北航数值分析大作业三

一、题目：关于x, y, t, u, v, w 的下列方程组0.5cos 2.670.5sin 1.070.5cos 3.740.5sin 0.79t u v w x t u v w y t u v w x t u v w y +++-=⎧⎪+++-=⎪⎨+++-=⎪⎪+++-=⎩1、试用数值方法求出f(x, y)在区域 {(,)|00.8,0.5 1.5}D x y x y =≤≤≤≤上的一个近似表达式,0(,)kr s rsr s p x y cx y ==∑要求(,)p x y 一最小的k 值达到以下的精度10202700((,)(,))10i j i j i j f x y p x y σ-===-≤∑∑其中，0.08,0.50.05i j x i y j ==+。

2、计算****(,),(,)i j i j f x y p x y （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，*i x =0.1i , *j y =0.5+0.2j 。

说明：1、用迭代方法求解非线性方程组时，要求近似解向量()k x 满足()(1)()12||||/||||10k k k x x x --∞∞-≤2、作二元插值时，要使用分片二次代数插值。

3、要由程序自动确定最小的k 值。

4、打印以下内容：●算法的设计方案。

●全部源程序（要求注明主程序和每个子程序的功能）。

●数表：,,i j x y (,)i j f x y （i = 0,1,2,…,10；j = 0,1,2,…,20）。

●选择过程的,k σ值。

●达到精度要求时的,k σ值以及(,)p x y 中的系数rs c （r = 0,1,…,k;s = 0,1,…,k ）。

●数表：**,,i j x y ****(,),(,)i j i j f x y p x y （i = 1, 2, ...,8；j = 1, 2, (5)。

5、采用f 型输出,,i j x y **,i j x y 的准确值，其余实型数采用e 型输出并且至少显示12位有效数字。

二、算法设计方案1.使用牛顿迭代法，对原题中给出的i x i 08.0=，j y j 05.05.0+=，(010,020i j ≤≤≤≤)的11*21组j i y x ,分别求出原题中方程组的一组解，于是得到一组和i i y x ,对应的j i t u ,。

2.对于已求出的j i t u ,，使用分片二次代数插值法对原题中关于u t z ,,的数表进行插值得到ij z 。

于是产生了z=f(x,y)的11*21个数值解。

3.从k=1开始逐渐增大k 的值，并使用最小二乘法曲面拟合法对z=f(x,y)进行拟合，得到每次的σ,k 。

当710-<σ时结束计算，输出拟合结果。

4.计算)5,,2,1,8,,2,1)(,(),,(****⋅⋅⋅=⋅⋅⋅=j i y x p y x f j i j i 的值并输出结果，以观察),(y x p 逼近),(y x f 的效果。

其中j y i x j i 2.05.0,1.0**+==。

三、算法实现方案 1、求(,)f x y ：（1）Newton 法解非线性方程组0.5cos 2.670.5sin 1.07(1)0.5cos 3.740.5sin 0.79t u v w x t u v w y t u v w x t u v w y +++-=⎧⎪+++-=⎪⎨+++-=⎪⎪+++-=⎩， 其中，t, u, v ,w 为待求的未知量，x, y 为代入的已知量。

设(,,,)Tt u v w ξ=，给定精度水平12110ε-=和最大迭代次数M ，则解该线性方程组的迭代格式为：*(0)(0)(0)(0)(0)(k+1)()()1()(,,,)()()0,1,T k k k t u v w F F k ξξξξξξ-⎧=⎪'=-⎨⎪=⎩在附近选取初值， 迭代终止条件为()(1)()1/k k k ξξξε-∞∞-≤，若k M >时仍未达到迭代精度，则迭代计算失败。

其中，雅可比矩阵0.5*cos(t) + u + v + w - x - 2.67t + 0.5*sin(u) + v + w - y - 1.07()0.5*t + u + cos(v) + w - x - 3.74t + 0.5*u + v + sin(w) - y - 0.79F ξ⎡⎤⎢⎥⎢⎥=⎢⎥⎢⎥⎣⎦， 0.5*sin()11110.5*cos()11()0.51sin()110.51cos()t u F v w ξ-⎡⎤⎢⎥⎢⎥'=⎢⎥-⎢⎥⎣⎦， （2）分片双二次插值：根据题目给出的表格，(0,1,,5)(0,1,,5)i i t ih i u j j τ= == =（其中，0.2h τ= =0.4，）对于给定的(,)t u ,如果(,)t u 满足,322,322i i j j h ht t t i u u u j ττ-<≤+ 2≤≤-<≤+ 2≤≤则选择(,)(1,,1;1,,1)k r t u k i i i r j j j =-+=-+为插值节点，相应的插值多项式为1111(,)()()(,)j i krkrk i r j h t u l t l u g t u ++=-=-=(2) ∑∑其中，1111()(1,,1)()(1,,1)i mk m i k mm kj nr n j r nn rt t l t k i i i t t u u l u r j j j u u +=-≠+=-≠-= =-+--==-+-∏∏如果1422h h t t t t ≤+>+或,则在式（2）中取i=1或i=4; 如果1422u u u ττ≤+>+或u ,则在式（2）中取u=1或u=4。

在区域{(,)|00.8,0.5 1.5}D x y x y =≤≤≤≤上，将(,)i j x y (0.08,i x i =i=0,1,…,10;j y =0.5+0.2j,j=0,1,…20)代入到非线性方程组（1）中，用Newton 法解出,,(,)i j i j t u ，再由分片双二次插值得,,,(,)i j i j i j z h t u =，则有,(,)i j i j z f x y =（i=0,1,…,10;j=0,1,…,20），即求出了(,)z f x y =。

2、求(,)p x y ：乘积型最小二乘拟合曲面： （1）求系数矩阵C ：11()()T T T C B B B UG G G --=其中，200021112101010111k k k x x x x x x B x x x ⎡⎤⎢⎥⎢⎥=⎢⎥⎢⎥⎢⎥⎣⎦ 200021112202020111k k k y y y y y y G y x y ⎡⎤⎢⎥⎢⎥=⎢⎥⎢⎥⎢⎥⎣⎦0,00,10,201,01,11,20,10,010,110,20,(,)(0,1,,10;0,1,,20)i j i jz z z z z z U z f x y i j z z z ⎡⎤⎢⎥⎢⎥== ==⎢⎥⎢⎥⎣⎦计算中涉及到对矩阵求逆，接着在后面将会具体说明列主元的高斯消去法求矩阵的逆的方法。

（2）确定最小的k 值，拟合曲面：,0(,)kr s rsr s p x y cx y ==∑设1020200((,)(,))i j i j i j f x y p x y σ===-∑∑，给定精度水平7210ε-=和最大迭代次数N ，则确定最小k 值的迭代格式为：(),01020()()200(,)(0,1,,10;0,1,,20)((,)(,))0,1,2,kk r s i j rs i j r s k k i j i j i j p x y c x y i j f x y p x y k σ===⎧= ==⎪⎪⎪=-⎨⎪⎪=⎪⎩∑∑∑迭代终止条件为()2k σε≤，若k N >时仍未达到迭代精度，则迭代计算失败。

待确定满足精度条件的最小k 值后，就可以进行曲面拟合计算了。

3、关于列主元的高斯消去法求矩阵的逆：设非奇异矩阵n n A C ⨯∈，且1B A -= ，则AB I =，对B 和I 列分块，有11(,,,,)(,,,,)j n j n A b b b e e e =即,1,2,,j j Ab e j n = =其中，(0,,0,1,0,,0)T j je =应用列主元的高斯消去法线性解方程组 ,1,2,,j j Ab e j n = =，则1(,,,,)j n B b b b =即为A 的逆。

注：若A 不可逆，则此算法失效。

四、源程序#include<iostream>// zhhfshzhfx3.cpp : 定义控制台应用程序的入口点。

//#include <stdio.h> #include <cmath>const int M= 500;//迭代最大次数const double E=1.0e-12;//确定牛顿迭代精度水平 const double E1=1.0e-7;//确定拟合精度水平 const int kmax=9;//k 的最大值 const double matrix[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}};const double mat_t[6]={0, 0.2, 0.4, 0.6, 0.8, 1.0};const double mat_u[6]={0, 0.4, 0.8, 1.2, 1.6, 2.0};double U[11][21];//对f(x,y)的值进行存储double C[kmax+1][kmax+1];//拟合系数矩阵////////////两数绝对值取大/////////////double max(double x,double y){return fabs(x)>fabs(y)?fabs(x):fabs(y);}/////////高斯消元法解线性方程组///////void linear_solution(double f[4],double ff[4][4], double (&delta)[4]) {int i, j, k, ik;double tmp, mik;for(k=0; k<3; k++){ik=k;for(i=k;i<4;i++)if(fabs(ff[ik][k])<fabs(ff[i][k]))ik=i;for(j=k;j<4;j++){tmp=ff[k][j];ff[k][j]=ff[ik][j];ff[ik][j]=tmp;}tmp=f[k];f[k]=f[ik];f[ik]=tmp;for(i=k+1;i<4;i++){mik=ff[i][k]/ff[k][k];for(j=k;j<4;j++)ff[i][j]=ff[i][j]-mik*ff[k][j];f[i]-=mik*f[k];}}delta[3]=f[3]/ff[3][3];for(k=2;k>=0;k--){tmp=0;for(j=k+1;j<4;j++)tmp+=ff[k][j]*delta[j];delta[k]=(f[k]-tmp)/ff[k][k];}}////////Newton法求解非线性方程组////////void equation_solution(double x, double y,double &u,double &t){double v=1.0,w=1.0;u=1.0;t=1.0;double delta[4],f[4],ff[4][4];int k=0;for(k=0;k<=M;k++){f[0]=-1.0*(0.5*cos(t)+u+v+w-x-2.67);f[1]=-1.0*(t+0.5*sin(u)+v+w-y-1.07);f[2]=-1.0*(0.5*t+u+cos(v)+w-x-3.74);f[3]=-1.0*(t+0.5*u+v+sin(w)-y-0.79);ff[0][0]=-0.5*sin(t);ff[0][1]=1.0;ff[0][2]=1.0;ff[0][3]=1.0;ff[1][0]=1.0;ff[1][1]=0.5*cos(u);ff[1][2]=1.0;ff[1][3]=1.0;ff[2][0]=0.5;ff[2][1]=1.0;ff[2][2]=-sin(v);ff[2][3]=1.0;ff[3][0]=1.0;ff[3][1]=0.5;ff[3][2]=1.0;ff[3][3]=cos(w);linear_solution(f,ff,delta);if(max(delta[3],max(delta[2],max(delta[0],delta[1])))/max(w,max(v,max(u,t)))<=E) break;else{t+=delta[0];u+=delta[1];v+=delta[2];w+=delta[3];if(k==M)printf("非线性方程组迭代不成功!\n");}}}/////////////分片双二次插值//////////double interpolation(double a,double b){int i,j,k,r,t1,t2;double z=0.0;i=int(fabs((a/0.2)+0.5));j=int(fabs((b/0.4)+0.5));if(i==0) i=1;if(i==5) i=4;if(j==0) j=1;if(j==5) j=4;for(k=i-1;k<=i+1; k++)for(r=j-1;r<=j+1;r++){double sum=1.0;sum*=matrix[k][r];for(t1=i-1;t1<=i+1;t1++)if(t1!=k)sum*=(a-mat_t[t1])/(mat_t[k]-mat_t[t1]);for(t2=j-1;t2<=j+1;t2++)if(t2!=r)sum*=(b-mat_u[t2])/(mat_u[r]-mat_u[t2]);z+=sum;}return z;}//////////////矩阵求逆/////////////void inverse(double (&matrix)[kmax+1][kmax+1],int k){double matrixT[kmax+1][kmax+1]={0.0};double b[kmax+1][kmax+1]={0.0},tmp,mik;int i,j,t,p,it;for(i=0;i<=kmax;i++)for(j=0;j<=kmax;j++)if(i==j)b[i][j]=1.0;else b[i][j]=0.0;for(t=0; t<k; t++){it=t;for(i=t+1;i<=k;i++)if(fabs(matrix[it][t])<fabs(matrix[i][t]))it=i;for(j=t;j<=k;j++){tmp=matrix[it][j];matrix[it][j]=matrix[t][j];matrix[t][j]=tmp;}for(p=0;p<=k;p++){tmp=b[t][p];b[t][p]=b[it][p];b[it][p]=tmp;}for(i=t+1;i<=k;i++){mik=matrix[i][t]/matrix[t][t];for(j=t+1;j<=k;j++)matrix[i][j]-=mik*matrix[t][j];for(p=0;p<=k;p++)b[i][p]-=mik*b[t][p];}}for(p=0;p<=k;p++){matrixT[k][p]=b[k][p]/matrix[k][k];for(i=k-1;i>=0;i--){for(tmp=0.0,j=i+1;j<=k;j++)tmp+=matrix[i][j]*matrixT[j][p];matrixT[i][p]=(b[i][p]-tmp)/matrix[i][i];}}for(i=0;i<=k;i++)for(j=0;j<=k;j++)matrix[i][j]=matrixT[i][j];}////////针对每个k值进行曲面拟合/////////double surface_fitting(int k, double x, double y){double B[11][kmax+1]={0.0},BT[kmax+1][11]={0.0};double G[21][kmax+1]={0.0},GT[kmax+1][21]={0.0};doubleBTB[kmax+1][kmax+1]={0.0},BTB1[kmax+1][kmax+1]={0.0},BTB2[kmax+1][kmax+1]={0.0},GTG[kmax+1][ kmax+1]={0.0};double matrix1[kmax+1][11]={0.0},matrix3[21][kmax+1]={0.0};double matrix2[kmax+1][21]={0.0};double p,sum;int i,j,t,r,s;for(i=0;i<=10;i++)for(j=0;j<=k;j++)B[i][j]=pow(0.08*i,j);for(i=0;i<=20;i++)for(j=0;j<=k;j++)G[i][j]=pow(0.5+0.05*i,j);for(i=0;i<=10;i++)for(j=0;j<=k;j++)BT[j][i]=B[i][j];for(i=0;i<=20;i++)for(j=0;j<=k;j++)GT[j][i]=G[i][j];for(i=0;i<=k;i++)for(j=0;j<=k;j++){for(sum=0.0,t=0;t<=10;t++)sum+=BT[i][t]*B[t][j];BTB[i][j]=sum;}inverse(BTB,k);for(i=0;i<=k;i++)for(j=0;j<=k;j++){for(sum=0.0,t=0;t<=20;t++)sum+=GT[i][t]*G[t][j];GTG[i][j]=sum;}inverse(GTG,k);for(i=0;i<=k;i++)for(j=0;j<=10;j++){for(sum=0.0,t=0;t<=k;t++)sum+=BTB[i][t]*BT[t][j];matrix1[i][j]=sum;}for(i=0;i<=20;i++)for(j=0;j<=k;j++){for(sum=0.0,t=0;t<=k;t++)sum+=G[i][t]*GTG[t][j];matrix3[i][j]=sum;}for(i=0;i<=k;i++)for(j=0;j<=20;j++){for(sum=0.0,t=0;t<=10;t++)sum+=matrix1[i][t]*U[t][j];matrix2[i][j]=sum;}for(i=0;i<=k;i++)for(j=0;j<=k;j++){for(sum=0.0,t=0;t<=20;t++)sum+=matrix2[i][t]*matrix3[t][j];C[i][j]=sum;}p=0.0;for(r=0;r<=k;r++)for(s=0;s<=k;s++)p+=C[r][s]*pow(x,r)*pow(y,s);return p;}////////////主函数///////////////int _tmain(int argc, _TCHAR* argv[]){double x,y,t,u,sum,sigma;int i,j,k,kmin;printf("数表：xi,yi,f(xi,yi)(i=0,1,2...10;j=0,1,2...,20):\n");for(i=0;i<=10;i++)for(j=0;j<=20;j++){x=0.08*i;y=0.5+0.05*j;equation_solution(x,y,u,t);U[i][j]=interpolation(t,u);printf("x%d=%f, y%d=%f, f(x%d, y%d)=%.12le\n",i,x,j,y,i,j,U[i][j]);}printf("选择过程的k,σ值分别为：\n");for(k=0;k<=kmax;k++){sum=0.0;for(i=0;i<=10;i++)for(j=0;j<=20;j++)sum+=pow(surface_fitting(k,0.08*i,0.5+0.05*j)-U[i][j],2);printf("%d %.12le\n",k,sum);if(sum<=E1){kmin=k;sigma=sum;printf("达到精度要求的k,σ值分别为：\nk=%d,σ=%.12le\n",kmin,sigma);printf("p(x,y)中的系数Crs(r=0,1,...,k;s=0,1,...,k):\n");for(i=0;i<=k;i++)for(j=0;j<=k;j++)printf("Crs[%d][%d]=%.12le\n",i,j,C[i][j]);break;}else if(k==kmax)printf("无法达到曲面拟合精度要求!");}printf("数表：x*[i],y*[j],f(x*[i],y*[j]),p(x*[i],y*[j])(i=1,2,...,8;j=1,2,...,5):\n");for(i=1;i<=8;i++)for(j=1;j<=5;j++){equation_solution(0.1*i,0.5+0.2*j,u,t);printf("x*[%d]=%f,y*[%d]=%f\n",i,0.1*i,j,0.5+0.2*j);printf("f(x*[%d],y*[%d])=%.12le,",i,j,interpolation(t,u));printf("p(x*[%d],y*[%d])=%.12le\n",i,j,surface_fitting(kmin,0.1*i,0.5+0.2*j));}return 0;}五、程序输出---------------------------------------------------------------------------------------------------------------------- 1. 数表：xi,yi,f(xi,yi)(i=0,1,2...10;j=0,1,2...,20):x0=0.000000, y0=0.500000, f(x0, y0)=4.465040184807e-001x0=0.000000, y1=0.550000, f(x0, y1)=3.246832629277e-001x0=0.000000, y2=0.600000, f(x0, y2)=2.101596866827e-001x0=0.000000, y4=0.700000, f(x0, y4)=3.401895562675e-003x0=0.000000, y5=0.750000, f(x0, y5)=-8.873581363800e-002 x0=0.000000, y6=0.800000, f(x0, y6)=-1.733716327497e-001 x0=0.000000, y7=0.850000, f(x0, y7)=-2.505346114666e-001 x0=0.000000, y8=0.900000, f(x0, y8)=-3.202765063876e-001 x0=0.000000, y9=0.950000, f(x0, y9)=-3.826680661097e-001 x0=0.000000, y10=1.000000, f(x0, y10)=-4.377957667384e-001 x0=0.000000, y11=1.050000, f(x0, y11)=-4.857589414438e-001 x0=0.000000, y12=1.100000, f(x0, y12)=-5.266672548835e-001 x0=0.000000, y13=1.150000, f(x0, y13)=-5.606384797965e-001 x0=0.000000, y14=1.200000, f(x0, y14)=-5.877965387677e-001 x0=0.000000, y15=1.250000, f(x0, y15)=-6.0826********e-001 x0=0.000000, y16=1.300000, f(x0, y16)=-6.221894528764e-001 x0=0.000000, y17=1.350000, f(x0, y17)=-6.296883781856e-001 x0=0.000000, y18=1.400000, f(x0, y18)=-6.308997600028e-001 x0=0.000000, y19=1.450000, f(x0, y19)=-6.259561525454e-001 x0=0.000000, y20=1.500000, f(x0, y20)=-6.149885466094e-001 x1=0.080000, y0=0.500000, f(x1, y0)=6.380152265113e-001x1=0.080000, y1=0.550000, f(x1, y1)=5.0661********e-001x1=0.080000, y2=0.600000, f(x1, y2)=3.821763692774e-001x1=0.080000, y3=0.650000, f(x1, y3)=2.648634911537e-001x1=0.080000, y4=0.700000, f(x1, y4)=1.547802002848e-001x1=0.080000, y5=0.750000, f(x1, y5)=5.199268349094e-002x1=0.080000, y6=0.800000, f(x1, y6)=-4.346804020490e-002 x1=0.080000, y7=0.850000, f(x1, y7)=-1.316010567885e-001 x1=0.080000, y8=0.900000, f(x1, y8)=-2.124310883088e-001 x1=0.080000, y9=0.950000, f(x1, y9)=-2.860045510580e-001 x1=0.080000, y10=1.000000, f(x1, y10)=-3.523860789794e-001 x1=0.080000, y11=1.050000, f(x1, y11)=-4.116554565222e-001x1=0.080000, y13=1.150000, f(x1, y13)=-5.092367247005e-001 x1=0.080000, y14=1.200000, f(x1, y14)=-5.477611179623e-001 x1=0.080000, y15=1.250000, f(x1, y15)=-5.795943883391e-001 x1=0.080000, y16=1.300000, f(x1, y16)=-6.048572588895e-001 x1=0.080000, y17=1.350000, f(x1, y17)=-6.236734213318e-001 x1=0.080000, y18=1.400000, f(x1, y18)=-6.361682484133e-001 x1=0.080000, y19=1.450000, f(x1, y19)=-6.424676566901e-001 x1=0.080000, y20=1.500000, f(x1, y20)=-6.426971026996e-001 x2=0.160000, y0=0.500000, f(x2, y0)=8.400813957666e-001x2=0.160000, y1=0.550000, f(x2, y1)=6.997641656732e-001x2=0.160000, y2=0.600000, f(x2, y2)=5.660614423517e-001x2=0.160000, y3=0.650000, f(x2, y3)=4.391716081176e-001x2=0.160000, y4=0.700000, f(x2, y4)=3.192421380408e-001x2=0.160000, y5=0.750000, f(x2, y5)=2.063761923874e-001x2=0.160000, y6=0.800000, f(x2, y6)=1.006385238914e-001x2=0.160000, y7=0.850000, f(x2, y7)=2.060740067837e-003x2=0.160000, y8=0.900000, f(x2, y8)=-8.935402476698e-002 x2=0.160000, y9=0.950000, f(x2, y9)=-1.736269688648e-001 x2=0.160000, y10=1.000000, f(x2, y10)=-2.507999561599e-001 x2=0.160000, y11=1.050000, f(x2, y11)=-3.209322694446e-001 x2=0.160000, y12=1.100000, f(x2, y12)=-3.840977350046e-001 x2=0.160000, y13=1.150000, f(x2, y13)=-4.403821754175e-001 x2=0.160000, y14=1.200000, f(x2, y14)=-4.898811523126e-001 x2=0.160000, y15=1.250000, f(x2, y15)=-5.326979655338e-001 x2=0.160000, y16=1.300000, f(x2, y16)=-5.689418792921e-001 x2=0.160000, y17=1.350000, f(x2, y17)=-5.987265495151e-001 x2=0.160000, y18=1.400000, f(x2, y18)=-6.221686297503e-001 x2=0.160000, y19=1.450000, f(x2, y19)=-6.393865356972e-001 x2=0.160000, y20=1.500000, f(x2, y20)=-6.504993507878e-001x3=0.240000, y1=0.550000, f(x3, y1)=9.029*********e-001x3=0.240000, y2=0.600000, f(x3, y2)=7.605802668596e-001x3=0.240000, y3=0.650000, f(x3, y3)=6.247151981456e-001x3=0.240000, y4=0.700000, f(x3, y4)=4.955197560009e-001x3=0.240000, y5=0.750000, f(x3, y5)=3.731340427746e-001x3=0.240000, y6=0.800000, f(x3, y6)=2.576567488723e-001x3=0.240000, y7=0.850000, f(x3, y7)=1.491505594102e-001x3=0.240000, y8=0.900000, f(x3, y8)=4.764698677337e-002x3=0.240000, y9=0.950000, f(x3, y9)=-4.684932320146e-002 x3=0.240000, y10=1.000000, f(x3, y10)=-1.343567603849e-001 x3=0.240000, y11=1.050000, f(x3, y11)=-2.149133449274e-001 x3=0.240000, y12=1.100000, f(x3, y12)=-2.885737006348e-001 x3=0.240000, y13=1.150000, f(x3, y13)=-3.554063647857e-001 x3=0.240000, y14=1.200000, f(x3, y14)=-4.154913964886e-001 x3=0.240000, y15=1.250000, f(x3, y15)=-4.689182499695e-001 x3=0.240000, y16=1.300000, f(x3, y16)=-5.157838831247e-001 x3=0.240000, y17=1.350000, f(x3, y17)=-5.561910752001e-001 x3=0.240000, y18=1.400000, f(x3, y18)=-5.902469305629e-001 x3=0.240000, y19=1.450000, f(x3, y19)=-6.180615482412e-001 x3=0.240000, y20=1.500000, f(x3, y20)=-6.397468392579e-001 x4=0.320000, y0=0.500000, f(x4, y0)=1.271246751483e+000x4=0.320000, y1=0.550000, f(x4, y1)=1.115002018147e+000x4=0.320000, y2=0.600000, f(x4, y2)=9.646077272157e-001x4=0.320000, y3=0.650000, f(x4, y3)=8.203473694751e-001x4=0.320000, y4=0.700000, f(x4, y4)=6.824476781795e-001x4=0.320000, y5=0.750000, f(x4, y5)=5.510852085975e-001x4=0.320000, y6=0.800000, f(x4, y6)=4.263923859018e-001x4=0.320000, y7=0.850000, f(x4, y7)=3.084629956332e-001x4=0.320000, y8=0.900000, f(x4, y8)=1.973571296919e-001x4=0.320000, y10=1.000000, f(x4, y10)=-4.285992234034e-003 x4=0.320000, y11=1.050000, f(x4, y11)=-9.483392529689e-002 x4=0.320000, y12=1.100000, f(x4, y12)=-1.785729903640e-001 x4=0.320000, y13=1.150000, f(x4, y13)=-2.555537790546e-001 x4=0.320000, y14=1.200000, f(x4, y14)=-3.258401501575e-001 x4=0.320000, y15=1.250000, f(x4, y15)=-3.895069883634e-001 x4=0.320000, y16=1.300000, f(x4, y16)=-4.466382045995e-001 x4=0.320000, y17=1.350000, f(x4, y17)=-4.973249517677e-001 x4=0.320000, y18=1.400000, f(x4, y18)=-5.416640326994e-001 x4=0.320000, y19=1.450000, f(x4, y19)=-5.797564797951e-001 x4=0.320000, y20=1.500000, f(x4, y20)=-6.117062881476e-001 x5=0.400000, y0=0.500000, f(x5, y0)=1.498321052482e+000x5=0.400000, y1=0.550000, f(x5, y1)=1.334998632066e+000x5=0.400000, y2=0.600000, f(x5, y2)=1.177125123739e+000x5=0.400000, y3=0.650000, f(x5, y3)=1.025*********e+000x5=0.400000, y4=0.700000, f(x5, y4)=8.789600231744e-001x5=0.400000, y5=0.750000, f(x5, y5)=7.391451087035e-001x5=0.400000, y6=0.800000, f(x5, y6)=6.0574********e-001x5=0.400000, y7=0.850000, f(x5, y7)=4.788838610666e-001x5=0.400000, y8=0.900000, f(x5, y8)=3.586506258818e-001x5=0.400000, y9=0.950000, f(x5, y9)=2.451022361964e-001x5=0.400000, y10=1.000000, f(x5, y10)=1.382683509285e-001 x5=0.400000, y11=1.050000, f(x5, y11)=3.815486540699e-002 x5=0.400000, y12=1.100000, f(x5, y12)=-5.525282116814e-002 x5=0.400000, y13=1.150000, f(x5, y13)=-1.419868808137e-001 x5=0.400000, y14=1.200000, f(x5, y14)=-2.220944390959e-001 x5=0.400000, y15=1.250000, f(x5, y15)=-2.956352324598e-001 x5=0.400000, y16=1.300000, f(x5, y16)=-3.626795115028e-001 x5=0.400000, y17=1.350000, f(x5, y17)=-4.233061642240e-001x5=0.400000, y19=1.450000, f(x5, y19)=-5.256554266672e-001 x5=0.400000, y20=1.500000, f(x5, y20)=-5.675647436551e-001 x6=0.480000, y0=0.500000, f(x6, y0)=1.731892740383e+000x6=0.480000, y1=0.550000, f(x6, y1)=1.562034577209e+000x6=0.480000, y2=0.600000, f(x6, y2)=1.397216918208e+000x6=0.480000, y3=0.650000, f(x6, y3)=1.237801006739e+000x6=0.480000, y4=0.700000, f(x6, y4)=1.0840********e+000x6=0.480000, y5=0.750000, f(x6, y5)=9.363227723149e-001x6=0.480000, y6=0.800000, f(x6, y6)=7.947044490537e-001x6=0.480000, y7=0.850000, f(x6, y7)=6.593871980282e-001x6=0.480000, y8=0.900000, f(x6, y8)=5.304875868400e-001x6=0.480000, y9=0.950000, f(x6, y9)=4.080886854542e-001x6=0.480000, y10=1.000000, f(x6, y10)=2.922442012295e-001 x6=0.480000, y11=1.050000, f(x6, y11)=1.829822068535e-001 x6=0.480000, y12=1.100000, f(x6, y12)=8.030849403543e-002 x6=0.480000, y13=1.150000, f(x6, y13)=-1.579041305164e-002 x6=0.480000, y14=1.200000, f(x6, y14)=-1.0534********e-001 x6=0.480000, y15=1.250000, f(x6, y15)=-1.883980906096e-001 x6=0.480000, y16=1.300000, f(x6, y16)=-2.650071493189e-001 x6=0.480000, y17=1.350000, f(x6, y17)=-3.352378389040e-001 x6=0.480000, y18=1.400000, f(x6, y18)=-3.991645038868e-001 x6=0.480000, y19=1.450000, f(x6, y19)=-4.568681433016e-001 x6=0.480000, y20=1.500000, f(x6, y20)=-5.084349932782e-001 x7=0.560000, y0=0.500000, f(x7, y0)=1.971221786400e+000x7=0.560000, y1=0.550000, f(x7, y1)=1.795329599501e+000x7=0.560000, y2=0.600000, f(x7, y2)=1.624067113228e+000x7=0.560000, y3=0.650000, f(x7, y3)=1.457830582708e+000x7=0.560000, y4=0.700000, f(x7, y4)=1.296954649752e+000x7=0.560000, y5=0.750000, f(x7, y5)=1.141718105447e+000x7=0.560000, y7=0.850000, f(x7, y7)=8.490326633294e-001x7=0.560000, y8=0.900000, f(x7, y8)=7.119113522641e-001x7=0.560000, y9=0.950000, f(x7, y9)=5.810941589219e-001x7=0.560000, y10=1.000000, f(x7, y10)=4.566585132335e-001 x7=0.560000, y11=1.050000, f(x7, y11)=3.386544961394e-001 x7=0.560000, y12=1.100000, f(x7, y12)=2.271082557696e-001 x7=0.560000, y13=1.150000, f(x7, y13)=1.220250891932e-001 x7=0.560000, y14=1.200000, f(x7, y14)=2.339221963760e-002 x7=0.560000, y15=1.250000, f(x7, y15)=-6.881870197104e-002 x7=0.560000, y16=1.300000, f(x7, y16)=-1.546493442129e-001 x7=0.560000, y17=1.350000, f(x7, y17)=-2.341526664587e-001 x7=0.560000, y18=1.400000, f(x7, y18)=-3.0739********e-001 x7=0.560000, y19=1.450000, f(x7, y19)=-3.744348623481e-001 x7=0.560000, y20=1.500000, f(x7, y20)=-4.353605565359e-001 x8=0.640000, y0=0.500000, f(x8, y0)=2.215667863688e+000x8=0.640000, y1=0.550000, f(x8, y1)=2.034201133607e+000x8=0.640000, y2=0.600000, f(x8, y2)=1.856955143619e+000x8=0.640000, y3=0.650000, f(x8, y3)=1.684358164161e+000x8=0.640000, y4=0.700000, f(x8, y4)=1.516776352400e+000x8=0.640000, y5=0.750000, f(x8, y5)=1.354519041151e+000x8=0.640000, y6=0.800000, f(x8, y6)=1.197844086673e+000x8=0.640000, y7=0.850000, f(x8, y7)=1.046963049419e+000x8=0.640000, y8=0.900000, f(x8, y8)=9.020*********e-001x8=0.640000, y9=0.950000, f(x8, y9)=7.632264776629e-001x8=0.640000, y10=1.000000, f(x8, y10)=6.306048219543e-001 x8=0.640000, y11=1.050000, f(x8, y11)=5.042528145972e-001 x8=0.640000, y12=1.100000, f(x8, y12)=3.842167155457e-001 x8=0.640000, y13=1.150000, f(x8, y13)=2.705204766410e-001 x8=0.640000, y14=1.200000, f(x8, y14)=1.631685723996e-001x8=0.640000, y16=1.300000, f(x8, y16)=-3.256661939682e-002 x8=0.640000, y17=1.350000, f(x8, y17)=-1.210165348444e-001 x8=0.640000, y18=1.400000, f(x8, y18)=-2.032513996228e-001 x8=0.640000, y19=1.450000, f(x8, y19)=-2.793303595584e-001 x8=0.640000, y20=1.500000, f(x8, y20)=-3.493199575400e-001 x9=0.720000, y0=0.500000, f(x9, y0)=2.464684222659e+000x9=0.720000, y1=0.550000, f(x9, y1)=2.278058979398e+000x9=0.720000, y2=0.600000, f(x9, y2)=2.0952********e+000x9=0.720000, y3=0.650000, f(x9, y3)=1.916718127997e+000x9=0.720000, y4=0.700000, f(x9, y4)=1.742854628776e+000x9=0.720000, y5=0.750000, f(x9, y5)=1.573998427334e+000x9=0.720000, y6=0.800000, f(x9, y6)=1.410434835231e+000x9=0.720000, y7=0.850000, f(x9, y7)=1.252401750608e+000x9=0.720000, y8=0.900000, f(x9, y8)=1.100094409628e+000x9=0.720000, y9=0.950000, f(x9, y9)=9.536698512613e-001x9=0.720000, y10=1.000000, f(x9, y10)=8.132510552489e-001 x9=0.720000, y11=1.050000, f(x9, y11)=6.789307429659e-001 x9=0.720000, y12=1.100000, f(x9, y12)=5.507748485043e-001 x9=0.720000, y13=1.150000, f(x9, y13)=4.288256769731e-001 x9=0.720000, y14=1.200000, f(x9, y14)=3.131047717398e-001 x9=0.720000, y15=1.250000, f(x9, y15)=2.036155140327e-001 x9=0.720000, y16=1.300000, f(x9, y16)=1.003454782409e-001 x9=0.720000, y17=1.350000, f(x9, y17)=3.268565186572e-003 x9=0.720000, y18=1.400000, f(x9, y18)=-8.765306591329e-002 x9=0.720000, y19=1.450000, f(x9, y19)=-1.724672478188e-001 x9=0.720000, y20=1.500000, f(x9, y20)=-2.512302207523e-001 x10=0.800000, y0=0.500000, f(x10, y0)=2.717811109467e+000 x10=0.800000, y1=0.550000, f(x10, y1)=2.526399501255e+000 x10=0.800000, y2=0.600000, f(x10, y2)=2.338411386860e+000x10=0.800000, y3=0.650000, f(x10, y3)=2.154329377280e+000x10=0.800000, y4=0.700000, f(x10, y4)=1.974574556652e+000x10=0.800000, y5=0.750000, f(x10, y5)=1.799510579099e+000x10=0.800000, y6=0.800000, f(x10, y6)=1.629448220554e+000x10=0.800000, y7=0.850000, f(x10, y7)=1.464650043751e+000x10=0.800000, y8=0.900000, f(x10, y8)=1.305334967651e+000x10=0.800000, y9=0.950000, f(x10, y9)=1.151682621307e+000x10=0.800000, y10=1.000000, f(x10, y10)=1.003837419906e+000x10=0.800000, y11=1.050000, f(x10, y11)=8.619123372279e-001x10=0.800000, y12=1.100000, f(x10, y12)=7.259923711112e-001x10=0.800000, y13=1.150000, f(x10, y13)=5.961377115201e-001x10=0.800000, y14=1.200000, f(x10, y14)=4.723866279136e-001x10=0.800000, y15=1.250000, f(x10, y15)=3.547580958979e-001x10=0.800000, y16=1.300000, f(x10, y16)=2.432541841813e-001x10=0.800000, y17=1.350000, f(x10, y17)=1.378622225247e-001x10=0.800000, y18=1.400000, f(x10, y18)=3.855677032640e-002x10=0.800000, y19=1.450000, f(x10, y19)=-5.469859593446e-002x10=0.800000, y20=1.500000, f(x10, y20)=-1.419496597088e-001---------------------------------------------------------------------------------------------------------------------- 2. 选择过程的k,σ值分别为：0 1.442880771836e+0021 3.220908973638e+0002 4.659960033271e-0033 1.721175379141e-0044 3.309534299251e-0065 2.541379696823e-008---------------------------------------------------------------------------------------------------------------------- 3. 达到精度要求的k,σ值分别为：k=5, σ=2.541379696823e-008----------------------------------------------------------------------------------------------------------------------4. p(x,y)中的系数Crs(r=0,1,...,k;s=0,1,...,k): Crs[0][0]=2.021*********e+000Crs[0][1]=-3.668426808518e+000Crs[0][2]=7.092486428705e-001Crs[0][3]=8.486053724111e-001Crs[0][4]=-4.158974209824e-001Crs[0][5]=6.743199179943e-002Crs[1][0]=3.191909003403e+000Crs[1][1]=-7.411103479266e-001Crs[1][2]=-2.697124651510e+000Crs[1][3]=1.631183476044e+000Crs[1][4]=-4.847200158207e-001Crs[1][5]=6.061429014963e-002Crs[2][0]=2.568898073659e-001Crs[2][1]=1.579918701245e+000Crs[2][2]=-4.634081949386e-001Crs[2][3]=-8.134430514622e-002Crs[2][4]=1.020*********e-001Crs[2][5]=-2.101522223263e-002Crs[3][0]=-2.692603350703e-001Crs[3][1]=-7.302476753732e-001Crs[3][2]=1.076145111339e+000Crs[3][3]=-8.0701********e-001Crs[3][4]=3.028*********e-001Crs[3][5]=-4.597263977688e-002Crs[4][0]=2.174597543212e-001Crs[4][1]=-1.783723777304e-001Crs[4][2]=-7.240580777274e-002Crs[4][3]=2.433304760580e-001Crs[4][4]=-1.413347404863e-001Crs[4][5]=2.651024147029e-002Crs[5][0]=-5.590326529323e-002Crs[5][1]=1.431992428176e-001Crs[5][2]=-1.362703704945e-001Crs[5][3]=4.0719********e-002Crs[5][4]=3.775031827767e-003Crs[5][5]=-2.667701063826e-003---------------------------------------------------------------------------------------------------------------------- 5. 数表：x*[i],y*[j],f(x*[i],y*[j]),p(x*[i],y*[j])(i=1,2,...,8;j=1,2,...,5):x*[1]=0.100000,y*[1]=0.700000f(x*[1],y*[1])=1.947204079177e-001,p(x*[1],y*[1])=1.947303555857e-001x*[1]=0.100000,y*[2]=0.900000f(x*[1],y*[2])=-1.830370791887e-001,p(x*[1],y*[2])=-1.830418420683e-001x*[1]=0.100000,y*[3]=1.100000f(x*[1],y*[3])=-4.454976469148e-001,p(x*[1],y*[3])=-4.455000471734e-001x*[1]=0.100000,y*[4]=1.300000f(x*[1],y*[4])=-5.975667076413e-001,p(x*[1],y*[4])=-5.975588652304e-001x*[1]=0.100000,y*[5]=1.500000f(x*[1],y*[5])=-6.464595939011e-001,p(x*[1],y*[5])=-6.464461231867e-001x*[2]=0.200000,y*[1]=0.700000f(x*[2],y*[1])=4.0597********e-001,p(x*[2],y*[1])=4.0598********e-001x*[2]=0.200000,y*[2]=0.900000f(x*[2],y*[2])=-2.251595837462e-002,p(x*[2],y*[2])=-2.252111784579e-002x*[2]=0.200000,y*[3]=1.100000f(x*[2],y*[3])=-3.382208160396e-001,p(x*[2],y*[3])=-3.382240255991e-001x*[2]=0.200000,y*[4]=1.300000f(x*[2],y*[4])=-5.444378315219e-001,p(x*[2],y*[4])=-5.444304560698e-001x*[2]=0.200000,y*[5]=1.500000f(x*[2],y*[5])=-6.473613385679e-001,p(x*[2],y*[5])=-6.473480192174e-001x*[3]=0.300000,y*[1]=0.700000。