Another widely used form for polynomial interpolation is the Newton polynomial. Unlike the standard general form, this representation is particularly convenient for constructing the interpolating polynomial incrementally.

The Newton polynomial of degree \( n \) is expressed as:

\( P_n(x) = a_0 + a_1 (x-x_0) + a_2 (x-x_0)(x-x_1) + \dots + a_n (x-x_0)(x-x_1)\dots(x-x_{n-1}) \)

In this form, each coefficient \( a_i \) is chosen such that the polynomial passes through the \( n+1 \) collocation points \( (x_0, f(x_0)), \dots, (x_n, f(x_n)) \). One key advantage of the Newton form is that it can be built progressively: adding a new interpolation point only requires computing an additional term, without recomputing the entire polynomial.

The first coefficient is simply the value of the function at the first node:

\( a_0 = f(x_0) \)

To determine the next coefficient \( a_1 \), we use the condition that the polynomial passes through the second point:

\( P_n(x_1) = a_0 + a_1 (x_1 - x_0) = f(x_1) \quad \Rightarrow \quad a_1 = \frac{f(x_1) - f(x_0)}{x_1 - x_0} \)

Subsequent coefficients \( a_2, a_3, \dots \) are determined similarly using divided differences. This recursive property makes the Newton polynomial particularly efficient for adding new points or updating the polynomial dynamically, unlike the Lagrange form which must be recomputed entirely.