from typing import Sequence
import numpy as np

"""Compute divided difference table in matrix form for Newton interpolation (Series 3)."""
def poly_mat(items: Sequence[tuple[float, float]]) -> np.ndarray:
    """
    Compute the divided differences table (Newton interpolation) as a NumPy array.

    Parameters:
        items: Sequence of (x, y) data points.
    Returns:
        A 2D NumPy array of shape (N, N) containing the divided differences table.
    """
    x_vals, y_vals = zip(*items)
    N = len(items)
    table = np.zeros((N, N), dtype=float)
    # Zeroth column: y values
    table[0, :] = y_vals

    # Iterate over every column
    for col in range(1, N):
        for row in range(1, col + 1):
            numerator = table[row - 1, col] - table[row - 1, col - 1]
            denominator = x_vals[col] - x_vals[col - row]
            table[row, col] = numerator / denominator
    return table