Numerics

Numerics, in the sense of numerical analysis, is the study of algorithms for approximating solutions to mathematical problems that can be described continuously but must be computed with finite precision. The field covers the design, analysis, and implementation of methods for performing numerical computations reliably and efficiently.

Core problems include solving systems of linear and nonlinear equations, numerical integration and differentiation, interpolation and function approximation, and the computation of eigenvalues and eigenvectors. It also includes numerical methods for solving ordinary and partial differential equations, as well as optimization and Monte Carlo techniques used for simulation and statistical estimation.

Methods in numerics are commonly grouped into direct methods, such as Gaussian elimination and matrix factorizations (LU, QR), and iterative methods, such as Jacobi, Gauss-Seidel, and conjugate gradient. Discretization strategies, quadrature rules, and spline or polynomial approximations underpin many techniques for function representation and integration. Numerical linear algebra focuses on the behavior of algorithms with respect to errors and conditioning.

Error analysis is central. Truncation errors arise from approximating a problem, while rounding errors come from finite-precision arithmetic. The concepts of stability, consistency, and convergence describe how errors propagate and whether a method yields accurate results as computation proceeds.

Floating-point arithmetic, often aligned with IEEE 754 standards, defines number representation and rounding behavior. Problem conditioning and algorithm stability influence reliability; poorly conditioned problems or unstable methods can produce large relative errors.

Applications of numerics span engineering, physics, computer science, finance, and data analysis, emphasizing robust, efficient algorithms that perform predictably across problems and hardware.