线性方程组void gauss(double a[4][4],double b[4],double x[4]) {intk,i,j,ik;double mik,temp,maxa,sum,tempb;for (k=0;k<3;k++){temp=fabs(a[k][k]);for(i=k;i<4;i++)if (fabs(a[i][k])>temp){ik=i;maxa=fabs(a[i][k]);}if(ik!=k){for (j=k;j<4;j++){temp=a[k][j];a[k][j]=a[ik][j];a[ik][j]=temp;}tempb=b[k];b[k]=b[ik];b[ik]=tempb;}for (i=k+1;i<4;i++){mik=a[i][k]/a[k][k];for (j=k+1;j<4;j++)a[i][j]=a[i][j]-mik*a[k][j];b[i]=b[i]-mik*b[k];}}x[3]=b[3]/a[3][3];for (k=2;k>=0;k--){sum=0;for (j=k+1;j<4;j++){sum+=a[k][j]*x[j];x[k]=(b[k]-sum)/a[k][k];}}}//求无穷范数double FindMax(double Arr[4]){inti;double MaxVal;MaxVal=fabs(Arr[0]);for (i=0;i<4;i++)if(fabs(Arr[i])>MaxVal)MaxVal=fabs(Arr[i]);return (MaxVal);}void newton(double x, double y,doubletuvw[4])//牛顿法解非线性方程组{double A[4][4],b[4],var[4]={1,2,1,2},D_var[4],MaxVar,MaxD_Var;inti;while(1){A[0][0]=-0.5*sin(var[0]);A[0][1]=1;A[0][2]=1;A[0][3]=1;A[1][0]=1;A[1][1]=0.5*cos(var[1]);A[1][2]=1;A[1][3]=1;A[2][0]=0.5;A[2][1]=1;A[2][2]=-sin(var[2]);A[2][3]=1;A[3][0]=1;A[3][1]=0.5;A[3][2]=1;A[3][3]=cos(var[3]);//系数矩阵（求导数）b[0]=-(0.5*cos(var[0])+var[1]+var[2]+var[3]-x-2.67);b[1]=-(var[0]+0.5*sin(var[1])+var[2]+var[3]-y-1.07);b[2]=-(0.5*var[0]+var[1]+cos(var[2])+var[3]-x-3.74);b[3]=-(var[0]+0.5*var[1]+var[2]+sin(var[3])-y-0.79);gauss(A,b,D_var);//高斯列主元法解线性方程组MaxVar=FindMax(var);MaxD_Var=FindMax(D_var);if ((MaxD_Var/MaxV ar)<(1e-12)) //精度判断{for (i=0;i<4;i++)tuvw[i]=var[i];break;}for (i=0;i<4;i++)var[i]=var[i]+D_var[i];}}//拉格朗日插值系数Lkdouble lagrange_t(double x,inti,int k){int m;double value=1,t[6]={0,0.2,0.4,0.6,0.8,1.0};for (m=i-1;m<=i+1;m++)if(m!=k)value=value*(x-t[m])/(t[k]-t[m]);return value;}//拉格朗日插值系数Lrdouble lagrange_u(double y,intj,int r){int m;double value=1;double u[6]={0,0.4,0.8,1.2,1.6,2};for (m=j-1;m<=j+1;m++)if(m!=r)value=value*(y-u[m])/(u[r]-u[m]);return value;}//二元二次分片代数插值double interpolation(double tt,doubleuu){inti,j,k,r;double zvalue;double z[6][6]={ {-0.5,-0.34,0.14,0.94,2.06,3.5},{-0.42,-0.5,-0.26,0.3,1.18,2.38},{-0.18,-0.5,-0.5,-0.18,0.46, 1.42},{0.22,-0.34,-0.58,-0.5,-0.1,0.62},{0.78,-0.02,-0.5,-0.66,-0.5,-0.02},{1.5,0.46,-0.26,-0.66,-0.74,-0.5}};switch (int(10*tt)){case 0:case 1:case 2:i=1;break;case 3:case 4:i=2;break;case 5:case 6:i=3;break;case 7:case 8:case 9:i=4;break;default: printf("error t\n");}switch (int(5*uu)){case 0:case 1:case 2:j=1;break;case 3:case 4:j=2;break;case 5:case 6:j=3;break;case 7:case 8:case 9:j=4;break;default:printf("error u\n");}zvalue=0;for (k=i-1;k<=i+1;k++)for (r=j-1;r<=j+1;r++)zvalue+=z[k][r]*lagrange_t(tt,i,k)*lagrange_u(uu,j,r);//拉格朗日插值系数Lk,Lr return zvalue;}//初始化矩阵Bvoid init_B(double B[][10],int k){double x[11],sum;inti,j,m;for (i=0;i<11;i++)x[i]=0.08*i;for (i=0;i<11;i++){B[i][0]=1;for (j=1;j<=k;j++){sum=1;for (m=1;m<=j;m++)sum=sum*x[i];B[i][j]=sum;}}}//初始化矩阵Gvoid init_G(double G[][10],int k) {inti,j,m;double y[21],sum;for (i=0;i<21;i++)y[i]=0.5+0.05*i;for (i=0;i<21;i++){G[i][0]=1;for (j=1;j<=k;j++){sum=1;for (m=1;m<=j;m++)sum=sum*y[i];G[i][j]=sum;}}}//求逆矩阵void inverse(double A[][10],int n) {double b[10][20],c[10][10],temp;inti,j,k;for (i=0;i<n;i++)for (j=0;j<2*n;j++)if(j<n)b[i][j]=A[i][j];else b[i][j]=0;for (i=0;i<n;i++)b[i][n+i]=1;for (k=0;k<n;k++){temp=b[k][k];i=k;while (b[k][k]==0){b[k][k]=b[i+1][k];i++;}if (i>k){b[i][k]=temp;for (j=0;j<k;j++){temp=b[k][j];b[k][j]=b[i][j];b[i][j]=temp;}for (j=k+1;j<2*n;j++){temp=b[k][j];b[k][j]=b[i][j];b[i][j]=temp;}}for (i=k+1;i<n;i++)for (j=2*n-1;j>=k;j--)b[i][j]=b[i][j]-b[i][k]*b[k][j]/b[k][k];for (j=2*n-1;j>=k;j--)b[k][j]=b[k][j]/b[k][k];}for (k=n-1;k>0;k--)for (i=0;i<k;i++)for(j=2*n-1;j>=k;j--)b[i][j]=b[i][j]-b[i][k]*b[k][j];for (i=0;i<n;i++)for (j=0;j<n;j++)c[i][j]=b[i][n+j];for (i=0;i<n;i++)for (j=0;j<n;j++)A[i][j]=c[i][j];}//最小二乘拟合曲面double surface_fitting(double x,doubley,double C[][10],intk,double Z[][21]) {double B[11][10],G[21][10],BTB[10][10],GTG[10][10],U[11][21],UU[21][21]; inti,j,m;double sum;init_B(B,k);init_G(G,k);//初始化矩阵B和Gfor (i=0;i<11;i++)for (j=0;j<21;j++)U[i][j]=Z[i][j];for (i=0;i<=k;i++)for (j=0;j<=k;j++){sum=0;for(m=0;m<11;m++)sum=sum+B[m][i]*B[m][j];BTB[i][j]=sum;}for (i=0;i<=k;i++)for (j=0;j<=k;j++){sum=0;for(m=0;m<21;m++)sum=sum+G[m][i]*G[m][j];GTG[i][j]=sum;}inverse(BTB,k+1);//求逆inverse(GTG,k+1);//求逆//求矩阵BT*Ufor (i=0;i<=k;i++)for (j=0;j<21;j++){sum=0;for(m=0;m<11;m++)sum+=B[m][i]*U[m][j];UU[i][j]=sum;}//求矩阵BT*U*Gfor (i=0;i<=k;i++)for (j=0;j<=k;j++){sum=0;for(m=0;m<21;m++)sum+=UU[i][m]*G[m][j];C[i][j]=sum;}//求矩阵BIB*BT*U*Gfor (i=0;i<=k;i++)for (j=0;j<=k;j++){sum=0;for(m=0;m<=k;m++)sum+=BTB[i][m]*C[m][j];UU[i][j]=sum;}//求矩阵BIB*BT*U*G*GTGfor (i=0;i<=k;i++)for (j=0;j<=k;j++){sum=0;for(m=0;m<=k;m++)sum+=UU[i][m]*GTG[m][j];C[i][j]=sum;}//求和，得出系数Cdouble result=0,r,s;for (i=0;i<=k;i++){r=1;for (m=1;m<=i;m++)r=r*x;for (j=0;j<=k;j++){s=1;for (m=1;m<=j;m++)s=s*y;result+=C[i][j]*r*s;}}return result;}//主函数void main(){inti,j;double x[11],y[21],z[11][21],C[10][10],tuvw[4];for (i=0;i<11;i++)x[i]=0.08*i;for (j=0;j<21;j++)y[j]=0.5+0.05*j; //初始化x、yprintf("数表xi,yi,f(xi,yi)的值：\n");for (i=0;i<11;i++)for (j=0;j<21;j++){newton(x[i],y[j],tuvw); //牛顿解非线性方程组z[i][j]=interpolation(tuvw[0],tuvw[1]);//二元二次插值printf("x[%d]=%2.2f,y[%d]=%2.2f,f(x%d,y%d)=%2.12e\n",i,x[i],j,y[j],i,j,z[i][j]);}//去曲面拟合double delta=0;int k=0;printf("\n");printf("选择过程中的k、σ：\n");while(1){delta=0;for (i=0;i<11;i++)for (j=0;j<21;j++)delta+=(z[i][j]-surface_fitting(x[i],y[j],C,k,z))*(z[i][j]-surface_fitting(x[i],y[j],C,k,z));printf("k=%d,σ=%2.12e\n",k,delta);if (delta<=1e-7)break;k++;}printf("\n达到精度要求时\nk=%d,σ=%2.12e\n",k,delta);printf("Crs：\n");for (i=0;i<=k;i++){for (j=0;j<=k;j++)printf("C[%d,%d]=%2.12e\n",i,j,C[i][j]);}//输出数表x*,y*,f*,p*double xx[8],yy[5];for (i=0;i<8;i++)xx[i]=0.1*(i+1);for(j=0;j<5;j++)yy[j]=0.5+0.2*(j+1);printf("\n数表x*i,y*j,f(x*i,y*j),p(x*i,y*j)：\n");for (i=0;i<8;i++)for (j=0;j<5;j++){newton(xx[i],yy[j],tuvw);printf("x*[%d]=%2.2f,y*[%d]=%2.2f,f*(x%d,y%d)=%2.12e,p*(x%d,y%d)=%2.12e\n", i+1,xx[i],j+1,yy[j],i+1,j+1,interpolation(tuvw[0],tuvw[1]),i+1,j+1,surface_fitting(xx[i], yy[j],C,k,z));}}四、程序运行结果：数表xi,yi,f(xi,yi)的值：x[0]=0.00, y[0]=0.50, f(x0,y0)=4.465040184799e-001x[0]=0.00, y[1]=0.55, f(x0,y1)=3.246832629274e-001x[0]=0.00, y[2]=0.60, f(x0,y2)=2.101596866825e-001x[0]=0.00, y[3]=0.65, f(x0,y3)=1.030436083159e-001x[0]=0.00, y[4]=0.70, f(x0,y4)=3.401895562659e-003x[0]=0.00, y[5]=0.75, f(x0,y5)=-8.873581363801e-002x[0]=0.00, y[6]=0.80, f(x0,y6)=-1.733716327497e-001x[0]=0.00, y[7]=0.85, f(x0,y7)=-2.505346114666e-001x[0]=0.00, y[8]=0.90, f(x0,y8)=-3.202765063876e-001x[0]=0.00, y[9]=0.95, f(x0,y9)=-3.826680661097e-001x[0]=0.00, y[10]=1.00, f(x0,y10)=-4.377957667384e-001x[0]=0.00, y[11]=1.05, f(x0,y11)=-4.857589414438e-001x[0]=0.00, y[12]=1.10, f(x0,y12)=-5.266672548835e-001x[0]=0.00, y[13]=1.15, f(x0,y13)=-5.606384797965e-001x[0]=0.00, y[14]=1.20, f(x0,y14)=-5.877965387677e-001x[0]=0.00, y[15]=1.25, f(x0,y15)=-6.0826********e-001x[0]=0.00, y[16]=1.30, f(x0,y16)=-6.221894528764e-001x[0]=0.00, y[17]=1.35, f(x0,y17)=-6.296883781856e-001x[0]=0.00, y[18]=1.40, f(x0,y18)=-6.308997600028e-001x[0]=0.00, y[19]=1.45, f(x0,y19)=-6.259561525454e-001x[0]=0.00, y[20]=1.50, f(x0,y20)=-6.149885466094e-001x[1]=0.08, y[0]=0.50, f(x1,y0)=6.380152265102e-001x[1]=0.08, y[1]=0.55, f(x1,y1)=5.0661********e-001x[1]=0.08, y[2]=0.60, f(x1,y2)=3.821763692772e-001x[1]=0.08, y[3]=0.65, f(x1,y3)=2.648634911536e-001x[1]=0.08, y[4]=0.70, f(x1,y4)=1.547802002848e-001x[1]=0.08, y[5]=0.75, f(x1,y5)=5.199268349093e-002x[1]=0.08, y[6]=0.80, f(x1,y6)=-4.346804020491e-002x[1]=0.08, y[7]=0.85, f(x1,y7)=-1.316010567885e-001x[1]=0.08, y[8]=0.90, f(x1,y8)=-2.124310883088e-001x[1]=0.08, y[9]=0.95, f(x1,y9)=-2.860045510580e-001x[1]=0.08, y[10]=1.00, f(x1,y10)=-3.523860789794e-001x[1]=0.08, y[11]=1.05, f(x1,y11)=-4.116554565222e-001x[1]=0.08, y[12]=1.10, f(x1,y12)=-4.639049115188e-001x[1]=0.08, y[13]=1.15, f(x1,y13)=-5.092367247005e-001x[1]=0.08, y[14]=1.20, f(x1,y14)=-5.477611179623e-001x[1]=0.08, y[16]=1.30, f(x1,y16)=-6.048572588895e-001 x[1]=0.08, y[17]=1.35, f(x1,y17)=-6.236734213318e-001 x[1]=0.08, y[18]=1.40, f(x1,y18)=-6.361682484133e-001 x[1]=0.08, y[19]=1.45, f(x1,y19)=-6.424676566901e-001 x[1]=0.08, y[20]=1.50, f(x1,y20)=-6.426971026996e-001 x[2]=0.16, y[0]=0.50, f(x2,y0)=8.400813957652e-001x[2]=0.16, y[1]=0.55, f(x2,y1)=6.997641656726e-001x[2]=0.16, y[2]=0.60, f(x2,y2)=5.660614423514e-001x[2]=0.16, y[3]=0.65, f(x2,y3)=4.391716081175e-001x[2]=0.16, y[4]=0.70, f(x2,y4)=3.192421380408e-001x[2]=0.16, y[5]=0.75, f(x2,y5)=2.063761923874e-001x[2]=0.16, y[6]=0.80, f(x2,y6)=1.006385238914e-001x[2]=0.16, y[7]=0.85, f(x2,y7)=2.060740067835e-003x[2]=0.16, y[8]=0.90, f(x2,y8)=-8.935402476698e-002 x[2]=0.16, y[9]=0.95, f(x2,y9)=-1.736269688648e-001 x[2]=0.16, y[10]=1.00, f(x2,y10)=-2.507999561599e-001 x[2]=0.16, y[11]=1.05, f(x2,y11)=-3.209322694446e-001 x[2]=0.16, y[12]=1.10, f(x2,y12)=-3.840977350046e-001 x[2]=0.16, y[13]=1.15, f(x2,y13)=-4.403821754175e-001 x[2]=0.16, y[14]=1.20, f(x2,y14)=-4.898811523126e-001 x[2]=0.16, y[15]=1.25, f(x2,y15)=-5.326979655338e-001 x[2]=0.16, y[16]=1.30, f(x2,y16)=-5.689418792921e-001 x[2]=0.16, y[17]=1.35, f(x2,y17)=-5.987265495151e-001 x[2]=0.16, y[18]=1.40, f(x2,y18)=-6.221686297503e-001 x[2]=0.16, y[19]=1.45, f(x2,y19)=-6.393865356972e-001 x[2]=0.16, y[20]=1.50, f(x2,y20)=-6.504993507878e-001 x[3]=0.24, y[0]=0.50, f(x3,y0)=1.0515********e+000x[3]=0.24, y[1]=0.55, f(x3,y1)=9.029*********e-001x[3]=0.24, y[2]=0.60, f(x3,y2)=7.605802668593e-001x[3]=0.24, y[3]=0.65, f(x3,y3)=6.247151981455e-001x[3]=0.24, y[4]=0.70, f(x3,y4)=4.955197560009e-001x[3]=0.24, y[5]=0.75, f(x3,y5)=3.731340427746e-001x[3]=0.24, y[6]=0.80, f(x3,y6)=2.576567488723e-001x[3]=0.24, y[7]=0.85, f(x3,y7)=1.491505594102e-001x[3]=0.24, y[8]=0.90, f(x3,y8)=4.764698677337e-002x[3]=0.24, y[9]=0.95, f(x3,y9)=-4.684932320146e-002 x[3]=0.24, y[10]=1.00, f(x3,y10)=-1.343567603849e-001 x[3]=0.24, y[11]=1.05, f(x3,y11)=-2.149133449274e-001 x[3]=0.24, y[12]=1.10, f(x3,y12)=-2.885737006348e-001 x[3]=0.24, y[13]=1.15, f(x3,y13)=-3.554063647857e-001 x[3]=0.24, y[14]=1.20, f(x3,y14)=-4.154913964886e-001 x[3]=0.24, y[15]=1.25, f(x3,y15)=-4.689182499695e-001 x[3]=0.24, y[16]=1.30, f(x3,y16)=-5.157838831247e-001x[3]=0.24, y[18]=1.40, f(x3,y18)=-5.902469305629e-001 x[3]=0.24, y[19]=1.45, f(x3,y19)=-6.180615482412e-001 x[3]=0.24, y[20]=1.50, f(x3,y20)=-6.397468392579e-001 x[4]=0.32, y[0]=0.50, f(x4,y0)=1.271246751481e+000x[4]=0.32, y[1]=0.55, f(x4,y1)=1.115002018146e+000x[4]=0.32, y[2]=0.60, f(x4,y2)=9.646077272154e-001x[4]=0.32, y[3]=0.65, f(x4,y3)=8.203473694749e-001x[4]=0.32, y[4]=0.70, f(x4,y4)=6.824476781794e-001x[4]=0.32, y[5]=0.75, f(x4,y5)=5.510852085975e-001x[4]=0.32, y[6]=0.80, f(x4,y6)=4.263923859018e-001x[4]=0.32, y[7]=0.85, f(x4,y7)=3.084629956332e-001x[4]=0.32, y[8]=0.90, f(x4,y8)=1.973571296919e-001x[4]=0.32, y[9]=0.95, f(x4,y9)=9.310562085941e-002x[4]=0.32, y[10]=1.00, f(x4,y10)=-4.285992234035e-003 x[4]=0.32, y[11]=1.05, f(x4,y11)=-9.483392529689e-002 x[4]=0.32, y[12]=1.10, f(x4,y12)=-1.785729903640e-001 x[4]=0.32, y[13]=1.15, f(x4,y13)=-2.555537790546e-001 x[4]=0.32, y[14]=1.20, f(x4,y14)=-3.258401501575e-001 x[4]=0.32, y[15]=1.25, f(x4,y15)=-3.895069883634e-001 x[4]=0.32, y[16]=1.30, f(x4,y16)=-4.466382045995e-001 x[4]=0.32, y[17]=1.35, f(x4,y17)=-4.973249517677e-001 x[4]=0.32, y[18]=1.40, f(x4,y18)=-5.416640326994e-001 x[4]=0.32, y[19]=1.45, f(x4,y19)=-5.797564797951e-001 x[4]=0.32, y[20]=1.50, f(x4,y20)=-6.117062881476e-001 x[5]=0.40, y[0]=0.50, f(x5,y0)=1.498321052481e+000x[5]=0.40, y[1]=0.55, f(x5,y1)=1.334998632066e+000x[5]=0.40, y[2]=0.60, f(x5,y2)=1.177125123739e+000x[5]=0.40, y[3]=0.65, f(x5,y3)=1.025*********e+000x[5]=0.40, y[4]=0.70, f(x5,y4)=8.789600231743e-001x[5]=0.40, y[5]=0.75, f(x5,y5)=7.391451087035e-001x[5]=0.40, y[6]=0.80, f(x5,y6)=6.0574********e-001x[5]=0.40, y[7]=0.85, f(x5,y7)=4.788838610666e-001x[5]=0.40, y[8]=0.90, f(x5,y8)=3.586506258818e-001x[5]=0.40, y[9]=0.95, f(x5,y9)=2.451022361964e-001x[5]=0.40, y[10]=1.00, f(x5,y10)=1.382683509285e-001 x[5]=0.40, y[11]=1.05, f(x5,y11)=3.815486540699e-002 x[5]=0.40, y[12]=1.10, f(x5,y12)=-5.525282116814e-002 x[5]=0.40, y[13]=1.15, f(x5,y13)=-1.419868808137e-001 x[5]=0.40, y[14]=1.20, f(x5,y14)=-2.220944390959e-001 x[5]=0.40, y[15]=1.25, f(x5,y15)=-2.956352324598e-001 x[5]=0.40, y[16]=1.30, f(x5,y16)=-3.626795115028e-001 x[5]=0.40, y[17]=1.35, f(x5,y17)=-4.233061642240e-001 x[5]=0.40, y[18]=1.40, f(x5,y18)=-4.776010361325e-001x[5]=0.40, y[20]=1.50, f(x5,y20)=-5.675647436551e-001 x[6]=0.48, y[0]=0.50, f(x6,y0)=1.731892740382e+000x[6]=0.48, y[1]=0.55, f(x6,y1)=1.562034577208e+000x[6]=0.48, y[2]=0.60, f(x6,y2)=1.397216918208e+000x[6]=0.48, y[3]=0.65, f(x6,y3)=1.237801006739e+000x[6]=0.48, y[4]=0.70, f(x6,y4)=1.0840********e+000x[6]=0.48, y[5]=0.75, f(x6,y5)=9.363227723149e-001x[6]=0.48, y[6]=0.80, f(x6,y6)=7.947044490537e-001x[6]=0.48, y[7]=0.85, f(x6,y7)=6.593871980282e-001x[6]=0.48, y[8]=0.90, f(x6,y8)=5.304875868399e-001x[6]=0.48, y[9]=0.95, f(x6,y9)=4.080886854542e-001x[6]=0.48, y[10]=1.00, f(x6,y10)=2.922442012295e-001 x[6]=0.48, y[11]=1.05, f(x6,y11)=1.829822068536e-001 x[6]=0.48, y[12]=1.10, f(x6,y12)=8.030849403542e-002 x[6]=0.48, y[13]=1.15, f(x6,y13)=-1.579041305164e-002 x[6]=0.48, y[14]=1.20, f(x6,y14)=-1.0534********e-001 x[6]=0.48, y[15]=1.25, f(x6,y15)=-1.883980906096e-001 x[6]=0.48, y[16]=1.30, f(x6,y16)=-2.650071493189e-001 x[6]=0.48, y[17]=1.35, f(x6,y17)=-3.352378389040e-001 x[6]=0.48, y[18]=1.40, f(x6,y18)=-3.991645038868e-001 x[6]=0.48, y[19]=1.45, f(x6,y19)=-4.568681433016e-001 x[6]=0.48, y[20]=1.50, f(x6,y20)=-5.084349932782e-001 x[7]=0.56, y[0]=0.50, f(x7,y0)=1.971221786400e+000x[7]=0.56, y[1]=0.55, f(x7,y1)=1.795329599501e+000x[7]=0.56, y[2]=0.60, f(x7,y2)=1.624067113228e+000x[7]=0.56, y[3]=0.65, f(x7,y3)=1.457830582708e+000x[7]=0.56, y[4]=0.70, f(x7,y4)=1.296954649752e+000x[7]=0.56, y[5]=0.75, f(x7,y5)=1.141718105447e+000x[7]=0.56, y[6]=0.80, f(x7,y6)=9.923495333243e-001x[7]=0.56, y[7]=0.85, f(x7,y7)=8.490326633294e-001x[7]=0.56, y[8]=0.90, f(x7,y8)=7.119113522641e-001x[7]=0.56, y[9]=0.95, f(x7,y9)=5.810941589219e-001x[7]=0.56, y[10]=1.00, f(x7,y10)=4.566585132334e-001 x[7]=0.56, y[11]=1.05, f(x7,y11)=3.386544961394e-001 x[7]=0.56, y[12]=1.10, f(x7,y12)=2.271082557696e-001 x[7]=0.56, y[13]=1.15, f(x7,y13)=1.220250891932e-001 x[7]=0.56, y[14]=1.20, f(x7,y14)=2.339221963760e-002 x[7]=0.56, y[15]=1.25, f(x7,y15)=-6.881870197104e-002 x[7]=0.56, y[16]=1.30, f(x7,y16)=-1.546493442129e-001 x[7]=0.56, y[17]=1.35, f(x7,y17)=-2.341526664587e-001 x[7]=0.56, y[18]=1.40, f(x7,y18)=-3.0739********e-001 x[7]=0.56, y[19]=1.45, f(x7,y19)=-3.744348623481e-001 x[7]=0.56, y[20]=1.50, f(x7,y20)=-4.353605565359e-001x[8]=0.64, y[1]=0.55, f(x8,y1)=2.034201133607e+000x[8]=0.64, y[2]=0.60, f(x8,y2)=1.856955143619e+000x[8]=0.64, y[3]=0.65, f(x8,y3)=1.684358164161e+000x[8]=0.64, y[4]=0.70, f(x8,y4)=1.516776352400e+000x[8]=0.64, y[5]=0.75, f(x8,y5)=1.354519041151e+000x[8]=0.64, y[6]=0.80, f(x8,y6)=1.197844086673e+000x[8]=0.64, y[7]=0.85, f(x8,y7)=1.046963049419e+000x[8]=0.64, y[8]=0.90, f(x8,y8)=9.020*********e-001x[8]=0.64, y[9]=0.95, f(x8,y9)=7.632264776629e-001x[8]=0.64, y[10]=1.00, f(x8,y10)=6.306048219543e-001 x[8]=0.64, y[11]=1.05, f(x8,y11)=5.042528145972e-001 x[8]=0.64, y[12]=1.10, f(x8,y12)=3.842167155457e-001 x[8]=0.64, y[13]=1.15, f(x8,y13)=2.705204766410e-001 x[8]=0.64, y[14]=1.20, f(x8,y14)=1.631685723996e-001 x[8]=0.64, y[15]=1.25, f(x8,y15)=6.214855811676e-002 x[8]=0.64, y[16]=1.30, f(x8,y16)=-3.256661939682e-002 x[8]=0.64, y[17]=1.35, f(x8,y17)=-1.210165348444e-001 x[8]=0.64, y[18]=1.40, f(x8,y18)=-2.032513996228e-001 x[8]=0.64, y[19]=1.45, f(x8,y19)=-2.793303595584e-001 x[8]=0.64, y[20]=1.50, f(x8,y20)=-3.493199575400e-001 x[9]=0.72, y[0]=0.50, f(x9,y0)=2.464684222660e+000x[9]=0.72, y[1]=0.55, f(x9,y1)=2.278058979399e+000x[9]=0.72, y[2]=0.60, f(x9,y2)=2.0952********e+000x[9]=0.72, y[3]=0.65, f(x9,y3)=1.916718127997e+000x[9]=0.72, y[4]=0.70, f(x9,y4)=1.742854628776e+000x[9]=0.72, y[5]=0.75, f(x9,y5)=1.573998427334e+000x[9]=0.72, y[6]=0.80, f(x9,y6)=1.410434835231e+000x[9]=0.72, y[7]=0.85, f(x9,y7)=1.252401750608e+000x[9]=0.72, y[8]=0.90, f(x9,y8)=1.100094409628e+000x[9]=0.72, y[9]=0.95, f(x9,y9)=9.536698512613e-001x[9]=0.72, y[10]=1.00, f(x9,y10)=8.132510552489e-001 x[9]=0.72, y[11]=1.05, f(x9,y11)=6.789307429659e-001 x[9]=0.72, y[12]=1.10, f(x9,y12)=5.507748485043e-001 x[9]=0.72, y[13]=1.15, f(x9,y13)=4.288256769731e-001 x[9]=0.72, y[14]=1.20, f(x9,y14)=3.131047717398e-001 x[9]=0.72, y[15]=1.25, f(x9,y15)=2.036155140327e-001 x[9]=0.72, y[16]=1.30, f(x9,y16)=1.003454782409e-001 x[9]=0.72, y[17]=1.35, f(x9,y17)=3.268565186572e-003 x[9]=0.72, y[18]=1.40, f(x9,y18)=-8.765306591329e-002 x[9]=0.72, y[19]=1.45, f(x9,y19)=-1.724672478188e-001 x[9]=0.72, y[20]=1.50, f(x9,y20)=-2.512302207523e-001 x[10]=0.80, y[0]=0.50, f(x10,y0)=2.717811109469e+000 x[10]=0.80, y[1]=0.55, f(x10,y1)=2.526399501256e+000x[10]=0.80, y[3]=0.65, f(x10,y3)=2.154329377280e+000x[10]=0.80, y[4]=0.70, f(x10,y4)=1.974574556652e+000x[10]=0.80, y[5]=0.75, f(x10,y5)=1.799510579099e+000x[10]=0.80, y[6]=0.80, f(x10,y6)=1.629448220554e+000x[10]=0.80, y[7]=0.85, f(x10,y7)=1.464650043751e+000x[10]=0.80, y[8]=0.90, f(x10,y8)=1.305334967651e+000x[10]=0.80, y[9]=0.95, f(x10,y9)=1.151682621307e+000x[10]=0.80, y[10]=1.00, f(x10,y10)=1.003837419906e+000 x[10]=0.80, y[11]=1.05, f(x10,y11)=8.619123372279e-001 x[10]=0.80, y[12]=1.10, f(x10,y12)=7.259923711112e-001 x[10]=0.80, y[13]=1.15, f(x10,y13)=5.961377115201e-001 x[10]=0.80, y[14]=1.20, f(x10,y14)=4.723866279136e-001 x[10]=0.80, y[15]=1.25, f(x10,y15)=3.547580958979e-001 x[10]=0.80, y[16]=1.30, f(x10,y16)=2.432541841813e-001 x[10]=0.80, y[17]=1.35, f(x10,y17)=1.378622225247e-001 x[10]=0.80, y[18]=1.40, f(x10,y18)=3.855677032640e-002 x[10]=0.80, y[19]=1.45, f(x10,y19)=-5.469859593446e-002 x[10]=0.80, y[20]=1.50, f(x10,y20)=-1.419496597088e-001选择过程中的k、σ：k=0,σ=1.442880771836e+002k=1,σ=3.220908973634e+000k=2,σ=4.659960033243e-003k=3,σ=1.721175379293e-004k=4,σ=3.309534300922e-006k=5,σ=2.541512318326e-008达到精度要求时k=5,σ=2.541512318326e-008Crs：C[0,0]=2.021*********e+000C[0,1]=-3.668423846364e+000C[0,2]=7.092421418056e-001C[0,3]=8.486122535542e-001C[0,4]=-4.159009451978e-001C[0,5]=6.743269192521e-002C[1,0]=3.191954374357e+000C[1,1]=-7.413689485984e-001C[1,2]=-2.696557302843e+000C[1,3]=1.630583174760e+000C[1,4]=-4.844127955148e-001C[1,5]=6.0553********e-002C[2,0]=2.564098287839e-001C[2,1]=1.582655804232e+000C[2,2]=-4.694162700325e-001C[2,3]=-7.498418726027e-002C[2,4]=9.883782174438e-002C[2,5]=-2.036812738515e-002C[3,0]=-2.676476564375e-001C[3,1]=-7.394417682663e-001C[3,2]=1.096321268007e+000C[3,3]=-8.283654926345e-001C[3,4]=3.138025505468e-001C[3,5]=-4.814400518080e-002C[4,0]=2.151515221121e-001C[4,1]=-1.652099197963e-001C[4,2]=-1.012971356977e-001C[4,3]=2.739136226010e-001C[4,4]=-1.569928317913e-001C[4,5]=2.962164662313e-002C[5,0]=-5.501011264460e-002C[5,1]=1.381011343328e-001C[5,2]=-1.250693502516e-001C[5,3]=2.885186714411e-002C[5,4]=9.856276497885e-003C[5,5]=-3.877033908793e-003数表x*i,y*j,f(x*i,y*j),p(x*i,y*j)：x*[1]=0.10, y*[1]=0.70, f*(x1,y1)=1.947204079177e-001, p*(x1,y1)=1.947303552407e-001 x*[1]=0.10, y*[2]=0.90, f*(x1,y2)=-1.830370791887e-001, p*(x1,y2)=-1.830418407586e-001 x*[1]=0.10, y*[3]=1.10, f*(x1,y3)=-4.454976469148e-001, p*(x1,y3)=-4.455000487616e-001 x*[1]=0.10, y*[4]=1.30, f*(x1,y4)=-5.975667076413e-001, p*(x1,y4)=-5.975588666853e-001 x*[1]=0.10, y*[5]=1.50, f*(x1,y5)=-6.464595939011e-001, p*(x1,y5)=-6.464461277567e-001 x*[2]=0.20, y*[1]=0.70, f*(x2,y1)=4.0597********e-001, p*(x2,y1)=4.0598********e-001 x*[2]=0.20, y*[2]=0.90, f*(x2,y2)=-2.251595837462e-002, p*(x2,y2)=-2.252111976633e-002 x*[2]=0.20, y*[3]=1.10, f*(x2,y3)=-3.382208160396e-001, p*(x2,y3)=-3.382240227895e-001 x*[2]=0.20, y*[4]=1.30, f*(x2,y4)=-5.444378315219e-001, p*(x2,y4)=-5.444304561199e-001 x*[2]=0.20, y*[5]=1.50, f*(x2,y5)=-6.473613385679e-001, p*(x2,y5)=-6.473480147123e-001 x*[3]=0.30, y*[1]=0.70, f*(x3,y1)=6.347771951509e-001, p*(x3,y1)=6.347874560151e-001 x*[3]=0.30, y*[2]=0.90, f*(x3,y2)=1.588011688394e-001, p*(x3,y2)=1.587962910079e-001 x*[3]=0.30, y*[3]=1.10, f*(x3,y3)=-2.0736********e-001, p*(x3,y3)=-2.0736********e-001 x*[3]=0.30, y*[4]=1.30, f*(x3,y4)=-4.653579068978e-001, p*(x3,y4)=-4.653499224584e-001 x*[3]=0.30, y*[5]=1.50, f*(x3,y5)=-6.202709530749e-001, p*(x3,y5)=-6.202571262457e-001 x*[4]=0.40, y*[1]=0.70, f*(x4,y1)=8.789600231743e-001, p*(x4,y1)=8.789698723457e-001 x*[4]=0.40, y*[2]=0.90, f*(x4,y2)=3.586506258818e-001, p*(x4,y2)=3.586460358660e-001 x*[4]=0.40, y*[3]=1.10, f*(x4,y3)=-5.525282116814e-002, p*(x4,y3)=-5.525541795963e-002 x*[4]=0.40, y*[4]=1.30, f*(x4,y4)=-3.626795115028e-001, p*(x4,y4)=-3.626710548561e-001x*[4]=0.40, y*[5]=1.50, f*(x4,y5)=-5.675647436551e-001, p*(x4,y5)=-5.675505482659e-001 x*[5]=0.50, y*[1]=0.70, f*(x5,y1)=1.136610910158e+000, p*(x5,y1)=1.136620368290e+000 x*[5]=0.50, y*[2]=0.90, f*(x5,y2)=5.749803409475e-001, p*(x5,y2)=5.749758252283e-001 x*[5]=0.50, y*[3]=1.10, f*(x5,y3)=1.159923767920e-001, p*(x5,y3)=1.159893591603e-001 x*[5]=0.50, y*[4]=1.30, f*(x5,y4)=-2.385683040123e-001, p*(x5,y4)=-2.385604036565e-001 x*[5]=0.50, y*[5]=1.50, f*(x5,y5)=-4.914343936557e-001, p*(x5,y5)=-4.914208309330e-001 x*[6]=0.60, y*[1]=0.70, f*(x6,y1)=1.406041798905e+000, p*(x6,y1)=1.406050718527e+000 x*[6]=0.60, y*[2]=0.90, f*(x6,y2)=8.0594********e-001, p*(x6,y2)=8.0593********e-001 x*[6]=0.60, y*[3]=1.10, f*(x6,y3)=3.044292210453e-001, p*(x6,y3)=3.044259035644e-001 x*[6]=0.60, y*[4]=1.30, f*(x6,y4)=-9.501613009962e-002, p*(x6,y4)=-9.500892391062e-002 x*[6]=0.60, y*[5]=1.50, f*(x6,y5)=-3.939023077456e-001, p*(x6,y5)=-3.938897085270e-001 x*[7]=0.70, y*[1]=0.70, f*(x7,y1)=1.685783515309e+000, p*(x7,y1)=1.685791278477e+000 x*[7]=0.70, y*[2]=0.90, f*(x7,y2)=1.049881153064e+000, p*(x7,y2)=1.049877642456e+000 x*[7]=0.70, y*[3]=1.10, f*(x7,y3)=5.082937839397e-001, p*(x7,y3)=5.082911724749e-001 x*[7]=0.70, y*[4]=1.30, f*(x7,y4)=6.614879670648e-002, p*(x7,y4)=6.615638110524e-002 x*[7]=0.70, y*[5]=1.50, f*(x7,y5)=-2.768343417776e-001, p*(x7,y5)=-2.768218193171e-001 x*[8]=0.80, y*[1]=0.70, f*(x8,y1)=1.974574556652e+000, p*(x8,y1)=1.974581369620e+000 x*[8]=0.80, y*[2]=0.90, f*(x8,y2)=1.305334967651e+000, p*(x8,y2)=1.305331820825e+000 x*[8]=0.80, y*[3]=1.10, f*(x8,y3)=7.259923711112e-001, p*(x8,y3)=7.259895225092e-001 x*[8]=0.80, y*[4]=1.30, f*(x8,y4)=2.432541841813e-001, p*(x8,y4)=2.432608163707e-001 x*[8]=0.80, y*[5]=1.50, f*(x8,y5)=-1.419496597088e-001, p*(x8,y5)=-1.419384254301e-001 Press any key to continue。