matlab三次样条插值例题

三次样条插值是一种常用的插值方法，它通过将插值区间分成多个小段，并在每个小段上拟合一个三次函数来逼近原始数据。下面是一个matlab的三次样条插值的例题：

% 例题
x = [0, 1, 2, 3, 4];  % 插值节点
y = [1, 3, 4, 1, 2];  % 插值节点对应的函数值

% 计算每个小段上的三次样条插值多项式的系数
n = length(x) - 1;
h = diff(x);
u = 6 * diff(y) ./ h;

alpha = zeros(1, n);
l = zeros(1, n + 1);
mu = zeros(1, n);
z = zeros(1, n + 1);

l(1) = 1;
mu(1) = 0;
z(1) = 0;

for i = 2:n
    l(i) = 2 * (x(i+1) - x(i-1)) - h(i-1) * mu(i-1);
    mu(i) = h(i) / l(i);
    z(i) = (alpha(i-1) - h(i-1) * z(i-1)) / l(i);
end

l(n + 1) = 1;
z(n + 1) = 0;
c = zeros(1, n + 1);
b = zeros(1, n);
d = zeros(1, n);

for j = n:-1:1
    c(j) = z(j) - mu(j) * c(j + 1);
    b(j) = (y(j + 1) - y(j)) / h(j) - h(j) * (c(j + 1) + 2 * c(j)) / 6;
    d(j) = (c(j + 1) - c(j)) / (6 * h(j));
end

% 输出每个小段上的三次样条插值多项式
for i = 1:n
    fprintf('第 %d 段的插值多项式为: S%d(x) = %.2f(x - %.2f)^3 + %.2f(x - %.2f)^2 + %.2f(x - %.2f) + %.2f\n', i, i, d(i), x(i+1), c(i), x(i+1), b(i), x(i+1), y(i+1));
end