文章对应视频讲解：(cubic spline)三次样条插值 Ⅰ、Ⅱ、Ⅲ 型

主要以实现为主,理论部分不做详细说明

一、三次样条插值

已知： $s(x_i)=f_i,i=1,2,\cdots ,n-1$ 。

Ⅰ型边界条件：$s(x_0)=f_0,s(x_n)=f_n{s}^{\prime }(x_0)=f_0^{\prime },s^{\prime }(x_n)=f_n^{\prime }$

Ⅱ型边界条件：$s(x_0)=f_0,s(x_n)=f_n{s}^{\prime \prime }(x_0)=f_0^{\prime \prime },s^{\prime \prime }(x_n)=f_n^{\prime \prime }$

Ⅲ型边界条件：$s(x_0)=f_0,s(x_n)=s(x_0)s^{\prime }(x_0)=s^{\prime }(x_n),s^{\prime \prime }(x_0)=s^{\prime \prime }(x_n)$

设 $h_i=x_i-x_{i-1},e_i$ 为 $[x_{i-1},x_i]$构成的小区间,$i=1,2,\cdots ,n$
$$M_i=s_I^{\prime \prime }(x_i),i=0,1,\cdots ,n$$

满足三次样条函数 $s(x)$
$$\begin{array}{rl}s(x)=& \frac{1}{6h_i}\left[{\left(x_i-x\right)}^3{M}_{i-1}+{\left(x-x_{i-1}\right)}^3{M}_i\right]\\ & +\frac{1}{h_i}\left[\left(x_i-x\right)f_{i-1}+\left(x-x_{i-1}\right)f_i\right]\\ & -\frac{h_i}{6}\left[\left(x_i-x\right)M_{i-1}+\left(x-x_{i-1}\right)M_i\right]\end{array}$$
其中$x\in e_i,i=1,2,\cdots ,n$
$${\lambda }_j=\frac{h_{j+1}}{h_j+h_{j+1}},{\mu }_j=1-{\lambda }_j=\frac{h_j}{h_j+h_{j+1}}$$
有
$$\begin{array}{c}{\mu }_j{M}_{j-1}+2M_j+{\lambda }_j{M}_{j+1}=d_j,j=1,2,\cdots ,n-1\end{array}$$
其中$d_j=6f\left[x_{j-1},x_j,x_{j+1}\right]$

Ⅰ型边界条件下有：

$$\begin{array}{c}\{\begin{array}{l}2M_0+M_1=\frac{6}{h_1}\left(\frac{f_1-f_0}{h_1}-f_0^{\prime }\right):=d_0\\ M_{n-1}+2M_n=\frac{6}{h_n}\left(f_n^{\prime }-\frac{f_n-f_{n-1}}{h_n}\right):=d_n\end{array}\end{array}$$
联立(1)和(2)，得线性方程组
$$\begin{array}{c}\left[\begin{array}{cccccc}2& 1& & & & \\ {\mu }_1& 2& {\lambda }_1& & & \\ & {\mu }_2& 2& {\lambda }_2& & \\ & & \ddots & \ddots & \ddots & \\ & & & {\mu }_{n-1}& 2& {\lambda }_{n-1}\\ & & & & 1& 2\end{array}\right]\left[\begin{array}{c}{M}_0\\ M_1\\ M_2\\ ⋮\\ M_{n-1}\\ M_n\end{array}\right]=\left[\begin{array}{c}{d}_0\\ d_1\\ d_2\\ ⋮\\ d_{n-1}\\ d_n\end{array}\right]\end{array}$$

Ⅱ 型边界条件下有：

$$\begin{array}{c}{M}_0=f_0^{\prime \prime },M_n=f_n^{\prime \prime }\end{array}$$
联立(1)和(4)，得线性方程组
$$\begin{array}{c}\left[\begin{array}{cccccc}1& & & & & \\ {\mu }_1& 2& {\lambda }_1& & & \\ & {\mu }_2& 2& {\lambda }_2& & \\ & & \ddots & \ddots & \ddots & \\ & & & {\mu }_{n-1}& 2& {\lambda }_{n-1}\\ & & & & & 1\end{array}\right]\left[\begin{array}{c}{M}_0\\ M_1\\ M_2\\ ⋮\\ M_{n-1}\\ M_n\end{array}\right]=\left[\begin{array}{c}{f}_0^{\prime \prime }\\ d_1\\ d_2\\ ⋮\\ d_{n-1}\\ f_n^{\prime \prime }\end{array}\right]\end{array}$$
进一步化简得 n-1阶三对角方程组
$$\begin{array}{c}\left[\begin{array}{ccccc}2& {\lambda }_1& & & \\ {\mu }_2& 2& {\lambda }_2& & \\ & \ddots & \ddots & \ddots & \\ & & {\mu }_{n-2}& 2& {\lambda }_{n-2}\\ & & & {\mu }_{n-1}& 2\end{array}\right]\left[\begin{array}{c}{M}_1\\ M_2\\ ⋮\\ M_{n-2}\\ M_{n-1}\end{array}\right]=\left[\begin{array}{c}{d}_1-{\mu }_1{f}_0^{\prime \prime }\\ d_2\\ ⋮\\ d_{n-2}\\ d_{n-1}-{\lambda }_{n-1}{f}_n^{\prime \prime }\end{array}\right]\end{array}$$

Ⅲ 型边界条件下有：

$$\begin{array}{c}{M}_0=M_n,{\lambda }_n{M}_1+{\mu }_n{M}_{n-1}+2M_n=d_n\end{array}$$
其中
$$\begin{gathered} {\lambda }_n=h_1/(h_1+h_n),{\mu }_n=h_n/(h_1+h_n) \\ {d}_n=6(f[x_0,x_1]-f[x_{n-1},x_n])/(h_1+h_n) \end{gathered}$$
联立(1)和(7),得线性方程组
$$\begin{array}{c}\left[\begin{array}{cccccc}1& & & & & -1\\ {\mu }_1& 2& {\lambda }_1& & & \\ & {\mu }_2& 2& {\lambda }_2& & \\ & & \ddots & \ddots & \ddots & \\ & & & {\mu }_{n-1}& 2& {\lambda }_{n-1}\\ & {\lambda }_n& & & {\mu }_n& 2\end{array}\right]\left[\begin{array}{c}{M}_0\\ M_1\\ M_2\\ ⋮\\ M_{n-1}\\ M_n\end{array}\right]=\left[\begin{array}{c}0\\ d_1\\ d_2\\ ⋮\\ d_{n-1}\\ d_n\end{array}\right]\end{array}$$
进一步化简得 n阶线性方程组
$$\begin{array}{c}\left[\begin{array}{ccccc}2& {\lambda }_1& & & {\mu }_1\\ {\mu }_2& 2& {\lambda }_2& & \\ & \ddots & \ddots & \ddots & \\ & & {\mu }_{n-1}& 2& {\lambda }_{n-1}\\ {\lambda }_n& & & {\mu }_n& 2\end{array}\right]\left[\begin{array}{c}{M}_1\\ M_2\\ ⋮\\ M_{n-1}\\ M_n\end{array}\right]=\left[\begin{array}{c}{d}_1\\ d_2\\ ⋮\\ d_{n-1}\\ d_n\end{array}\right]\end{array}$$
最终，通过解线性方程组，可以得到$M_0,M_1,\cdots ,M_n$，代入到$s(x)$中，得三次样条插值函数

二、算法

对I型进行详细介绍, II,III型是类似的,不作详细说明

Ⅰ型三次样条插值

输入:

• $n+1$ 个插值节点$(x_i,y_i),i=0,1,\cdots ,n$ 构成向量 $x_0,y_0$
• I型边界条件 $f_0^{\prime },f_n^{\prime }$
• 目标近似点 $x$

输出

• 近似点的值 $y$

实现步骤

• 步骤 $1$: 计算 $h_i=x_i-x_{i-1},i=1,2,\cdots ,n$$${\lambda }_j=\frac{h_{j+1}}{h_j+h_{j+1}},{\mu }_j=1-{\lambda }_j$$$$d_j=6f\left[x_{j-1},x_j,x_{j+1}\right]\text{(二阶差商)}j=1,2\cdots ,n-1$$ $$\begin{gathered} d_0=\frac{6}{h_1}\left(\frac{f_1-f_0}{h_1}-f_0^{\prime }\right) \\ {d}_n=\frac{6}{h_n}\left(f_n^{\prime }-\frac{f_n-f_{n-1}}{h_n}\right) \end{gathered}$$

• 步骤 $2$: 代入线性方程组(3),并用追赶法解三对角方程组，得$M_0,M_1,\cdots ,M_n$

• 步骤 $3$: (侧重点:如何把分段的效果表示出来)
  
  得到 $x$点处的插值近似值$s(x)$

Ⅱ型三次样条插值

输入

• $n+1$ 个插值节点$(x_i,y_i),i=0,1,\cdots ,n$ 构成向量 $x_0,y_0$
• II 型边界条件 $f_0^{\prime \prime },f_n^{\prime \prime }$

追赶法解三对角方程组(6)

Ⅲ型三次样条插值

前提:判断满足III型条件

输入

• $n+1$ 个插值节点$(x_i,y_i),i=0,1,\cdots ,n$ 构成向量 $x_0,y_0$

另外
$${\lambda }_n=h_1/(h_1+h_n),{\mu }_n=h_n/(h_1+h_n)$$$$d_n=6(f[x_0,x_1]-f[x_{n-1},x_n])/(h_1+h_n)$$

解线性方程组(9)

三、北太天元 or matlab实现

Ⅰ型

function [s,M] = spline1_interp(x0,y0,df0,dfn,x)
% I型三次样条插值 
% Input: 节点向量x0,y0,两个端点的一阶导 df0,df1
%        目标点 x
% Output: 插值结果 s , M
%   子函数：divided_differences,tridiag_chase
%   Version:            1.0
%   last modified:      04/14/2024    
    n = length(x0);
    h = zeros(1,n-1); h = x0(2:n)-x0(1:n-1);
    nh = length(h);
    lamda = zeros(1,nh-1); 
    lamda = h(2:nh)./(h(1:nh-1)+h(2:nh));
    mu = 1-lamda;
    d= zeros(1,n);
    % 计算差商
    D = divided_differences(x0,y0,4)
    d(2:n-1) = D(3:n,4); %取二阶差商
    d(1) = 6/h(1) * ((y0(2)-y0(1))/h(1) - df0);
    d(n) = 6/h(n-1) * (dfn - (y0(n)-y0(n-1))/h(n-1));
    % 表示三对角方程组
    A = diag(2*ones(1,n),0) + diag([mu,1],-1) +diag([1,lamda],1);
    % 解三对角方程组
    [M]=tridiag_chase(A,d);
    % 分段表示
    nx = length(x);
    s = zeros(1,nx);
% 对于每个 x
for j = 1:1:nx 
    % 判断在哪个小区间
    for i = 1:n-1
        if x(j) >= x0(i) && x(j) <= x0(i+1)
            hi = h(i); t2 = x0(i+1);t1 = x0(i); 
            M2 = M(i+1);M1 = M(i);
            s(j) =1/(6*hi) * ((t2-x(j))^3 *M1 +(x(j) -t1)^3*M2);
            s(j) = s(j) + 1/hi *( (t2-x(j)) * y0(i) + (x(j)- t1)*y0(i+1) );
            s(j) = s(j) - hi/6 * ( (t2-x(j))*M1 + (x(j)-t1)*M2 );
            break;
        end
    end
end
end

保存为spline1_interp.m文件

Ⅱ型

function [s,M] = spline2_interp(x0,y0,dff0,dffn,x)
% II型三次样条插值 
% Input: 节点向量x0,y0,两个端点的二阶导 dff0,dff1
%        目标点 x
% Output: 插值结果 s , M
%   子函数：divided_differences,tridiag_chase
%   Version:            1.0
%   last modified:      04/14/2024    
    n = length(x0);
    h = zeros(1,n-1); h = x0(2:n)-x0(1:n-1);
    nh = length(h);
    lamda = zeros(1,nh-1); 
    lamda = h(2:nh)./(h(1:nh-1)+h(2:nh));
    mu = 1-lamda;
    d= zeros(1,n-2);
    % 计算差商
    D = divided_differences(x0,y0,4);
    d = D(3:n,4); %取二阶差商
    d(1) = d(1) - mu(1)*dff0;
    d(n-2) = d(n-2) - lamda(n-2)*dffn;
    % 表示三对角方程组
    A = diag(2*ones(1,length(d)),0) + diag(mu(2:nh-1),-1) +diag(lamda(1:nh-2),1);
    % 解三对角方程组
    [M]=tridiag_chase(A,d); % 得到的M 是列向量
    M = [dff0;M;dffn];
    % 分段表示
    nx = length(x);
    s = zeros(1,nx);
% 对于每个 x
for j = 1:1:nx 
    % 判断在哪个小区间
    for i = 1:n-1
        if x(j) >= x0(i) && x(j) <= x0(i+1)
            hi = h(i); t2 = x0(i+1);t1 = x0(i); 
            M2 = M(i+1);M1 = M(i);
            s(j) =1/(6*hi) * ((t2-x(j))^3 *M1 +(x(j) -t1)^3*M2);
            s(j) = s(j) + 1/hi *( (t2-x(j)) * y0(i) + (x(j)- t1)*y0(i+1) );
            s(j) = s(j) - hi/6 * ( (t2-x(j))*M1 + (x(j)-t1)*M2 );
            break;
        end
    end
end
end

保存为 spline2_interp.m 文件

Ⅲ型

function [s,M] = spline3_interp(x0,y0,x)
% III 型三次样条插值 
% Input: 节点向量x0,y0
%        目标点 x
% Output: 插值结果 x , M
%   子函数：divided_differences,myJocabi
%   Version:            1.0
%   last modified:      04/14/2024    
    n = length(x0);
    h = zeros(1,n-1); h = x0(2:n)-x0(1:n-1);
    nh = length(h);
    lamda = zeros(1,nh-1); 
    lamda = h(2:nh)./(h(1:nh-1)+h(2:nh));
    mu = 1-lamda;
    d= zeros(1,n-1);
    % 计算差商
    D = divided_differences(x0,y0,4);
    d(1:n-2) = D(3:n,4) %取二阶差商
    % 从D中取差商时,注意位置和n的关系
    d(n-1) = 6*(D(2,3)-D(n,3))/(h(1)+h(nh));
    mu = [mu,h(nh)/(h(1)+h(nh))];
    % 表示方程组
    A = diag(2*ones(1,length(d)),0) + diag(mu(2:nh),-1) +diag(lamda,1);
    A(1,n-1) = mu(1);
    A(n-1,1) = h(1)/(h(1)+h(nh));
    % 用Jacobi迭代法进行求解
    [M]=myJacobi(A,d',zeros(1,length(d)),10e-8,100); % 得到的M 是列向量
    M = M(:,end);
    M=[M(1);M];
    % 分段表示
    nx = length(x);
    s = zeros(1,nx);
% 对于每个 x
for j = 1:1:nx 
    % 判断在哪个小区间
    for i = 1:n-1
        if x(j) >= x0(i) && x(j) <= x0(i+1)
            hi = h(i); t2 = x0(i+1);t1 = x0(i); 
            M2 = M(i+1);M1 = M(i);
            s(j) =1/(6*hi) * ((t2-x(j))^3 *M1 +(x(j) -t1)^3*M2);
            s(j) = s(j) + 1/hi *( (t2-x(j)) * y0(i) + (x(j)- t1)*y0(i+1) );
            s(j) = s(j) - hi/6 * ( (t2-x(j))*M1 + (x(j)-t1)*M2 );
            break;
        end
    end
end
end

保存为 spline3_interp.m 文件

四、数值算例

1. 课本上一个例子

求三次插值样条函数$s(x)$,边界条件$$s^{\prime }(x_0)=f_0^{\prime }=1,s^{\prime }(x_3)=f_3^{\prime }=0$$
插值条件

满足Ⅰ型插值条件，使用Ⅰ次样条插值：

%% 1 I 型三次样条插值 课本P55 例2.13
clc;clear all;
x0 = [0 1 2 3];
y0 = [0 0 0 0];
df0 = 1;dfn = 0;
x = 1/2;
[s,M] = spline1_interp(x0,y0,df0,dfn,x)

x= linspace(0,3);
s = spline1_interp(x0,y0,df0,dfn,x);
plot(x,s)
hold on
plot (x0,y0,'o')

得到

M = 
   -3.4667
    0.9333
   -0.2667
    0.1333

图像如下

2. Runge函数下的表现

$$f(x)=\frac{1}{x^2+1},x\in [-5,5]$$

%% 4 Runge函数下的表现 
clc;clear all;format long;
Runge = @(x)1./(x.^2+1);
x0 = linspace(-5,5,50);
y0 = Runge(x0);
df0 = 10/(25+1)^2; dfn = -10/(25+1)^2;

x=linspace(-5,5,200);
y = Runge(x);
s = spline1_interp(x0,y0,df0,dfn,x);
plot(x,y,'r');
hold on
    plot(x,s,'b');
    plot(x0,y0,'o');
    legend('f(x)','s(x)')
    hold off

令 delta = max(abs(y- s));
delta随 n 的变化

可以看到,随着n的增大,误差越来越小,$s(x)$ 越趋近于 $f(x)$

3. Ⅱ型

%% 2 II 型三次样条插值  y = x^3
clc;clear all;
x0 = linspace(0,3,10);
y0 = x0.^3;
dff0 = 0;dffn = 18;
x= linspace(0,3,200);
y = x.^3;
[s,M] = spline2_interp(x0,y0,dff0,dffn,x);
plot(x,y,'r');
hold on
    plot(x,s,'b');
hold off

所编写的Ⅱ型三次样条插值代码也是可以正常运行的.

4. Ⅲ型

%% 3 III 型三次样条插值  y =sin(x)
clc;clear all;
x0 = linspace(0,4*pi,50);
y0 = sin(x0);
x= linspace(0,4*pi,200);
y = sin(x);
[s,M] = spline3_interp(x0,y0,x);
plot(x,y,'r');
hold on
    plot(x,s,'b');
    plot(x0,y0,'o');
    legend('sin(x)','s(x)')
    hold off

Ⅲ型一般用于循环的情况. 可以看到,我们的代码实现效果是正常的.

三次样条插值的主要优点:

• 平滑性： 二阶连续可微
• 全局性： 考虑了所有的数据点，能够反映整体数据的趋势和特征

在实现过程中用到

• 二阶差商的计算
• 追赶法解三对角
• Jacobi迭代法解线性方程组