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 - Divided Difference Table:

Create a divided difference table, which is a triangular table where the first column contains the y_i values. Compute the differences recursively using the formula:

[y_i, y_i+1, …, y_j] = [y_i+1, y_i+2, …, y_j] - [y_i, y_i+1, …, y_j-1

f[x_i, x_i+1, …, x_j] = x_j - x_i

f[x_i+1, x_i+2, …, x_j] - f[x_i, x_i+1, …, x_j-1]

Step 3 - Construct the Interpolating Polynomial:

The Newton interpolating polynomial P(x) can be expressed as:

P(x) = y₀ + (x - x₀₀, x₁] + (x - x₀)(x - x₁)f[x₀, x₁, x₂] + …

where f[x₀, x₁, …, x_k] are the divided differences.

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.

Newton interpolation with divided differences provides a polynomial that passes through the given data points. It is a flexible method for interpolation and is especially useful when additional data points need to be added.