quadratuurimetodit

Quadratuurimetodit, often translated as quadrature methods or numerical integration techniques, are a class of algorithms used to approximate the definite integral of a function. These methods are essential when an exact analytical solution for the integral is difficult or impossible to find, or when the function is only known at discrete points.

The fundamental idea behind quadrature methods is to replace the continuous function with a simpler, approximative function whose integral is easy to compute. This simpler function is typically a polynomial. Different quadrature rules are distinguished by the degree of the polynomial used and how the function's values are sampled.

Commonly used quadrature methods include Newton-Cotes formulas, which are based on interpolating the function with a polynomial of degree n at equally spaced points. Examples include the trapezoidal rule (n=1) and Simpson's rule (n=2). These methods are generally easy to implement but can suffer from convergence issues for high-degree polynomials.

Another important class of methods is Gaussian quadrature. Instead of using equally spaced points, Gaussian quadrature selects optimal points (nodes) and weights that maximize the degree of the polynomial that can be integrated exactly. This often leads to higher accuracy for a given number of function evaluations compared to Newton-Cotes formulas.

The choice of quadrature method depends on factors such as the smoothness of the function, the desired accuracy, and computational cost. Error estimation is also a crucial aspect of numerical integration, allowing users to gauge the reliability of the approximation.