计算方法上机报告（姚威）

Px =- (4824449974797021*x^3)/576460752303423488 (4095195769401027*x)/18014398509481984 + 7832/585

估算误差e =131.0618

+ (6142*x^2)/29601 -

拟合四次多项式如下及相应的系数

p =0.0009 -0.0532 0.8909 -3.7050 16.7939 Px =(4311222046072087*x^4)/4611686018427387904 - (959118195350923*x^3)/18014398509481984 + (8024210619242803*x^2)/9007199254740992 - (8342830140798239*x)/2251799813685248 + 590881761455037/35184372088832

估算误差e =59.0412 拟合五次多项式如下及相应的系数

p =0.0001 -0.0045 0.0623 -0.1202 -0.4997 14.9504 Px =(3360357177283195*x^5)/36893488147419103232 - (5222864195889131*x^4)/1152921504606846976 + (4487849523002499*x^3)/72057594037927936 - (8657808537992275*x^2)/72057594037927936 - (4500908134762693*x)/9007199254740992 + 8416302560899915/562949953421312

估算误差e =33.1446 拟合六次多项式如下及相应的系数

p =-0.0000 0.0005 -0.0160 0.2069 -0.9506 1.2804 14.3314 Px =- (6967708126795581*x^6)/1180591620717411303424 + (2379712557858713*x^5)/4611686018427387904 - (2311817739804005*x^4)/144115188075855872 + (3726966007444799*x^3)/18014398509481984 - (4280910250199293*x^2)/4503599627370496 + (5766485188992375*x)/4503599627370496 + 8067842731307123/562949953421312

估算误差e =29.1139 拟合七次多项式如下及相应的系数

p =-0.0000 0.0001 -0.0035 0.0507 -0.3597 1.3563 -2.2558 15.0741 Px =- (6901230370958207*x^7)/4722366482869645213696 + (8622383106910967*x^6)/73786976294838206464 - (4084605813955375*x^5)/1152921504606846976 + (7309441973132019*x^4)/144115188075855872 - (6479824387411487*x^3)/18014398509481984 + (6108410266124833*x^2)/4503599627370496 - (2539858743572183*x)/1125899906842624 + 8485987341360957/562949953421312

估算误差e =20.1716 拟合八次多项式如下及相应的系数

p =0.0000 -0.0000 0.0003 -0.0083 0.1043 -0.6909 2.3648 -3.4106 15.2169

Px = (2341429260780705*x^8)/37778931862957161709568 - (4374797679237899*x^7)/590295810358705651712 + (3220983078808941*x^6)/9223372036854775808 - (4772103004746209*x^5)/576460752303423488 + (1879800419770211*x^4)/18014398509481984 - (3111697315246825*x^3)/4503599627370496 + (41601748837387*x^2)/17592186044416 - (7679972061905371*x)/2251799813685248 + 8566378569142467/562949953421312

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估算误差e =19.6054 Matlab输出图形如图所示

最小二乘法一阶拟合近似曲线35353025252020151510505101520251515202025最小二乘法二阶拟合近似曲线最小二乘法三阶拟合近似曲线3535最小二乘法四阶拟合近似曲线30303025101005101520251005101520250510152025最小二乘法五阶拟合近似曲线3535最小二乘法六阶拟合近似曲线35最小二乘法八阶拟合近似曲线最小二乘法七阶拟合近似曲线3530303030252520201515100510152025100510152025252520201515100510152025100510152025

法二：多项式拟合使用一个多项式逼近一组给定的数据，是数据分析上常用的方法，使用polyfit函数实现。拟合的准则是最小二乘法，即找出使?f(xi)?yi最小的f(x)。

i?1n23.4.2 基于最小二乘法ployfit函数法

%法方程法最小二乘法 clear clc x=0:24;

y=[15 14 14 14 14 15 16 18 20 20 23 25 28 31 34 31 29 27 25 24 22 20 18 17 16]; for m=1:4

p=polyfit(x,y,m); p1=polyval(p,x); y1=polyval(p,x); xi=0:0.01:24; yi=polyval(p,xi); figure(1)

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subplot(2,2,m); plot(xi,yi);

plot(x,y,'.',xi,yi,'r','LineWidth',2); grid on

disp('各阶估算误差') e(m)=sum((y-y1).^2); end

subplot(2,2,1);

title('最小二乘法一阶拟合近似曲线'); subplot(2,2,2);

title('最小二乘法二阶拟合近似曲线'); subplot(2,2,3);

title('最小二乘法三阶拟合近似曲线'); subplot(2,2,4);

title('最小二乘法四阶拟合近似曲线');

for m=5:8

p=polyfit(x,y,m); p1=polyval(p,x); y1=polyval(p,x); xi=0:0.01:24; yi=polyval(p,xi); figure(2) m=m-4;

subplot(2,2,m); plot(xi,yi);

plot(x,y,'.',xi,yi,'r','LineWidth',2); grid on

disp('各阶估算误差') e(m+4)=sum((y-y1).^2); end

subplot(2,2,1);

title('最小二乘法五阶拟合近似曲线'); subplot(2,2,2);

title('最小二乘法六阶拟合近似曲线'); subplot(2,2,3);

title('最小二乘法七阶拟合近似曲线'); subplot(2,2,4);

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title('最小二乘法八阶拟合近似曲线');

for m=9:12

p=polyfit(x,y,m) p1=polyval(p,x); y1=polyval(p,x); xi=0:0.01:24; yi=polyval(p,xi); figure(3) m=m-8;

subplot(2,2,m); plot(xi,yi);

plot(x,y,'.',xi,yi,'r','LineWidth',2); grid on

disp('各阶估算误差') e(m+8)=sum((y-y1).^2); end

subplot(2,2,1);

title('最小二乘法九阶拟合近似曲线'); subplot(2,2,2);

title('最小二乘法十阶拟合近似曲线'); subplot(2,2,3);

title('最小二乘法十一阶拟合近似曲线'); subplot(2,2,4);

title('最小二乘法十二阶拟合近似曲线');

for m=13:16 p=polyfit(x,y,m); p1=polyval(p,x); y1=polyval(p,x); xi=0:0.01:24; yi=polyval(p,xi); figure(4) m=m-12;

subplot(2,2,m); plot(xi,yi);

plot(x,y,'.',xi,yi,'r','LineWidth',2); grid on

disp('各阶估算误差')

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