数值分析-曲线拟合

数学与计算科学学院《数值分析》实验报告题目：曲线拟合专业：信息与计算科学学号：姓名：指导教师：成绩：二零一五年四月二十四日《数值分析》实验报告（2）学号：姓名：班级：成绩：实验名称：曲线拟合实验地点：化工楼317所使用的工具软件及环境：Matlab实验任务与目的学习使用最小二乘法的原理了解法方程的特性理解和分析生成的过程以及结果分析二、实验内容或问题描述通过实验的观察和测量给出离散的一些点，再来求出近似的函数解析式。最小二乘法是求解这一问题的很好的方法，本实验运用这一方法实现对给定数据的拟合。解题思路与方法，或解题详细步骤1.拟合曲线三次多项式nm=3+1,基函数｛1，x，x^2，x^3｝2.拟合节点数：np=10+13.拟合数据：xp[np],yp[np],nm矩阵阶数4.基函数在各拟合节点上值构成向量xe(i)，全部基函数向量构成二维矩阵xe5.正则方程系数矩阵G,常数项b6.解方程组，求基函数的系数a(1),a(2),a(3),a(4)四、程序设计代码%%拟合曲线【-1，1】，f(x)=1/(1+25x^2）%准确解x=-1:0.01:1,n=2/0.01+1;y=f(x)x=-1:0.01:1;n=2/0.01+1;fori=1:ny(i)=1/(1+25*x(i)^2);end;%%拟合曲线三次多项式nm=3+1,基函数｛1，x，x^2，x^3｝，拟合节点数：np=10+1%%拟合数据：xp[np],yp[np],nm矩阵阶数nm=4;np=10+1;%nm=2;%np=4+1;%xp=1:5;%yp=[4,4.5,6,8,8.5];%w=[2,1,3,1,1];%xp=-1:2/(np-1):1;fori=1:npxp(i)=-1+0.2*(i-1);yp(i)=1/(1+25*xp(i)^2);end;w=ones(1,np);%拟合曲线步骤一，基函数在各拟合节点上值构成向量xe(i)，全部基函数向量构成二维矩阵xexe=ones(nm,np);t=xe(1,:);fori=2:nmt=t.*xp;xe(i,:)=t;end;xe%拟合曲线步骤二，正则方程系数矩阵G,常数项bfori=1:nmforj=1:iG(i,j)=sum(xe(i,:).*xe(j,:).*w);G(j,i)=G(i,j);end;b(i)=sum(xe(i,:).*yp.*w);end;%拟合曲线步骤三，解方程组，求基函数的系数a(1),a(2),a(3),a(4)a=b/G;Gbapause%%求拟合曲线上各点x=-1:0.01:1的近似值：s(x),点下标k=1:n:近似值s(x(k))fork=1:ns(k)=a(1);xk=1;fori=2:nm;xk=xk*x(k);s(k)=s(k)+a(i)*xk;end;end;%%绘制函数曲线y=f(x)和拟合曲线s=s(x)figure(1);plot(x,y,'*');holdon;plot(x,s,'-');holdon;plot(xp,yp,'.');%%绘制误差曲线figure(2);plot(x,y-s,'r');五、运行结果>>quxianyihexe=Columns1through81.00001.00001.00001.00001.00001.00001.00001.0000-1.0000-0.8000-0.6000-0.4000-0.200000.20000.40001.00000.64000.36000.16000.040000.04000.1600-1.0000-0.5120-0.2160-0.0640-0.008000.00800.0640Columns9through111.00001.00001.00000.60000.80001.00000.36000.64001.00000.21600.51201.0000G=11.00000.00004.40000.00000.00004.40000.00003.13284.40000.00003.132800.00003.132802.6259b=2.7946-0.00000.32820.0000a=0.4841-0.0000-0.57520.0000quxianyihexe=Columns1through81.00001.00001.00001.00001.00001.00001.00001.0000-1.0000-0.8000-0.6000-0.4000-0.200000.20000.40001.00000.64000.36000.16000.040000.04000.1600-1.0000-0.5120-0.2160-0.0640-0.008000.00800.0640Columns9through111.00001.00001.00000.60000.80001.00000.36000.64001.00000.21600.51201.0000G=11.00000.00004.40000.00000.00004.40000.00003.13284.40000.00003.132800.00003.132802.6259b=2.7946-0.00000.32820.0000a=0.4841-0.0000-0.57520.0000quxianyihexe=Columns1through81.00001.00001.00001.00001.00001.00001.00001.0000-1.0000-0.8000-0.6000-0.4000-0.200000.20000.40001.00000.64000.36000.16000.040000.04000.1600-1.0000-0.5120-0.2160-0.0640-0.008000.00800.0640Columns9through111.00001.00001.00000.60000.80001.00000.36000.64001.00000.21600.51201.0000G=11.00000.00004.40000.00000.00004.40000.00003.13284.40000.00003.132800.00003.132802.6259b=2.7946-0.00000.32820.0000a=0.4841-0.0000-0.57520.0000quxianyihexe=Columns1through81.00001.00001.00001.00001.00001.00001.00001.0000-1.0000-0.8000-0.6000-0.4000-0.200000.20000.40001.00000.64000.36000.16000.040000.04000.1600-1.0000-0.5120-0.2160-0.0640-0.008000.00800.0640Columns9through111.00001.00001.00000.60000.80001.00000.36000.64001.00000.21600.51201.0000G=11.00000.00004.40000.00000.00004.40000.00003.13284.40000.00003.132800.00003.132802.6259b=2.7946-0.00000.32820.0000a=0.4841-0.0000-0.57520.0000quxianyihexe=Columns1through81.00001.00001.00001.00001.00001.00001.00001.0000-1.0000-0.8000-0.6000-0.4000-0.200000.20000.40001.00000.64000.36000.16000.040000.04000.1600-1.0000-0.5120-0.2160-0.0640-0.008000.00800.0640Columns9through111.00001.00001.00000.60000.80001.00000.36000.64001.00000.21600.51201.0000G=11.00000.00004.40000.00000.00004.40000.00003.13284.40000.00003.132800.00003.132802.6259b=2.7946-0.00000.32820.0000a=0.4841-0.0000-0.57520.0000quxianyihexe=Columns1through81.00001.00001.00001.00001.00001.00001.00001.0000-1.0000-0.8000-0.6000-0.4000-0.200000.20000.40001.00000.64000.36000.16000.040000.04000.1600-1.0000-0.5120-0.2160-0.0640-0.008000.00800.0640Columns9through111.00001.00001.00000.60000.80001.00000.36000.64001.00000.21600.51201.0000G=11.00000.00004.40000.00000.00004.40000.00003.13284.40000.00003.132800.00003.132802.6259b=2.7946-0.00000.32820.0000a=0.4841-0.0000-0.57520.0000quxianyihexe=Columns1through81.00001.00001.00001.00001.00001.00001.00001.0000-1.0000-0.8000-0.6000-0.4000-0.200000.20000.40001.00000.64000.36000.16000.040000.04000.1600-1.0000-0.5120-0.2160-0.0640-0.008000.00800.0640Columns9through111.00001.00001.00000.60000.80001.00000.36000.64001.00000.21600.51201.0000G=11.00000.00004.40000.00000.00004.40000.00003.13284.40000.0000