Lagrangeformájú

Lagrangeformájú refers to a concept within mathematics, specifically related to polynomial interpolation. It is a method for constructing a polynomial that passes through a given set of data points. This method is named after the mathematician Joseph-Louis Lagrange.

The Lagrange form of a polynomial provides a direct way to express the interpolating polynomial. Instead of finding the coefficients of a polynomial in the standard basis (e.g., $a_0 + a_1x + a_2x^2 + ...$), the Lagrange form expresses the polynomial as a sum of terms, where each term is associated with one of the data points.

Each term in the Lagrange form is constructed such that it is zero at all data points except for one, where it takes the value of the corresponding y-coordinate. This is achieved using products of linear factors. For a set of $n+1$ data points $(x_0, y_0), (x_1, y_1), ..., (x_n, y_n)$, the Lagrange interpolating polynomial $P(x)$ can be written as:

$P(x) = \sum_{j=0}^{n} y_j L_j(x)$

where $L_j(x)$ are the Lagrange basis polynomials defined as:

$L_j(x) = \prod_{i=0, i \neq j}^{n} \frac{x - x_i}{x_j - x_i}$

This form guarantees that $P(x_k) = y_k$ for all $k$ from 0 to $n$. While elegant, the direct computation of the Lagrange form can be computationally intensive for a large number of points, and it may suffer from Runge's phenomenon, which is the oscillatory behavior of high-degree interpolating polynomials.