Numerical

Numerical is an adjective meaning relating to numbers or numerical data. In mathematics and computing, it designates methods that use approximate arithmetic to obtain solutions, typically when exact symbolic solutions are impractical or impossible.

In numerical analysis, the focus is on algorithms for approximate solutions of mathematical problems and on understanding the sources and behavior of errors. Core concerns include accuracy, stability, convergence, and efficiency. Errors are often split into truncation error, from model discretization, and rounding error, from finite-precision arithmetic.

Common problem classes include solving linear systems, computing eigenvalues, numerical integration and differentiation, and the numerical solution of ordinary and partial differential equations. Algorithms are grouped as direct methods (Gaussian elimination, LU/QR decompositions) or iterative methods (Jacobi, Gauss-Seidel, Krylov subspace methods). Discretization strategies for differential equations include finite difference, finite element, and finite volume methods.

A crucial topic is floating-point arithmetic, which dictates how real numbers are represented, rounded, and operated on by computers. This affects precision, rounding modes, overflow/underflow, and the reliability of results, making error control and conditioning essential.

Applications span science, engineering, economics, and data analysis. Numerical methods enable simulation and modeling across industries, balancing speed, memory usage, and accuracy. The field intersects with software engineering, statistics, and applied mathematics, continually evolving with advances in algorithms and hardware.