NumberVerfahren

NumberVerfahren is a term used in computational mathematics to denote numerical methods and algorithms for computing with numbers. In German-language contexts, it serves as a generic label for procedures that produce approximate numerical results when exact solutions are unavailable or impractical. NumberVerfahren covers a wide range of tasks, including solving equations, approximating integrals and functions, and performing linear and nonlinear algebra on digital computers. The common themes are discretization, iteration, and error analysis, with attention to convergence and numerical stability.

Typical families of NumberVerfahren include root-finding methods such as the bisection method and Newton-Raphson; linear-system solvers such as Gaussian elimination and LU/QR decompositions; and eigenvalue algorithms such as the power method and the QR algorithm. Numerical integration and interpolation methods such as the trapezoidal rule, Simpson's rule, and polynomial or spline fitting are also part of NumberVerfahren. In optimization, NumberVerfahren refers to procedures for finding extrema of functions using numerical evaluation.

Key considerations in NumberVerfahren are accuracy, stability, and computational efficiency. Errors can arise from truncation and rounding, and the conditioning of problems affects reliability. Implementations often exploit vectorization, parallel processing, and, where appropriate, arbitrary-precision arithmetic to control error growth.

History and scope: The development of computer-based numerical methods in the 20th century expanded the applicability of NumberVerfahren across science, engineering, economics, and data analysis. Related fields include numerical analysis, scientific computing, and computational mathematics.