三次插值样条方程的函数
时间: 2023-09-24 10:11:13 浏览: 46
以下是用于计算三次插值样条方程的函数:
```matlab
function [s,coeff] = cubic_spline(x,y)
% x和y分别是数据点的横纵坐标,输出s是插值函数的函数句柄,coeff是插值函数的系数矩阵
n = length(x);
dx = zeros(1,n-1);
dy = zeros(1,n-1);
vl = 0;
vn = 0;
A = zeros(n,n);
r = zeros(n,1);
for i = 1:n-1 %定义deltas
dx(i) = x(i+1) - x(i);
dy(i) = y(i+1) - y(i);
end
for i=2:n-1 %加载A矩阵
A(i,i-1:i+1) = [dx(i-1) 2*(dx(i-1) +dx(i)) dx(i)];
r(i) = 3*(dy(i)/dx(i) - dy(i-1)/dx(i-1));%右侧端点
end
%设置端点条件
%仅仅使用5对点中的一对;
A(1, 1) = 1;%自然样条条件
A(n, n) = 1;
% A(1,1) = 2;%曲率-相邻条件
% r(1) = vl;
% A(n,n) = 2;
% r(n) = vn;
% A(1,1:2) = [2*dx(1) dx(1)];
% r(1) = 3*(dy(1)/dx(1)-vl);%钳制
% A(n,n-1:n) = [dx(n-1) 2*dx(n-1)];
% r(n) = 3*(vn-dy(n-1)/dx(n-1));
% A(1,1:2) = [1,-1];
% a(n,n-1:n) = [1, -1];
% A(1,1:3) = [dx(2) -(dx(1)+dx(2)) dx(1)]; %当n>=4,非纽结
% A(n,n-2:n) = [dx(n-1) -(dx(n-2)+dx(n-1)) dx(n-2)];
coeff = zeros(n,3);
coeff(:,2) = A\r; %求解系数C
for i=1:n-1 %求解b和d
coeff(i,3) = (coeff(i+1,2) - coeff(i,2))/(3*dx(i));
coeff(i,1) = dy(i)/dx(i)-dx(i)*(2*coeff(i,2)+coeff(i+1,2))/3;
end
coeff = coeff(1:n-1,1:3);%三个系数
%构造插值函数
s = @(xq) interp_spline(x,y,coeff,xq);
end
function yq = interp_spline(x,y,coeff,xq)
%根据系数矩阵和插值点的横坐标计算插值函数在该点的值yq
n = length(x);
yq = zeros(size(xq));
for i = 1:length(xq)
if xq(i) < x(1) || xq(i) > x(n)
yq(i) = NaN;
else
for j = 1:n-1
if xq(i) >= x(j) && xq(i) <= x(j+1)
yq(i) = coeff(j,1) + coeff(j,2)*(xq(i)-x(j)) + ...
coeff(j,3)*(xq(i)-x(j))^2 + ...
coeff(j+1,3)*(xq(i)-x(j))^3/(3*(x(j+1)-x(j)));
break;
end
end
end
end
end
```
其中,`cubic_spline`函数用于计算样条插值的系数矩阵,并返回一个插值函数的函数句柄;`interp_spline`函数用于根据系数矩阵和插值点的横坐标计算插值函数在该点的值。使用该函数时,只需要调用`cubic_spline`函数即可,例如:
```matlab
x = [0 1 2 3 4];
y = [10 5 3 4 8];
[s,coeff] = cubic_spline(x,y);
xq = linspace(0,4,100);
yq = s(xq);
plot(x,y,'o',xq,yq);
```
上述代码将生成一条经过数据点的三次插值样条曲线。