Newtoninterpolun

Newton interpolation, also known as Newton's forward difference formula or divided differences interpolation, is a method for polynomial interpolation used in numerical analysis and approximation theory. It provides a way to construct a polynomial that passes through a given set of data points by iteratively building up the solution using divided differences. This approach is particularly useful for efficiently updating the interpolating polynomial when new data points are added.

The method is based on the concept of divided differences, which generalize the notion of finite differences to arbitrary sets of points. Given a set of *n* + 1 data points \((x_0, y_0), (x_1, y_1), \dots, (x_n, y_n)\), Newton interpolation constructs a polynomial \(P(x)\) of degree at most *n* that interpolates these points. The polynomial is expressed as a sum of terms involving binomial coefficients and divided differences:

\[ P(x) = y_0 + (x - x_0) \cdot f[x_0, x_1] + (x - x_0)(x - x_1) \cdot f[x_0, x_1, x_2] + \dots + (x - x_0)\dots(x - x_{n-1}) \cdot f[x_0, x_1, \dots, x_n] \]

Here, \(f[x_i, x_{i+1}, \dots, x_j]\) represents the divided difference of the function values at the points \(x_i, x_{i+1}, \dots, x_j\). Divided differences can be computed recursively using a table, which simplifies the process of constructing the polynomial.

One advantage of Newton interpolation is its efficiency in updating the polynomial when new data points are incorporated. By adding a new term corresponding to the latest divided difference, the polynomial can be adjusted without recomputing the entire interpolation. This property makes it particularly useful in dynamic or incremental data scenarios.

Newton interpolation is closely related to Lagrange interpolation but often offers computational advantages, especially when dealing with large datasets or when points are added sequentially. However, it may suffer from numerical instability for certain configurations of input points, particularly when the \(x_i\) values are not uniformly spaced. Despite this, it remains a fundamental tool in numerical analysis and is widely taught in introductory courses on approximation theory.