Newtonformában

Newtonformában is a Hungarian term that translates to "in Newton's form" and refers to a specific way of expressing polynomial functions. This representation is also known as Newton's divided difference polynomial or Newton's interpolating polynomial. It is a method for finding a polynomial that passes through a given set of data points.

The key idea behind Newtonformában is to build the interpolating polynomial incrementally. Instead of calculating all coefficients simultaneously, it starts with a simple polynomial and adds successive terms that correct the errors at each data point. These corrections are based on divided differences, which are recursively defined quantities derived from the function values at specific points.

The Newton form of a polynomial is particularly useful because it allows for easy addition of new data points without recalculating the entire polynomial. If a new point is added, a new term is simply appended to the existing polynomial. This makes it efficient for situations where data is acquired sequentially or when a polynomial approximation needs to be refined.

The general form of a Newton polynomial of degree n for data points $(x_0, y_0), (x_1, y_1), \dots, (x_n, y_n)$ is given by:

$P_n(x) = f[x_0] + f[x_0, x_1](x-x_0) + f[x_0, x_1, x_2](x-x_0)(x-x_1) + \dots + f[x_0, x_1, \dots, x_n](x-x_0)(x-x_1)\dots(x-x_{n-1})$.

Here, $f[x_0, \dots, x_k]$ represents the $k$-th divided difference. This form provides a structured way to construct the interpolating polynomial and is a fundamental concept in numerical analysis and approximation theory.