Numerical Analysis is the branch of mathematics concerned with developing and studying algorithms for obtaining approximate numerical solutions to mathematical problems. It addresses problems for which exact analytical solutions are unavailable, inconvenient or too expensive to compute directly. Core subjects include numerical solution of linear and nonlinear equations, interpolation, approximation, numerical differentiation and integration, ordinary and partial differential equations, eigenvalue problems, optimisation and numerical linear algebra. A central concern is not merely whether an algorithm produces a result but whether the result is accurate, stable, convergent and computationally efficient. Numerical Analysis therefore connects mathematical theory with executable algorithms and provides the foundations for much scientific and engineering computation.
Approximation error is fundamental to Numerical Analysis. Truncation or discretisation error arises when a continuous or exact mathematical operation is replaced by a finite approximation, while round-off error arises from finite-precision computer arithmetic. An algorithm can be mathematically convergent but still behave poorly at realistic precision if round-off errors become amplified. Stability describes how errors introduced during computation affect the resulting solution, while conditioning describes how sensitive the mathematical problem itself is to perturbations in its inputs. These concepts must be distinguished because an ill-conditioned problem can be difficult even when an excellent algorithm is used, whereas an unstable algorithm can cause severe error amplification even for a well-conditioned problem.
Root-finding illustrates several fundamental numerical methods. Bisection repeatedly halves an interval containing a sign change and provides guaranteed convergence under appropriate continuity conditions, although it can converge relatively slowly. Newton's method uses derivative information to obtain faster local convergence near suitable roots but can fail or diverge when initial conditions are poor or derivatives are problematic. Secant methods approximate derivative information using previous function evaluations. Fixed-point iteration transforms an equation into an equivalent iterative form, with convergence depending on properties of the transformation. These methods demonstrate the trade-off among convergence speed, robustness, derivative requirements and computational cost that recurs throughout Numerical Analysis.
Numerical linear algebra is central because scientific models frequently produce systems of linear equations and matrix eigenvalue problems. Gaussian elimination solves linear systems through elimination and back substitution, while LU factorisation separates a matrix into triangular factors and can efficiently support repeated solves. QR factorisation is important for least-squares problems and eigenvalue algorithms. Singular value decomposition provides a powerful factorisation used in least squares, dimensionality reduction and analysis of ill-conditioned matrices. Iterative methods such as conjugate gradients can be advantageous for large sparse systems under appropriate conditions. Conditioning, sparsity, storage requirements and preconditioning strongly influence which algorithm is suitable for a particular computational problem.
Numerical integration approximates definite integrals when analytical antiderivatives are unavailable or inconvenient. Quadrature rules such as the trapezoidal and Simpson methods approximate integrals using weighted function evaluations, while Gaussian quadrature chooses nodes and weights to achieve high accuracy for broad classes of functions. Adaptive methods adjust computational effort according to estimated local error. Numerical differentiation is generally more sensitive to noise and round-off error because differentiation amplifies high-frequency perturbations, particularly when finite differences use excessively small step sizes. This illustrates that taking a smaller numerical step is not universally better; truncation error and round-off error can compete, producing an intermediate step size with lower total error.
Numerical methods for differential equations approximate solutions across time or space. Euler's method is simple but often less accurate than higher-order methods such as Runge-Kutta schemes. Stability is especially important for stiff differential equations, where explicit methods may require impractically small time steps. Boundary-value problems and partial differential equations can be discretised using finite differences, finite elements or finite volume methods.
Finite element methods represent solutions using basis functions over a mesh, while finite volume methods emphasise conservation across control volumes. Mesh quality, timestep selection, boundary conditions and solver tolerances can all affect accuracy. Convergence studies compare solutions at increasing resolution to assess whether numerical results approach a stable limiting value.
Numerical optimisation seeks approximate minimisers or maximisers of objective functions, potentially subject to equality or inequality constraints. Gradient-based methods use derivatives to determine search directions, while derivative-free methods can be useful when derivatives are unavailable or unreliable. Convex optimisation has particularly strong theoretical properties because local minima of convex objectives under suitable constraints are global minima. Nonconvex optimisation can contain multiple local minima, saddle points and flat regions, making initial conditions and algorithm selection important. Numerical optimisation underlies parameter estimation, machine learning, engineering design and many inverse problems. Stopping criteria typically consider changes in objective values, parameter values, gradient norms or constraint violations, but numerical tolerances must be chosen relative to the scale and accuracy requirements of the problem.
Numerical Analysis provides the mathematical foundation for reliable computational science. Its development has progressed alongside computing hardware, from hand and mechanical calculation through early electronic computers to modern parallel architectures. Contemporary research addresses large-scale sparse systems, uncertainty quantification, adaptive algorithms, multiphysics problems and numerical methods designed for heterogeneous hardware. The field also studies computational complexity, backward and forward error, iterative refinement and automatic differentiation.
In practical scientific work, Numerical Analysis determines whether a computed answer is a credible approximation rather than merely a number produced by software. Its principles explain why discretisation, precision, solver choice, conditioning, convergence and error estimation must be considered whenever mathematical problems are transformed into finite computational procedures.
