Polinominterpoláció

Polinominterpoláció is a method used in numerical analysis to find a polynomial that passes through a given set of data points. Given a set of $n+1$ distinct data points $(x_0, y_0), (x_1, y_1), \dots, (x_n, y_n)$, the goal of polynomial interpolation is to find a unique polynomial $P(x)$ of degree at most $n$ such that $P(x_i) = y_i$ for all $i = 0, 1, \dots, n$. This polynomial is called the interpolating polynomial.

There are several ways to construct the interpolating polynomial. The Lagrange form of the interpolating polynomial is given by $P(x) = \sum_{j=0}^{n} y_j L_j(x)$, where $L_j(x)$ are the Lagrange basis polynomials, defined as $L_j(x) = \prod_{i=0, i \neq j}^{n} \frac{x - x_i}{x_j - x_i}$. Another common method is using Newton's divided differences, which provides a recursive way to build the polynomial. This form is particularly useful for adding new data points without recalculating the entire polynomial.

Polynomial interpolation is a fundamental concept with applications in curve fitting, function approximation, and solving differential equations. However, it's important to be aware of potential issues like Runge's phenomenon, where the interpolating polynomial can oscillate wildly between data points, especially for high-degree polynomials and unevenly spaced data.