give me an Python example using natural cubic spline,do not use Scipy

Sure, here is an example using natural cubic spline in Python without using Scipy:

import numpy as np

def natural_cubic_spline(x, y):
    n = len(x)
    a = y.copy()
    b = np.zeros(n)
    d = np.zeros(n)
    h = np.diff(x)
    alpha = np.zeros(n-1)
    for i in range(1, n-1):
        alpha[i] = 3*(a[i+1]-a[i])/h[i] - 3*(a[i]-a[i-1])/h[i-1]
    l = np.zeros(n)
    mu = np.zeros(n)
    z = np.zeros(n)
    l[0] = 1
    mu[0] = z[0] = 0
    for i in range(1, n-1):
        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]-h[i-1]*z[i-1])/l[i]
    l[n-1] = 1
    z[n-1] = c[n-1] = 0
    for j in range(n-2, -1, -1):
        c[j] = z[j] - mu[j]*c[j+1]
        b[j] = (a[j+1]-a[j])/h[j] - h[j]*(c[j+1]+2*c[j])/3
        d[j] = (c[j+1]-c[j])/(3*h[j])
    return a, b, c, d

# Example usage
x = np.array([1, 2, 3, 4, 5])
y = np.array([2, 3, 5, 1, 4])
a, b, c, d = natural_cubic_spline(x, y)
print("a: ", a)
print("b: ", b)
print("c: ", c)
print("d: ", d)

This function takes in two arrays `x` and `y` representing the x and y coordinates of a set of points. It returns the coefficients `a`, `b`, `c`, and `d` for the natural cubic spline that interpolates these points. The `a` coefficients are simply the y values of the input points. The `b`, `c`, and `d` coefficients are computed using the algorithm for natural cubic spline interpolation.