Step 1 - Data and Initialization:

You have a set of data points (x₀, y₀), (x₁, y₁), …, (x_n, y_n) that you want to interpolate. These points represent the values of a function at specific x-coordinates.

Step 2 - Lagrange Basis Polynomials:

For each data point x_i, define a Lagrange basis polynomial L_i(x) as follows:

L_i(x) = (x_i - x₀)·(x_i - x₁)·…·(x_i - x_i-1)·(x_i - x_i+1)·…·(x_i - x_n) / (x_i - x₀)·(x_i - x₁)·…·(x_i - x_i-1)·(x_i - x_i+1)·…·(x_i - x_n

L_i(x) = (x - x₀)·(x - x₁)·…·(x - x_i-1)·(x - x_i+1)·…·(x - x_n) / (x_i - x₀)·(x_i - x₁)·…·(x_i - x_i-1)·(x_i - x_i+1)·…·(x_i - x_n)

Step 3 - Construct the Interpolating Polynomial:

The Lagrange interpolating polynomial P(x) is constructed as the weighted sum of the basis polynomials:

P(x) = y₀·L₀(x) + y₁·L₁(x) + … + y_n·L_n(x)

Step 4 - Evaluate the Interpolating Polynomial:

Now, you can use the interpolating polynomial P(x) to approximate the function's values at any point x within the range of your data points.

Lagrange interpolation provides a polynomial that passes through all the given data points. It's simple to implement and useful for approximation, but it can suffer from Runge's phenomenon, where oscillations can occur if too many data points are used.