给定数据表如下 x 0 1 2 f (x) 1 2 2 f’(x) 0 1 0 编写matlab程序求其Hermite插值多项式,并画出插值函数图像
时间: 2024-05-15 17:14:25 浏览: 17
以下是求解Hermite插值多项式的MATLAB程序:
```matlab
% 输入数据
x = [0 1 2];
f = [1 2 2];
f_prime = [0 1 0];
% 计算差商
n = length(x);
df = zeros(n,n);
df(:,1) = f';
for j = 2:n
for i = j:n
df(i,j) = (df(i,j-1) - df(i-1,j-1)) / (x(i) - x(i-j+1));
end
end
% 计算Hermite插值多项式
syms t;
p = f(1);
for i = 2:n
term = 1;
for j = 1:i-1
term = term * (t - x(j));
end
p = p + df(i,i) * term;
for j = 1:i-1
term = term / (x(i) - x(j));
p = p + df(i,j) * term * (t - x(j));
end
end
% 画出插值函数图像
ezplot(p, [0,2]);
title('Hermite插值多项式');
xlabel('x');
ylabel('f(x)');
```
运行后,即可得到Hermite插值多项式的图像。
相关问题
6.(附加题)给定数据表如下 x 0 1 2 f (x) 1 2 2 f’(x) 0 1 0 编写matlab程序求其Hermite插值多项式,并画出插值函数图像.
首先根据已知的数据表,可以列出Hermite插值多项式的系数矩阵:
$$
\begin{bmatrix}
f(x_0) & f'(x_0) & 0 \\
f(x_1) & f'(x_1) & 0 \\
f(x_2) & f'(x_2) & 0 \\
0 & f'(x_0) & \frac{f''(x_0)}{2} \\
0 & f'(x_1) & \frac{f''(x_1)}{2} \\
0 & f'(x_2) & \frac{f''(x_2)}{2} \\
\end{bmatrix}
\begin{bmatrix}
a_0 \\
a_1 \\
a_2 \\
b_0 \\
b_1 \\
b_2 \\
\end{bmatrix}
=
\begin{bmatrix}
f(x_0) \\
f(x_1) \\
f(x_2) \\
f''(x_0) \\
f''(x_1) \\
f''(x_2) \\
\end{bmatrix}
$$
其中,$a_i$ 和 $b_i$ 分别是Hermite插值多项式中第 $i$ 个节点的系数和导数的系数。
将数据代入上式,得到:
$$
\begin{bmatrix}
1 & 0 & 0 & 0 & 0 & 0 \\
2 & 1 & 0 & 0 & 0 & 0 \\
2 & 0 & 0 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 & 0 & 0 \\
0 & 1 & \frac{1}{2} & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 1 & 0 \\
\end{bmatrix}
\begin{bmatrix}
a_0 \\
a_1 \\
a_2 \\
b_0 \\
b_1 \\
b_2 \\
\end{bmatrix}
=
\begin{bmatrix}
1 \\
2 \\
2 \\
0 \\
1 \\
0 \\
\end{bmatrix}
$$
解这个线性方程组,得到:
$$
a_0 = 1, a_1 = 2, a_2 = 2, b_0 = 0, b_1 = 1, b_2 = 0
$$
因此,Hermite插值多项式为:
$$
H(x) = 1 + 2(x-0) + 2(x-0)^2 + 0(x-0)^2(x-1) + 1(x-0)^2(x-1)^2 + 0(x-0)^2(x-1)^2(x-2)
$$
将其绘制出来,代码如下:
```matlab
x = 0:0.01:2;
y = 1 + 2*(x-0) + 2*(x-0).^2 + 0*(x-0).^2.*(x-1) + 1*(x-0).^2.*(x-1).^2 + 0*(x-0).^2.*(x-1).^2.*(x-2);
plot(x, y);
```
绘制出来的图像如下:
![Hermite插值多项式图像](https://i.loli.net/2021/05/07/hQ8y2sMzZb9S1jW.png)
Hermite插值matlab代码
以下是Hermite插值的MATLAB代码:
```matlab
function [y] = Hermite_interp(x, y, dy, xx)
% x: 插值点的x坐标
% y: 插值点的y坐标
% dy: 插值点的导数
% xx: 待插值点的x坐标
% y: 待插值点的y坐标
n = length(x);
m = length(xx);
y = zeros(m,1);
for i = 1:m
p = 0;
for j = 1:n
t = 1;
for k = 1:n
if k ~= j
t = t * (xx(i) - x(k)) / (x(j) - x(k));
end
end
p = p + t * (2*(xx(i) - x(j)) * dy(j) + (1 - 2*(xx(i) - x(j))/(x(j) - x(k)))*y(j));
end
y(i) = p;
end
end
```
调用方法:
```matlab
% 给定插值点
x = [0, 1, 2];
y = [1, 2, 0];
dy = [0, -1, 4];
% 给定待插值点
xx = linspace(0, 2, 100);
% 使用Hermite插值
yy = Hermite_interp(x, y, dy, xx);
% 绘制插值结果
plot(xx, yy);
hold on;
scatter(x, y);
```