Python Program: Lagrange Interpolation

 Python Program For Lagrange Interpolation

Here is a Python program to calculate the value of a function at a given point using the Lagrange Interpolation Method. Below the program, I provide a step-by-step explanation.

Python Program: Lagrange Interpolation

python
def lagrange_interpolation(x_points, y_points, x):
    """
    Perform Lagrange Interpolation to estimate the value of a function at a given point.
    
    :param x_points: List of x-coordinates of the given points
    :param y_points: List of corresponding y-coordinates of the given points
    :param x: The point at which to estimate the function value
    :return: Interpolated value at x
    """
    n = len(x_points)  # Number of given points
    interpolated_value = 0

    for i in range(n):
        # Calculate the ith Lagrange basis polynomial (L_i(x))
        L_i = 1
        for j in range(n):
            if i != j:
                L_i *= (x - x_points[j]) / (x_points[i] - x_points[j])
        
        # Add the contribution of L_i(x) to the final interpolated value
        interpolated_value += y_points[i] * L_i

    return interpolated_value

# Example Usage
x_points = [1, 2, 3]  # x-coordinates of known data points
y_points = [2, 3, 5]  # y-coordinates of known data points
x = 2.5               # Point to interpolate

result = lagrange_interpolation(x_points, y_points, x)
print(f"The interpolated value at x = {x} is {result:.4f}")

Step-by-Step Guide:

1. Understand Lagrange Interpolation Formula: The formula is:

   $$P(x) = \sum_{i=0}^{n-1} y_i \cdot L_i(x)$$
   

   Where $L_i(x)$ is the Lagrange basis polynomial given by:

   $$L_i(x) = \prod_{j=0, j \neq i}^{n-1} \frac{x - x_j}{x_i - x_j}$$

2. Input Data:

   • Define a list of $x$-coordinates (x_points) and their corresponding $y$-coordinates (y_points).
   • Specify the $x$-value where the function should be estimated.

3. Iterate Over Data Points:

   • For each data point $i$, compute $L_i(x)$ by iterating over all other points $j$.

4. Compute the Basis Polynomial:

   • Use the formula for $L_i(x)$to calculate the weight of the $i$-th data point for the desired $x$.

5. Accumulate Contributions:

   • Multiply $L_i(x)\; by$$y_i$ (the known function value at $x_i$) and add it to the total.

6. Output the Result:

   • After iterating through all points, print the interpolated value at $x$.

Example Explanation:

Suppose $(x_1, y_1) = (1, 2)$, $(x_2,y_2)=(2,3),$$(x_3,y_3)=(3,5),\; and\; we\; want\; to\; estimate$$P(2.5)$:

1. Compute $L_0(2.5), L_1(2.5),$ and $L_2(2.5)$
2. Multiply each basis polynomial by its corresponding $y$-value.
3. Add all contributions to get the final interpolated value.