曲线拟合的最小二乘法实验

1、Lab04曲线拟合的最小二乘法实验【实验目的和要求】1让学生体验曲线拟合的最小二乘法，加深对曲线拟合的最小二乘法的理解；2掌握函数ployfit和函数lsqcurvefit功能和使用方法，分别用这两个函数进行多项式拟合和非多项式拟合。【实验内容】1在Matlab命令窗口，用help命令查询函数polyfit和函数lsqcurvefit功能和使用方法。2用多项式y=x3-6x2+5x-3，产生一组数据(xi，yi)(i=1，2，n)，再在yi上添加随机干扰(可用rand产生(0，1)均匀分布随机数，或用randn产生N(0，1)均匀分布随机数)，然后对xi和添加了随机干扰的yi用Matlab提供

2、的函数ployfit用3次多项式拟合，将结果与原系数比较。再作2或4次多项式拟合，分析所得结果。3用电压V=10伏的电池给电容器充电，电容器上t时刻的电压为，其中V0是电容器的初始电压，是充电常数。对于下面的一组t，v数据，用Matlab提供的函数lsqcurvefit确定V0和。t(秒)0.51234579v(伏)6.366.487.268.228.668.999.439.63【实验仪器与软件】 1CPU主频在1GHz以上，内存在128Mb以上的PC；2Matlab 6.0及以上版本。实验讲评：实验成绩： 评阅教师：200 年 月 日问题及算法分析：1、利用help命令，在MATLAB中查找

3、polyfit和lsqcurvefit函数的用法。2、在一组数据(xi，yi)(i=1，2，n)上，对yi上添加随机干扰，运用多项式拟合函数，对数据进行拟合（分别用2次，3次，4次拟合），分析拟合的效果。 3、根据t和V的关系画散点图，再根据给定的函数运用最小二乘拟合函数，确定其相应参数。第一题：（1） help polyfit POLYFIT Fit polynomial to data. P = POLYFIT(X,Y,N) finds the coefficients of a polynomial P(X) of degree N that fits the data Y best i

4、n a least-squares sense. P is a row vector of length N+1 containing the polynomial coefficients in descending powers, P(1)*XN + P(2)*X(N-1) +.+ P(N)*X + P(N+1). P,S = POLYFIT(X,Y,N) returns the polynomial coefficients P and a structure S for use with POLYVAL to obtain error estimates for predictions

5、. S contains fields for the triangular factor (R) from a QR decomposition of the Vandermonde matrix of X, the degrees of freedom (df), and the norm of the residuals (normr). If the data Y are random, an estimate of the covariance matrix of P is (Rinv*Rinv)*normr2/df, where Rinv is the inverse of R.

6、P,S,MU = POLYFIT(X,Y,N) finds the coefficients of a polynomial in XHAT = (X-MU(1)/MU(2) where MU(1) = MEAN(X) and MU(2) = STD(X). This centering and scaling transformation improves the numerical properties of both the polynomial and the fitting algorithm. Warning messages result if N is = length(X),

7、 if X has repeated, or nearly repeated, points, or if X might need centering and scaling. Class support for inputs X,Y: float: double, single See also poly, polyval, roots. Reference page in Help browser doc polyfit（2） help lsqcurvefit LSQCURVEFIT solves non-linear least squares problems. LSQCURVEFI

8、T attempts to solve problems of the form: min sum (FUN(X,XDATA)-YDATA).2 where X, XDATA, YDATA and the values X returned by FUN can be vectors or matrices. X=LSQCURVEFIT(FUN,X0,XDATA,YDATA) starts at X0 and finds coefficients X to best fit the nonlinear functions in FUN to the data YDATA (in the lea

9、st-squares sense). FUN accepts inputs X and XDATA and returns a vector (or matrix) of function values F, where F is the same size as YDATA, evaluated at X and XDATA. NOTE: FUN should return FUN(X,XDATA) and not the sum-of-squares sum(FUN(X,XDATA)-YDATA).2). (FUN(X,XDATA)-YDATA) is squared and summed

10、 implicitly in the algorithm.) X=LSQCURVEFIT(FUN,X0,XDATA,YDATA,LB,UB) defines a set of lower and upper bounds on the design variables, X, so that the solution is in the range LB = X 第二题： 三次线性拟合clear allx=0:0.5:5;y=x.3-6*x.2+5*x-3;y1=y;for i=1:length(y)y1(i)=y1(i)+rand;enda=polyfit(x,y1,3);b=polyval

11、(a,x);plot(x,y,*,x,b),aa =1.0121 -6.1033 5.1933 -2.4782 二次线性拟合clear all x=0:0.5:20; y=x.3-6*x.2+5*x-3; y1=y; for i=1:length(y) y1(i)=y1(i)+rand; end a=polyfit(x,y1,2); b=polyval(a,x); plot(x,y,*,x,b),aa = 23.9982 -232.0179 367.9756 四次线性拟合clear allx=0:0.5:20;y=x.3-6*x.2+5*x-3;y1=y;for j=1:length(y)y1

12、(j)=y1(j)+rand;enda=polyfit(x,y1,4);b=polyval(a,x);plot(x,y,*,x,b),aa = -0.0001 1.0038 -6.0561 5.2890 -2.8249第三题： 拟合曲线为：f(x)= 定义函数：function f=fun(a,x)f=a(1)-(a(1)-a(2)*exp(-a(3)*x);主程序：clear allclcx=0.5 1 2 3 4 5 7 9;y=6.36 6.48 7.26 8.22 8.66 8.99 9.43 9.63;a0=1 1 1;a=lsqcurvefit(fun,a0,x,y);y1=a(1)-(a(1)-a(2)*exp(-a(3)*x);plot(x,y,r*,x,y1,b)V1=a(2)tei=1/a(3)Optimization terminated: rel