x=x0-f(x0)/subs(df(),x0);
ifabs(x-x0)>errorlim
n=n+1;
else
break;
end
x0=x;
end
disp(['迭代次数:
n=',num2str(n)])
disp(['所求非零根:
正根x1=',num2str(x),'负根x2=',num2str(-x)])
(2)子函数非线性函数f
functiony=f(x)
y=log((513+0.6651*x)/(513-0.6651*x))-x/(1400*0.0918);
end
(3)子函数非线性函数的一阶导数df
functiony=df()
symsx1
y=log((513+0.6651*x1)/(513-0.6651*x1))-x1/(1400*0.0918);
y=diff(y);
end
运行结果如下:
迭代次数:
n=5
所求非零根:
正根x1=767.3861负根x2=-767.3861
大作业四
分析:
(1)输出插值多项式。
(2)在区间[-5,5]均匀插入99个节点,计算这些节点上函数f(x)的近似值,并在同一图上画出原函数和插值多项式的图形。
(3)观察龙格现象,计算插值函数在各节点处的误差,并画出误差图。
解:
Matlab程序代码如下:
%此函数实现y=1/(1+4*x^2)的n次Newton插值,n由调用函数时指定
%函数输出为插值结果的系数向量(行向量)和插值多项式
function[ty]=func5(n)
x0=linspace(-5,5,n+1)';
y0=1./(1.+4.*x0.^2);
b=zeros(1,n+1);
fori=1:
n+1
s=0;
forj=1:
i
t=1;
fork=1:
i
ifk~=j
t=(x0(j)-x0(k))*t;
end;
end;
s=s+y0(j)/t;
end;
b(i)=s;
end;
t=linspace(0,0,n+1);
fori=1:
n
s=linspace(0,0,n+1);
s(n+1-i:
n+1)=b(i+1).*poly(x0(1:
i));
t=t+s;
end;
t(n+1)=t(n+1)+b
(1);
y=poly2sym(t);
10次插值运行结果:
[bY]=func5(10)
b=
Columns1through4
-0.00000.00000.0027-0.0000
Columns5through8
-0.0514-0.00000.3920-0.0000
Columns9through11
-1.14330.00001.0000
Y=
-(10035*x^10)/2928+x^9/616+(256*x^8)/93425-x^7/76-(693*x^6)/1312-(3*x^5)/+(36624*x^4)/93425-(5*x^3)/-(32311*x^2)/70496+(7*x)/+1
b为插值多项式系数向量,Y为插值多项式。
插值近似值:
x1=linspace(-5,5,101);
x=x1(2:
100);
y=polyval(b,x)
y=
Columns1through12
2.70033.99944.35154.09743.49262.72371.92111.17150.52740.0154-0.3571-0.5960
Columns13through24
-0.7159-0.7368-0.6810-0.5709-0.4278-0.2704-0.11470.02700.14580.23600.29490.3227
Columns25through36
0.32170.29580.25040.19150.12550.0588-0.0027-0.0537-0.0900-0.1082-0.1062-0.0830
Columns37through48
-0.03900.02450.10520.20000.30500.41580.52800.63690.73790.82690.90020.9549
Columns49through60
0.98861.00000.98860.95490.90020.82690.73790.63690.52800.41580.30500.2000
Columns61through72
0.10520.0245-0.0390-0.0830-0.1062-0.1082-0.0900-0.0537-0.00270.05880.12550.1915
Columns73through84
0.25040.29580.32170.32270.29490.23600.14580.0270-0.1147-0.2704-0.4278-0.5709
Columns85through96
-0.6810-0.7368-0.7159-0.5960-0.35710.01540.52741.17151.92112.72373.49264.0974
Columns97through99
4.35153.99942.7003
绘制原函数和拟合多项式的图形代码:
plot(x,1./(1+4.*x.^2))
holdall
plot(x,y,'r')
xlabel('X')
ylabel('Y')
title('Runge现象')
gtext('原函数')
gtext('十次牛顿插值多项式')
绘制结果:
误差计数并绘制误差图:
holdoff
ey=1./(1+4.*x.^2)-y
ey=
Columns1through12
-2.6900-3.9887-4.3403-4.0857-3.4804-2.7109-1.9077-1.1575-0.5128-0.00000.37330.6130
Columns13through24
0.73390.75580.70100.59210.45020.29430.14010.0000-0.1169-0.2051-0.2617-0.2870
Columns25through36
-0.2832-0.2542-0.2053-0.1424-0.0719-0.00000.06740.12540.16960.19710.20620.1962
Columns37through48
0.16790.12340.06600.0000-0.0691-0.1349-0.1902-0.2270-0.2379-0.2171-0.1649-0.0928
Columns49through60
-0.02710-0.0271-0.0928-0.1649-0.2171-0.2379-0.2270-0.1902-0.1349-0.06910.0000
Columns61through72
0.06600.12340.16790.19620.20620.19710.16960.12540.06740.0000-0.0719-0.1424
Columns73through84
-0.2053-0.2542-0.2832-0.2870-0.2617-0.2051-0.11690.00000.14010.29430.45020.5921
Columns85through96
0.70100.75580.73390.61300.37330.0000-0.5128-1.1575-1.9077-2.7109-3.4804-4.0857
Columns97through99
-4.3403-3.9887-2.6900
plot(x,ey)
xlabel('X')
ylabel('ey')
title('Runge现象误差图')
输出结果为:
大作业五
解:
Matlab程序为:
x=[-520,-280,-156.6,-78,-39.62,-3.1,0,3.1,39.62,78,156.6,280,520]';
y=[0,-30,-36,-35,-28.44,-9.4,0,9.4,28.44,35,36,30,0]';
n=13;
%求解M
fori=1:
1:
n-1
h(i)=x(i+1)-x(i);
end
fori=2:
1:
n-1
a(i)=h(i-1)/(h(i-1)+h(i));
b(i)=1-a(i);
c(i)=6*((y(i+1)-y(i))/h(i)-(y(i)-y(i-1))/h(i-1))/(h(i-1)+h(i));
end
a(n)=h(n-1)/(h
(1)+h(n-1));
b(n)=h
(1)/(h
(1)+h(n-1));
c(n)=6/(h
(1)+h(n-1))*((y
(2)-y
(1))/h
(1)-(y(n)-y(n-1))/h(n-1));
A(1,1)=2;
A(1,2)=b
(2);
A(1,n-1)=a
(2);
A(n-1,n-2)=a(n);
A(n-1,n-1)=2;
A(n-1,1)=b(n);
fori=2:
1:
n-2
A(i,i)=2;
A(i,i+1)=b(i+1);
A(i,i-1)=a(i+1);
end
C=c(2:
n);
C=C';
m=A\C;
M
(1)=m(n-1);
M(2:
n)=m;
xx=-520:
10.4:
520;
fori=1:
51
forj=1:
1:
n-1
ifx(j)<=xx(i)&&xx(i)break;
end
end
yy(i)=M(j+1)*(xx(i)-x(j))^3/(6*h(j))-M(j)*(xx(i)-x(j+1))^3/(6*h(j))+(y(j+1)-M(j+1)*h(j)^2/6)*(xx(i)-x(j))/h(j)-(y(j)-M(j)*h(j)^2/6)*(xx(i)-x(j+1))/h(j);
end;
fori=52:
101
yy(i)=-yy(102-i);
end;
fori=1:
50
xx(i)=-xx(i);
end
plot(xx,yy);
holdon;
fori=1:
1:
n/2
x(i)=-x(i);
end
plot(x,y,'bd');
title('机翼外形曲线');
输出结果:
运行jywx.m文件,得到
2.
(1)编制求第一型3次样条插值函数的通用程序;
(2)已知汽车门曲线型值点的数据如下:
解:
(1)Matlab编制求第一型3次样条插值函数的通用程序:
function[Sx]=Threch(X,Y,dy0,dyn)
%X为输入变量x的数值
%Y为函数值y的数值
%dy0为左端一阶导数值
%dyn为右端一阶导数值
%Sx为输出的函数表达式
n=length(X)-1;
d=zeros(n+1,1);
h=zeros(1,n-1);
f1=zeros(1,n-1);
f2=zeros(1,n-2);
fori=1:
n%求函数的一阶差商
h(i)=X(i+1)-X(i);
f1(i)=(Y(i+1)-Y(i))/h(i);
end
fori=2:
n%求函数的二阶差商
f2(i)=(f1(i)-f1(i-1))/(X(i+1)-X(i-1));
d(i)=6*f2(i);
end
d
(1)=6*(f1
(1)-dy0)/h
(1);
d(n+1)=6*(dyn-f1(n-1))/h(n-1);
%赋初值
A=zeros(n+1,n+1);
B=zeros(1,n-1);
C=zeros(1,n-1);
fori=1:
n-1
B(i)=h(i)/(h(i)+h(i+1));
C(i)=1-B(i);
end
A(1,2)=1;
A(n+1,n)=1;
fori=1:
n+1
A(i,i)=2;
end
fori=2:
n
A(i,i-1)=B(i-1);
A(i,i+1)=C(i-1);
end
M=A\d;
symsx;
fori=1:
n
Sx(i)=collect(Y(i)+(f1(i)-(M(i)/3+M(i+1)/6)*h(i))*(x-X(i))+M(i)/2*(x-X(i))^2+(M(i+1)-M(i))/(6*h(i))*(x-X(i))^3);
digits(4);
Sx(i)=vpa(Sx(i));
end
fori=1:
n
disp('S(x)=');
fprintf('%s(%d,%d)\n',char(Sx(i)),X(i),X(i+1));
end
S=zeros(1,n);
fori=1:
n
x=X(i)+0.5;
S(i)=Y(i)+(f1(i)-(M(i)/3+M(i+1)/6)*h(i))*(x-X(i))+M(i)/2*(x-X(i))^2+(M(i+1)-M(i))/(6*h(i))*(x-X(i))^3;
end
disp('S(i+0.5)');
disp('iX(i+0.5)S(i+0.5)');
fori=1:
n
fprintf('%d%.4f%.4f\n',i,X(i)+0.5,S(i));
end
end
输出结果:
>>X=[012345678910];
>>Y=[2.513.304.044.705.225.545.785.405.575.705.80];
>>Threch(X,Y,0.8,0.2)
S(x)=
0.8*x-0.001486*x^2-0.008514*x^3+2.51(0,1)
S(x)=
0.8122*x-0.01365*x^2-0.004458*x^3+2.506(1,2)
S(x)=
0.8218*x-0.01849*x^2-0.003652*x^3+2.499(2,3)
S(x)=
0.317*x^2-0.1847*x-0.04093*x^3+3.506(3,4)
S(x)=
6.934*x-1.463*x^2+0.1074*x^3-5.986(4,5)
S(x)=
4.177*x^2-21.26*x-0.2686*x^3+41.01(5,6)
S(x)=
53.86*x-8.344*x^2+0.427*x^3-109.2(6,7)
S(x)=
6.282*x^2-48.52*x-0.2694*x^3+129.6(7,8)
S(x)=
14.88*x-1.643*x^2+0.06076*x^3-39.42(8,9)
S(x)=
8.966*x-0.986*x^2+0.03641*x^3-21.67(9,10)
S(i+0.5)
iX(i+0.5)S(i+0.5)
10.50002.9086
21.50003.6784
32.50004.3815
43.50004.9882
54.50005.3833
65.50005.7237
76.50005.5943
87.50005.4302
98.50005.6585
109.50005.7371
ans=
[-0.008514*x^3-0.001486*x^2+0.8*x+2.51,-0.004458*x^3-0.01365*x^2+0.8122*x+2.506,-0.003652*x^3-0.01849*x^2+0.8218*x+2.499,-0.04093*x^3+0.317*x^2-0.1847*x+3.506,0.1074*x^3-1.463*x^2+6.934*x-5.986,-0.2686*x^3+4.177*x^2-21.26*x+41.01,0.427*x^3-8.344*x^2+53.86*x-109.2,-0.2694*x^3+6.282*x^2-48.52*x+129.6,0.06076*x^3-1.643*x^2+14.88*x-39.42,0.03641*x^3-0.986*x^2+8.966*x-21.67]
大作业六
1、炼钢厂出钢时所用的圣刚睡的钢包,在使用过程中由于钢液及炉渣对包衬耐火材料的侵蚀,使其容积不断增大,经试验,钢包的容积与相应的使用次数的数据如下:
(使用次数x,容积y)
x
y
x
y
2
106.42
9
110.59
3
108.26
10
110.60
5
109.58
14
110.72
6
109.50
16
110.90
7
109.86
17
110.76
9
110.00
19
111.10
10
109.93
20
111.30
选用双曲线
对使用最小二乘法数据进行拟合。
解:
Matlab程序如下:
functiona=nihehanshu()
x0=[2356791011121416171920];
y0=[106.42108.26109.58109.50109.86110.00109.93110.59110.60110.72110.90110.76111.10111.30];
A=zeros(2,2);
B=zeros(2,1);
a=zeros(2,1);
x=1./x0;
y=1./y0;
A(1,1)=14;
A(1,2)=sum(x);
A(2,1)=A(1,2);
A(2,2)=sum(x.^2);
B
(1)=sum(y);
B
(2)=sum(x.*y);
a=A\B;
y=1./(a
(1)+a
(2)*1./x0);
subplot(1,2,2);
plot(x0,y0-y,'bd-');
title('拟合曲线误差');
subplot(1,2,1);
plot(x0,y0,'go');
holdon;
x=2:
0.5:
20;
y=1./(a
(1)+a
(2)*1./x);
plot(x,y,'r*-');
legend('散点','拟合曲线图1/y=a
(1)+a
(2)*1/x');
title('最小二乘法拟合曲线');
试验所求的系数为:
nihehanshu
ans=
0.2446
0.9705
则拟合曲线为
拟合曲线图、散点图、误差图如下:
2、下面给出的是乌鲁木齐最近1个月早晨7:
00左右(时间)的天气预报所得到的温度,按照数据找出任意次曲线拟合方程和它的图像。
用MATLAB编程对上述数据进行最小二乘拟合。
2008年10月--11月26日
天数
1
2
3
4
5
6
7
8
9
10
温度
9
10
11
12
13
14
13
12
11
9
天数
11
12
13
14
15
16
17
18
19
20
温度
10
11
12
13
14
12
11
10
9
8
天数
21
22
23
24
25
26
27
28
29
30
温度
7
8
9
11
9
7
6
5
3
1
解:
Matlab的程序如下:
x=[1:
1:
30];
y=[9,10,11,12,13,14,13,12,11,9,10,11,12,13,14,12,11,10,9,8,7,8,9,11,9,7,6,5,3,1];
a1=polyfit(x,y,3)%三次多项式拟合%
a2=polyfit(x,y,9)%九次多项式拟合