interpolatsioonpolünoomi

Interpolatsioonpolünoom (or interpolating polynomial in English) is a polynomial that passes exactly through a given set of data points. Given $n+1$ distinct points $(x_0, y_0), (x_1, y_1), \dots, (x_n, y_n)$, the interpolating polynomial $P(x)$ of degree at most $n$ is the unique polynomial such that $P(x_i) = y_i$ for all $i = 0, 1, \dots, n$.

There are several ways to construct the interpolating polynomial. Two common methods are using the Lagrange form and the Newton form. The Lagrange form expresses the polynomial as a sum of basis polynomials, each designed to be 1 at one of the data points and 0 at all others. The Newton form builds the polynomial incrementally, adding terms that incorporate new data points, and often utilizes divided differences.

The existence and uniqueness of the interpolating polynomial of degree at most $n$ for $n+1$ distinct points are guaranteed by algebraic principles. However, for a large number of points, the interpolating polynomial can exhibit undesirable oscillations between the data points, a phenomenon known as Runge's phenomenon. This means that while the polynomial passes through the given points, it may not accurately represent the underlying function that generated those points, especially in regions away from the data.