Lagrange interpolation method + python implementation

This article will introduce the mathematical principles of Lagrange interpolation and implement it in Python.

1. Mathematical principles

First of all, we have to know a theorem: given n+1 different nodes, its n-degree interpolation polynomial is unique.

1.1 Linear interpolation

Linear interpolation is an interpolation
Known:

Solve: $$L_1(x)=a_{1}x+a_0$$
According to the point slope formula:
$$L_1\left(x\right)=y_0+\frac{(y_1-y_0)}{(x_1-x_0)}(x-x_0)$$
即：
$$L_1\left(x\right)=\frac{(x-x1}{(x_0-x_1)}{y}_0+\frac{(x-x0}{(x_1-x_0)}{y}_1$$
This is a Lagrangian polynomial
Let
$$l_0(x)=\frac{(x-x1}{(x_0-x_1)}$$
$$l_1\left(x\right)=\frac{(x-x0}{(x_1-x_0)}$$
Then the first-order Lagrange polynomial can be expressed as
$$l_1\left(x\right)=y_0{l}_0\left(x\right)+y_1{l}_1\left(x\right)$$

1.2 Secondary interpolation

Knowledge:

Solving:
$$L_2\left(x\right)=a_2{x}^2+a_{1}x+a_0$$
Constructed in the following way, let
$$L_2\left(x\right)=A(x-x_1)(x-x_2)+B(x-x_0)(x-x_2)+C(x-x_0)(x-x_1)$$
can be solved:
$$A=\frac{y_0}{(x_0-x_1)(x_0-x_2)}$$
$$B=\frac{y_1}{(x_1-x_0)(x_1-x_2)}$$
$$C=\frac{y_2}{(x_2-x_0)(x_2-x_1)}$$
于是可以得到:
$$L_2(x)=\sum _{i=1}^2(\prod _{i=0,i\ne j}^2\frac{x-x_i}{x_j-x_i})y_j$$
This formula is called a quadratic Lagrange polynomial
If let:
$$l_0(x)=\frac{(x-x1)(x-x_2）}{(x_0-x_1)(x_0-x_2)}$$
$$l_1\left(x\right)=\frac{(x-x0)(x-x_2)}{(x_1-x_0)(x_1-x_2)}$$
$$l_2\left(x\right)=\frac{(x-x0)(x-x_1)}{(x_2-x_0)(x_2-x_1)}$$
则
$$L_2(x)=y_0{l}_0\left(x\right)+y_1{l}_1\left(x\right)+y_2{l}_2\left(x\right)$$

1.3 Lagrangian interpolation polynomial of degree n

According to the above law, we can get the n-degree Lagrangian interpolation polynomial:
$$L_2\left(x\right)=\sum _{i=1}^n(\prod _{i=0,i\ne j}^n\frac{x-x_i}{x_j-x_i})y_j$$

2. Python implementation

import numpy as np

def LagrangeInterpolation(x):
    grid_x = np.array([4, 5, 6])
    k = len(grid_x)
    value = np.array([10, 5.25, 1])
    result = 0
    for j in range(k):
        result_l = 1
        for i in range(k):
            if i != j:
                result_l = result_l * (x - grid_x[i]) / (grid_x[j] - grid_x[i])

        result = result + value[j] * result_l

    return result


if __name__ == '__main__':
    x=18
    result=LagrangeInterpolation(x)
    print(result)

The result is as follows:

-11.0

参考：https://blog.csdn.net/u011699626/article/details/120394802