kvadraturmetode

The kvadraturmetode, or quadrature method, is a numerical technique used to approximate definite integrals. It is based on the idea of estimating the area under a curve by summing the areas of geometrically simple shapes, typically rectangles or trapezoids, that closely match the shape of the function over a specified interval.

Quadrature methods are widely used in computational mathematics and engineering, especially when analytical integration is challenging or impossible. These methods involve selecting specific points within the integration interval and assigning weights to the function values at those points. The most basic form is the rectangle (or midpoint) method, where the function is sampled at the midpoint, and the area is approximated by multiplying the function value by the interval length.

More sophisticated techniques, such as the trapezoidal rule and Simpson's rule, improve accuracy by considering the slope or curvature of the function. The trapezoidal rule approximates the area as a trapezoid, connecting the interval endpoints with straight lines. Simpson's rule fits a quadratic polynomial through three points and integrates it exactly, providing higher precision for smooth functions.

Numerical quadrature is essential in many scientific computations, as it enables the evaluation of integrals in situations where the antiderivative is unknown or difficult to compute analytically. The accuracy of quadrature methods depends on the number of sample points and the smoothness of the function. Adaptive quadrature techniques dynamically adjust sampling points to improve approximation efficiency.

Overall, the kvadraturmetode provides a flexible and practical approach to numerical integration, forming a foundation for various advanced algorithms in computational analysis.