Algorithms which solve mathematical problems by means of numerical approximation (as opposed to symbolic computation).
Numerical methods include the study of algorithms that use numerical approximation (as opposed to general symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). Numerical methods naturally find applications in all fields of science and engineering, and include implementations of many important aspects of computation including: solving ordinary and partial differential equations, numerical linear algebra, stochastic differential equations, Markov chains, and so forth.
Numerical methods use several approaches to calculate observables. For example, iterative methods that form successive approximations that converge to the exact solution only in a limit. A convergence test, often involving the residual, is specified in order to decide when a sufficiently accurate solution has (hopefully) been found. Examples include Newton's method, the bisection method, and Jacobi iteration. Another example is the use of discretization, a procedure that is used when continuous problems must sometimes be replaced by a discrete problem whose solution is known to approximate that of the continuous problem.
The field of numerical methods includes many sub-disciplines. Some of the major ones are:
Computing values of functions
Interpolation, extrapolation, and regression
Solving equations and systems of equations
Solving eigenvalue or singular value problems
Optimization
Evaluating integrals
Differential equations
