# List of numerical analysis topics

This is a list of numerical analysis topics.

## General

* Validated numerics
* Iterative method
* Rate of convergence — the speed at which a convergent sequence approaches its limit
** Order of accuracy — rate at which numerical solution of differential equation converges to exact solution
* Series acceleration — methods to accelerate the speed of convergence of a series
** Aitken's delta-squared process — most useful for linearly converging sequences
** Minimum polynomial extrapolation — for vector sequences
** Richardson extrapolation
** Shanks transformation — similar to Aitken's delta-squared process, but applied to the partial sums
** Van Wijngaarden transformation — for accelerating the convergence of an alternating series
* Abramowitz and Stegun — book containing formulas and tables of many special functions
** Digital Library of Mathematical Functions — successor of book by Abramowitz and Stegun
* Curse of dimensionality
* Local convergence and global convergence — whether you need a good initial guess to get convergence
* Superconvergence
* Discretization
* Difference quotient
* Complexity:
** Computational complexity of mathematical operations
** Smoothed analysis — measuring the expected performance of algorithms under slight random perturbations of worst-case inputs
* Symbolic-numeric computation — combination of symbolic and numeric methods
* Cultural and historical aspects:
** History of numerical solution of differential equations using computers
** Hundred-dollar, Hundred-digit Challenge problems — list of ten problems proposed by Nick Trefethen in 2002
** International Workshops on Lattice QCD and Numerical Analysis
** Timeline of numerical analysis after 1945
* General classes of methods:
** Collocation method — discretizes a continuous equation by requiring it only to hold at certain points
** Level-set method
*** Level set (data structures) — data structures for representing level sets
** Sinc numerical methods — methods based on the sinc function, sinc(x) = sin(x) / x
** ABS methods

## Error

Error analysis (mathematics)
* Approximation
* Approximation error
* Catastrophic cancellation
* Condition number
* Discretization error
* Floating point number
** Guard digit — extra precision introduced during a computation to reduce round-off error
** Truncation — rounding a floating-point number by discarding all digits after a certain digit
** Round-off error
*** Numeric precision in Microsoft Excel
** Arbitrary-precision arithmetic
* Interval arithmetic — represent every number by two floating-point numbers guaranteed to have the unknown number between them
** Interval contractor — maps interval to subinterval which still contains the unknown exact answer
** Interval propagation — contracting interval domains without removing any value consistent with the constraints
*** See also: Interval boundary element method, Interval finite element
* Loss of significance
* Numerical error
* Numerical stability
* Error propagation:
** Propagation of uncertainty
** Residual (numerical analysis)
* Relative change and difference — the relative difference between x and y is |x − y| / max(|x|, |y|)
* Significant figures
** Artificial precision — when a numerical value or semantic is expressed with more precision than was initially provided from measurement or user input
** False precision — giving more significant figures than appropriate
* Sterbenz lemma
* Truncation error — error committed by doing only a finite numbers of steps
* Well-posed problem
* Affine arithmetic

## Elementary and special functions

* Unrestricted algorithm
* Summation:
** Kahan summation algorithm
** Pairwise summation — slightly worse than Kahan summation but cheaper
** Binary splitting
** 2Sum
* Multiplication:
** Multiplication algorithm — general discussion, simple methods
** Karatsuba algorithm — the first algorithm which is faster than straightforward multiplication
** Toom–Cook multiplication — generalization of Karatsuba multiplication
** Schönhage–Strassen algorithm — based on Fourier transform, asymptotically very fast
** Fürer's algorithm — asymptotically slightly faster than Schönhage–Strassen
* Division algorithm — for computing quotient and/or remainder of two numbers
** Long division
** Restoring division
** Non-restoring division
** SRT division
** Newton–Raphson division: uses Newton's method to find the reciprocal of D, and multiply that reciprocal by N to find the final quotient Q.
** Goldschmidt division
* Exponentiation:
** Exponentiation by squaring
** Addition-chain exponentiation
* Multiplicative inverse Algorithms: for computing a number's multiplicative inverse (reciprocal).
** Newton's method
* Polynomials:
** Horner's method
** Estrin's scheme — modification of the Horner scheme with more possibilities for parallelization
** Clenshaw algorithm
** De Casteljau's algorithm
* Square roots and other roots:
** Integer square root
** Methods of computing square roots
** nth root algorithm
** hypot — the function (x$^2$ + y$^2$)$^{1/2}$
** Alpha max plus beta min algorithm — approximates hypot(x,y)
** Fast inverse square root — calculates 1 / √x using details of the IEEE floating-point system
* Elementary functions (exponential, logarithm, trigonometric functions):
** Trigonometric tables — different methods for generating them
** CORDIC — shift-and-add algorithm using a table of arc tangents
** BKM algorithm — shift-and-add algorithm using a table of logarithms and complex numbers
* Gamma function:
** Lanczos approximation
** Spouge's approximation — modification of Stirling's approximation; easier to apply than Lanczos
* AGM method — computes arithmetic–geometric mean; related methods compute special functions
* FEE method (Fast E-function Evaluation) — fast summation of series like the power series for e$^x$
* Gal's accurate tables — table of function values with unequal spacing to reduce round-off error
* Spigot algorithm — algorithms that can compute individual digits of a real number
* Approximations of π:
** Liu Hui's π algorithm — first algorithm that can compute π to arbitrary precision
** Leibniz formula for π — alternating series with very slow convergence
** Wallis product — infinite product converging slowly to π/2
** Viète's formula — more complicated infinite product which converges faster
** Gauss–Legendre algorithm — iteration which converges quadratically to π, based on arithmetic–geometric mean
** Borwein's algorithm — iteration which converges quartically to 1/π, and other algorithms
** Chudnovsky algorithm — fast algorithm that calculates a hypergeometric series
** Bailey–Borwein–Plouffe formula — can be used to compute individual hexadecimal digits of π
** Bellard's formula — faster version of Bailey–Borwein–Plouffe formula
** List of formulae involving π

## Numerical linear algebra

Numerical linear algebra — study of numerical algorithms for linear algebra problems

### Basic concepts

* Types of matrices appearing in numerical analysis:
** Sparse matrix
*** Band matrix
*** Bidiagonal matrix
*** Tridiagonal matrix
*** Pentadiagonal matrix
*** Skyline matrix
** Circulant matrix
** Triangular matrix
** Diagonally dominant matrix
** Block matrix — matrix composed of smaller matrices
** Stieltjes matrix — symmetric positive definite with non-positive off-diagonal entries
** Hilbert matrix — example of a matrix which is extremely ill-conditioned (and thus difficult to handle)
** Wilkinson matrix — example of a symmetric tridiagonal matrix with pairs of nearly, but not exactly, equal eigenvalues
** Convergent matrix — square matrix whose successive powers approach the zero matrix
* Algorithms for matrix multiplication:
** Strassen algorithm
** Coppersmith–Winograd algorithm
** Cannon's algorithm — a distributed algorithm, especially suitable for processors laid out in a 2d grid
** Freivalds' algorithm — a randomized algorithm for checking the result of a multiplication
* Matrix decompositions:
** LU decomposition — lower triangular times upper triangular
** QR decomposition — orthogonal matrix times triangular matrix
*** RRQR factorization — rank-revealing QR factorization, can be used to compute rank of a matrix
** Polar decomposition — unitary matrix times positive-semidefinite Hermitian matrix
** Decompositions by similarity:
*** Eigendecomposition — decomposition in terms of eigenvectors and eigenvalues
*** Jordan normal form — bidiagonal matrix of a certain form; generalizes the eigendecomposition
**** Weyr canonical form — permutation of Jordan normal form
*** Jordan–Chevalley decomposition — sum of commuting nilpotent matrix and diagonalizable matrix
*** Schur decomposition — similarity transform bringing the matrix to a triangular matrix
** Singular value decomposition — unitary matrix times diagonal matrix times unitary matrix
* Matrix splitting — expressing a given matrix as a sum or difference of matrices

### Solving systems of linear equations

* Gaussian elimination
** Row echelon form — matrix in which all entries below a nonzero entry are zero
** Bareiss algorithm — variant which ensures that all entries remain integers if the initial matrix has integer entries
** Tridiagonal matrix algorithm — simplified form of Gaussian elimination for tridiagonal matrices
* LU decomposition — write a matrix as a product of an upper- and a lower-triangular matrix
** Crout matrix decomposition
** LU reduction — a special parallelized version of a LU decomposition algorithm
* Block LU decomposition
* Cholesky decomposition — for solving a system with a positive definite matrix
** Minimum degree algorithm
** Symbolic Cholesky decomposition
* Iterative refinement — procedure to turn an inaccurate solution in a more accurate one
* Direct methods for sparse matrices:
** Frontal solver — used in finite element methods
** Nested dissection — for symmetric matrices, based on graph partitioning
* Levinson recursion — for Toeplitz matrices
* SPIKE algorithm — hybrid parallel solver for narrow-banded matrices
* Cyclic reduction — eliminate even or odd rows or columns, repeat
* Iterative methods:
** Jacobi method
** Gauss–Seidel method
*** Successive over-relaxation (SOR) — a technique to accelerate the Gauss–Seidel method
**** Symmetric successive over-relaxation (SSOR) — variant of SOR for symmetric matrices
*** Backfitting algorithm — iterative procedure used to fit a generalized additive model, often equivalent to Gauss–Seidel
** Modified Richardson iteration
** Conjugate gradient method (CG) — assumes that the matrix is positive definite
*** Derivation of the conjugate gradient method
*** Nonlinear conjugate gradient method — generalization for nonlinear optimization problems
** Biconjugate gradient method (BiCG)
*** Biconjugate gradient stabilized method (BiCGSTAB) — variant of BiCG with better convergence
** Conjugate residual method — similar to CG but only assumed that the matrix is symmetric
** Generalized minimal residual method (GMRES) — based on the Arnoldi iteration
** Chebyshev iteration — avoids inner products but needs bounds on the spectrum
** Stone's method (SIP — Strongly Implicit Procedure) — uses an incomplete LU decomposition
** Kaczmarz method
** Preconditioner
*** Incomplete Cholesky factorization — sparse approximation to the Cholesky factorization
*** Incomplete LU factorization — sparse approximation to the LU factorization
** Uzawa iteration — for saddle node problems
* Underdetermined and overdetermined systems (systems that have no or more than one solution):
** Numerical computation of null space — find all solutions of an underdetermined system
** Moore–Penrose pseudoinverse — for finding solution with smallest 2-norm (for underdetermined systems) or smallest residual
** Sparse approximation — for finding the sparsest solution (i.e., the solution with as many zeros as possible)

### Eigenvalue algorithms

Eigenvalue algorithm — a numerical algorithm for locating the eigenvalues of a matrix
* Power iteration
* Inverse iteration
* Rayleigh quotient iteration
* Arnoldi iteration — based on Krylov subspaces
* Lanczos algorithm — Arnoldi, specialized for positive-definite matrices
** Block Lanczos algorithm — for when matrix is over a finite field
* QR algorithm
* Jacobi eigenvalue algorithm — select a small submatrix which can be diagonalized exactly, and repeat
** Jacobi rotation — the building block, almost a Givens rotation
** Jacobi method for complex Hermitian matrices
* Divide-and-conquer eigenvalue algorithm
* Folded spectrum method
* LOBPCG — Locally Optimal Block Preconditioned Conjugate Gradient Method
* Eigenvalue perturbation — stability of eigenvalues under perturbations of the matrix

### Other concepts and algorithms

* Orthogonalization algorithms:
** Gram–Schmidt process
** Householder transformation
*** Householder operator — analogue of Householder transformation for general inner product spaces
** Givens rotation
* Krylov subspace
* Block matrix pseudoinverse
* Bidiagonalization
* Cuthill–McKee algorithm — permutes rows/columns in sparse matrix to yield a narrow band matrix
* In-place matrix transposition — computing the transpose of a matrix without using much additional storage
* Pivot element — entry in a matrix on which the algorithm concentrates
* Matrix-free methods — methods that only access the matrix by evaluating matrix-vector products

## Interpolation and approximation

Interpolation — construct a function going through some given data points
* Nearest-neighbor interpolation — takes the value of the nearest neighbor

### Polynomial interpolation

Polynomial interpolation — interpolation by polynomials
* Linear interpolation
* Runge's phenomenon
* Vandermonde matrix
* Chebyshev polynomials
* Chebyshev nodes
* Lebesgue constants
* Different forms for the interpolant:
** Newton polynomial
*** Divided differences
*** Neville's algorithm — for evaluating the interpolant; based on the Newton form
** Lagrange polynomial
** Bernstein polynomial — especially useful for approximation
** Brahmagupta's interpolation formula — seventh-century formula for quadratic interpolation
* Extensions to multiple dimensions:
** Bilinear interpolation
** Trilinear interpolation
** Bicubic interpolation
** Tricubic interpolation
** Padua points — set of points in R$^2$ with unique polynomial interpolant and minimal growth of Lebesgue constant
* Hermite interpolation
* Birkhoff interpolation
* Abel–Goncharov interpolation

### Spline interpolation

Spline interpolation — interpolation by piecewise polynomials
* Spline (mathematics) — the piecewise polynomials used as interpolants
* Perfect spline — polynomial spline of degree m whose mth derivate is ±1
* Cubic Hermite spline
** Centripetal Catmull–Rom spline — special case of cubic Hermite splines without self-intersections or cusps
* Monotone cubic interpolation
* Hermite spline
* Bézier curve
** De Casteljau's algorithm
** composite Bézier curve
** Generalizations to more dimensions:
*** Bézier triangle — maps a triangle to R$^3$
*** Bézier surface — maps a square to R$^3$
* B-spline
** Box spline — multivariate generalization of B-splines
** Truncated power function
** De Boor's algorithm — generalizes De Casteljau's algorithm
* Non-uniform rational B-spline (NURBS)
** T-spline — can be thought of as a NURBS surface for which a row of control points is allowed to terminate
* Kochanek–Bartels spline
* Coons patch — type of manifold parametrization used to smoothly join other surfaces together
* M-spline — a non-negative spline
* I-spline — a monotone spline, defined in terms of M-splines
* Smoothing spline — a spline fitted smoothly to noisy data
* Blossom (functional) — a unique, affine, symmetric map associated to a polynomial or spline
* See also: List of numerical computational geometry topics

### Trigonometric interpolation

Trigonometric interpolation — interpolation by trigonometric polynomials
* Discrete Fourier transform — can be viewed as trigonometric interpolation at equidistant points
** Relations between Fourier transforms and Fourier series
* Fast Fourier transform (FFT) — a fast method for computing the discrete Fourier transform
** Bluestein's FFT algorithm
** Bruun's FFT algorithm
** Cooley–Tukey FFT algorithm
** Split-radix FFT algorithm — variant of Cooley–Tukey that uses a blend of radices 2 and 4
** Goertzel algorithm
** Prime-factor FFT algorithm
** Rader's FFT algorithm
** Bit-reversal permutation — particular permutation of vectors with 2$^m$ entries used in many FFTs.
** Butterfly diagram
** Twiddle factor — the trigonometric constant coefficients that are multiplied by the data
** Cyclotomic fast Fourier transform — for FFT over finite fields
** Methods for computing discrete convolutions with finite impulse response filters using the FFT:
*** Overlap–add method
*** Overlap–save method
* Sigma approximation
* Dirichlet kernel — convolving any function with the Dirichlet kernel yields its trigonometric interpolant
* Gibbs phenomenon

### Other interpolants

* Simple rational approximation
** Polynomial and rational function modeling — comparison of polynomial and rational interpolation
* Wavelet
** Continuous wavelet
** Transfer matrix
** See also: List of functional analysis topics, List of wavelet-related transforms
* Inverse distance weighting
* Radial basis function (RBF) — a function of the form ƒ(x) = φ(|x−x$_0$|)
** Polyharmonic spline — a commonly used radial basis function
** Thin plate spline — a specific polyharmonic spline: r$^2$ log r
** Hierarchical RBF
* Subdivision surface — constructed by recursively subdividing a piecewise linear interpolant
** Catmull–Clark subdivision surface
** Doo–Sabin subdivision surface
** Loop subdivision surface
* Slerp (spherical linear interpolation) — interpolation between two points on a sphere
** Generalized quaternion interpolation — generalizes slerp for interpolation between more than two quaternions
* Irrational base discrete weighted transform
* Nevanlinna–Pick interpolation — interpolation by analytic functions in the unit disc subject to a bound
** Pick matrix — the Nevanlinna–Pick interpolation has a solution if this matrix is positive semi-definite
* Multivariate interpolation — the function being interpolated depends on more than one variable
** Barnes interpolation — method for two-dimensional functions using Gaussians common in meteorology
** Coons surface — combination of linear interpolation and bilinear interpolation
** Lanczos resampling — based on convolution with a sinc function
** Natural neighbor interpolation
** PDE surface
** Transfinite interpolation — constructs function on planar domain given its values on the boundary
** Trend surface analysis — based on low-order polynomials of spatial coordinates; uses scattered observations
** Method based on polynomials are listed under Polynomial interpolation

### Approximation theory

Approximation theory
* Orders of approximation
* Lebesgue's lemma
* Curve fitting
** Vector field reconstruction
* Modulus of continuity — measures smoothness of a function
* Least squares (function approximation) — minimizes the error in the L$^2$-norm
* Minimax approximation algorithm — minimizes the maximum error over an interval (the L$^∞$-norm)
** Equioscillation theorem — characterizes the best approximation in the L$^∞$-norm
* Unisolvent point set — function from given function space is determined uniquely by values on such a set of points
* Stone–Weierstrass theorem — continuous functions can be approximated uniformly by polynomials, or certain other function spaces
* Approximation by polynomials:
** Linear approximation
** Bernstein polynomial — basis of polynomials useful for approximating a function
** Bernstein's constant — error when approximating |x| by a polynomial
** Remez algorithm — for constructing the best polynomial approximation in the L$^∞$-norm
** Bernstein's inequality (mathematical analysis) — bound on maximum of derivative of polynomial in unit disk
** Mergelyan's theorem — generalization of Stone–Weierstrass theorem for polynomials
** Müntz–Szász theorem — variant of Stone–Weierstrass theorem for polynomials if some coefficients have to be zero
** Bramble–Hilbert lemma — upper bound on L$^p$ error of polynomial approximation in multiple dimensions
** Discrete Chebyshev polynomials — polynomials orthogonal with respect to a discrete measure
** Favard's theorem — polynomials satisfying suitable 3-term recurrence relations are orthogonal polynomials
* Approximation by Fourier series / trigonometric polynomials:
** Jackson's inequality — upper bound for best approximation by a trigonometric polynomial
*** Bernstein's theorem (approximation theory) — a converse to Jackson's inequality
** Fejér's theorem — Cesàro means of partial sums of Fourier series converge uniformly for continuous periodic functions
** Erdős–Turán inequality — bounds distance between probability and Lebesgue measure in terms of Fourier coefficients
* Different approximations:
** Moving least squares
** Padé approximant
*** Padé table — table of Padé approximants
** Hartogs–Rosenthal theorem — continuous functions can be approximated uniformly by rational functions on a set of Lebesgue measure zero
** Szász–Mirakyan operator — approximation by e$^{−n}$ x$^k$ on a semi-infinite interval
** Szász–Mirakjan–Kantorovich operator
** Baskakov operator — generalize Bernstein polynomials, Szász–Mirakyan operators, and Lupas operators
** Favard operator — approximation by sums of Gaussians
* Surrogate model — application: replacing a function that is hard to evaluate by a simpler function
* Constructive function theory — field that studies connection between degree of approximation and smoothness
* Universal differential equation — differential–algebraic equation whose solutions can approximate any continuous function
* Fekete problem — find N points on a sphere that minimize some kind of energy
* Carleman's condition — condition guaranteeing that a measure is uniquely determined by its moments
* Krein's condition — condition that exponential sums are dense in weighted L$^2$ space
* Lethargy theorem — about distance of points in a metric space from members of a sequence of subspaces
* Wirtinger's representation and projection theorem
* Journals:
** Constructive Approximation
** Journal of Approximation Theory

### Miscellaneous

* Extrapolation
** Linear predictive analysis — linear extrapolation
* Unisolvent functions — functions for which the interpolation problem has a unique solution
* Regression analysis
** Isotonic regression
* Curve-fitting compaction
* Interpolation (computer graphics)

## Finding roots of nonlinear equations

    See #Numerical linear algebra for linear equations
Root-finding algorithm — algorithms for solving the equation f(x) = 0
* General methods:
** Bisection method — simple and robust; linear convergence
*** Lehmer–Schur algorithm — variant for complex functions
** Fixed-point iteration
** Newton's method — based on linear approximation around the current iterate; quadratic convergence
*** Kantorovich theorem — gives a region around solution such that Newton's method converges
*** Newton fractal — indicates which initial condition converges to which root under Newton iteration
*** Quasi-Newton method — uses an approximation of the Jacobian:
**** Broyden's method — uses a rank-one update for the Jacobian
**** Symmetric rank-one — a symmetric (but not necessarily positive definite) rank-one update of the Jacobian
**** Davidon–Fletcher–Powell formula — update of the Jacobian in which the matrix remains positive definite
**** Broyden–Fletcher–Goldfarb–Shanno algorithm — rank-two update of the Jacobian in which the matrix remains positive definite
**** Limited-memory BFGS method — truncated, matrix-free variant of BFGS method suitable for large problems
*** Steffensen's method — uses divided differences instead of the derivative
** Secant method — based on linear interpolation at last two iterates
** False position method — secant method with ideas from the bisection method
** Muller's method — based on quadratic interpolation at last three iterates
** Sidi's generalized secant method — higher-order variants of secant method
** Inverse quadratic interpolation — similar to Muller's method, but interpolates the inverse
** Brent's method — combines bisection method, secant method and inverse quadratic interpolation
** Ridders' method — fits a linear function times an exponential to last two iterates and their midpoint
** Halley's method — uses f, f' and f''; achieves the cubic convergence
** Householder's method — uses first d derivatives to achieve order d + 1; generalizes Newton's and Halley's method
* Methods for polynomials:
** Aberth method
** Bairstow's method
** Durand–Kerner method
** Graeffe's method
** Jenkins–Traub algorithm — fast, reliable, and widely used
** Laguerre's method
** Splitting circle method
* Analysis:
** Wilkinson's polynomial
* Numerical continuation — tracking a root as one parameter in the equation changes
** Piecewise linear continuation

## Optimization

Mathematical optimization — algorithm for finding maxima or minima of a given function

### Basic concepts

* Active set
* Candidate solution
* Constraint (mathematics)
** Constrained optimization — studies optimization problems with constraints
** Binary constraint — a constraint that involves exactly two variables
* Corner solution
* Feasible region — contains all solutions that satisfy the constraints but may not be optimal
* Global optimum and Local optimum
* Maxima and minima
* Slack variable
* Continuous optimization
* Discrete optimization

### Linear programming

Linear programming (also treats integer programming) — objective function and constraints are linear
* Algorithms for linear programming:
** Simplex algorithm
*** Bland's rule — rule to avoid cycling in the simplex method
*** Klee–Minty cube — perturbed (hyper)cube; simplex method has exponential complexity on such a domain
*** Criss-cross algorithm — similar to the simplex algorithm
*** Big M method — variation of simplex algorithm for problems with both "less than" and "greater than" constraints
** Interior point method
*** Ellipsoid method
*** Karmarkar's algorithm
*** Mehrotra predictor–corrector method
** Column generation
** k-approximation of k-hitting set — algorithm for specific LP problems (to find a weighted hitting set)
* Linear complementarity problem
* Decompositions:
** Benders' decomposition
** Dantzig–Wolfe decomposition
** Theory of two-level planning
** Variable splitting
* Basic solution (linear programming) — solution at vertex of feasible region
* Fourier–Motzkin elimination
* Hilbert basis (linear programming) — set of integer vectors in a convex cone which generate all integer vectors in the cone
* LP-type problem
* Linear inequality
* Vertex enumeration problem — list all vertices of the feasible set

### Convex optimization

Convex optimization
* Quadratic programming
** Linear least squares (mathematics)
** Total least squares
** Frank–Wolfe algorithm
** Sequential minimal optimization — breaks up large QP problems into a series of smallest possible QP problems
** Bilinear program
* Basis pursuit — minimize L$_1$-norm of vector subject to linear constraints
** Basis pursuit denoising (BPDN) — regularized version of basis pursuit
*** In-crowd algorithm — algorithm for solving basis pursuit denoising
* Linear matrix inequality
* Conic optimization
** Semidefinite programming
** Second-order cone programming
** Sum-of-squares optimization
** Quadratic programming (see above)
* Bregman method — row-action method for strictly convex optimization problems
* Proximal gradient method — use splitting of objective function in sum of possible non-differentiable pieces
* Subgradient method — extension of steepest descent for problems with a non-differentiable objective function
* Biconvex optimization — generalization where objective function and constraint set can be biconvex

### Nonlinear programming

Nonlinear programming — the most general optimization problem in the usual framework
* Special cases of nonlinear programming:
** See Linear programming and Convex optimization above
** Geometric programming — problems involving signomials or posynomials
*** Signomial — similar to polynomials, but exponents need not be integers
*** Posynomial — a signomial with positive coefficients
** Quadratically constrained quadratic program
** Linear-fractional programming — objective is ratio of linear functions, constraints are linear
*** Fractional programming — objective is ratio of nonlinear functions, constraints are linear
** Nonlinear complementarity problem (NCP) — find x such that x ≥ 0, f(x) ≥ 0 and x$^T$ f(x) = 0
** Least squares — the objective function is a sum of squares
*** Non-linear least squares
*** Gauss–Newton algorithm
**** BHHH algorithm — variant of Gauss–Newton in econometrics
**** Generalized Gauss–Newton method — for constrained nonlinear least-squares problems
*** Levenberg–Marquardt algorithm
*** Iteratively reweighted least squares (IRLS) — solves a weighted least-squares problem at every iteration
*** Partial least squares — statistical techniques similar to principal components analysis
**** Non-linear iterative partial least squares (NIPLS)
** Mathematical programming with equilibrium constraints — constraints include variational inequalities or complementarities
** Univariate optimization:
*** Golden section search
*** Successive parabolic interpolation — based on quadratic interpolation through the last three iterates
* General algorithms:
** Concepts:
*** Descent direction
*** Guess value — the initial guess for a solution with which an algorithm starts
*** Line search
**** Backtracking line search
**** Wolfe conditions
** Gradient method — method that uses the gradient as the search direction
*** Gradient descent
**** Stochastic gradient descent
*** Landweber iteration — mainly used for ill-posed problems
** Successive linear programming (SLP) — replace problem by a linear programming problem, solve that, and repeat
** Sequential quadratic programming (SQP) — replace problem by a quadratic programming problem, solve that, and repeat
** Newton's method in optimization
*** See also under Newton algorithm in the section Finding roots of nonlinear equations
** Nonlinear conjugate gradient method
** Derivative-free methods
*** Coordinate descent — move in one of the coordinate directions
**** Adaptive coordinate descent — adapt coordinate directions to objective function
**** Random coordinate descent — randomized version
*** Nelder–Mead method
*** Pattern search (optimization)
*** Powell's method — based on conjugate gradient descent
*** Rosenbrock methods — derivative-free method, similar to Nelder–Mead but with guaranteed convergence
** Augmented Lagrangian method — replaces constrained problems by unconstrained problems with a term added to the objective function
** Ternary search
** Tabu search
** Guided Local Search — modification of search algorithms which builds up penalties during a search
** Reactive search optimization (RSO) — the algorithm adapts its parameters automatically
** MM algorithm — majorize-minimization, a wide framework of methods
** Least absolute deviations
*** Expectation–maximization algorithm
**** Ordered subset expectation maximization
** Nearest neighbor search
** Space mapping — uses "coarse" (ideal or low-fidelity) and "fine" (practical or high-fidelity) models

### Optimal control and infinite-dimensional optimization

Optimal control
* Pontryagin's minimum principle — infinite-dimensional version of Lagrange multipliers
** Costate equations — equation for the "Lagrange multipliers" in Pontryagin's minimum principle
** Hamiltonian (control theory) — minimum principle says that this function should be minimized
* Types of problems:
** Linear-quadratic regulator — system dynamics is a linear differential equation, objective is quadratic
** Linear-quadratic-Gaussian control (LQG) — system dynamics is a linear SDE with additive noise, objective is quadratic
*** Optimal projection equations — method for reducing dimension of LQG control problem
* Algebraic Riccati equation — matrix equation occurring in many optimal control problems
* Bang–bang control — control that switches abruptly between two states
* Covector mapping principle
* Differential dynamic programming — uses locally-quadratic models of the dynamics and cost functions
* DNSS point — initial state for certain optimal control problems with multiple optimal solutions
* Legendre–Clebsch condition — second-order condition for solution of optimal control problem
* Pseudospectral optimal control
** Bellman pseudospectral method — based on Bellman's principle of optimality
** Chebyshev pseudospectral method — uses Chebyshev polynomials (of the first kind)
** Flat pseudospectral method — combines Ross–Fahroo pseudospectral method with differential flatness
** Gauss pseudospectral method — uses collocation at the Legendre–Gauss points
** Legendre pseudospectral method — uses Legendre polynomials
** Pseudospectral knotting method — generalization of pseudospectral methods in optimal control
** Ross–Fahroo pseudospectral method — class of pseudospectral method including Chebyshev, Legendre and knotting
* Ross–Fahroo lemma — condition to make discretization and duality operations commute
* Ross' π lemma — there is fundamental time constant within which a control solution must be computed for controllability and stability
* Sethi model — optimal control problem modelling advertising
Infinite-dimensional optimization
* Semi-infinite programming — infinite number of variables and finite number of constraints, or other way around
* Shape optimization, Topology optimization — optimization over a set of regions
** Topological derivative — derivative with respect to changing in the shape
* Generalized semi-infinite programming — finite number of variables, infinite number of constraints

### Uncertainty and randomness

* Approaches to deal with uncertainty:
** Markov decision process
** Partially observable Markov decision process
** Robust optimization
*** Wald's maximin model
** Scenario optimization — constraints are uncertain
** Stochastic approximation
** Stochastic optimization
** Stochastic programming
** Stochastic gradient descent
* Random optimization algorithms:
** Random search — choose a point randomly in ball around current iterate
** Simulated annealing
*** Adaptive simulated annealing — variant in which the algorithm parameters are adjusted during the computation.
*** Great Deluge algorithm
*** Mean field annealing — deterministic variant of simulated annealing
** Bayesian optimization — treats objective function as a random function and places a prior over it
** Evolutionary algorithm
*** Differential evolution
*** Evolutionary programming
*** Genetic algorithm, Genetic programming
**** Genetic algorithms in economics
*** MCACEA (Multiple Coordinated Agents Coevolution Evolutionary Algorithm) — uses an evolutionary algorithm for every agent
*** Simultaneous perturbation stochastic approximation (SPSA)
** Luus–Jaakola
** Particle swarm optimization
** Stochastic tunneling
** Harmony search — mimicks the improvisation process of musicians
** see also the section Monte Carlo method

### Theoretical aspects

* Convex analysis — function f such that f(tx + (1 − t)y) ≥ tf(x) + (1 − t)f(y) for t ∈ [0,1]
** Pseudoconvex function — function f such that ∇f · (y − x) ≥ 0 implies f(y) ≥ f(x)
** Quasiconvex function — function f such that f(tx + (1 − t)y) ≤ max(f(x), f(y)) for t ∈ [0,1]
** Subderivative
** Geodesic convexity — convexity for functions defined on a Riemannian manifold
* Duality (optimization)
** Weak duality — dual solution gives a bound on the primal solution
** Strong duality — primal and dual solutions are equivalent
** Shadow price
** Dual cone and polar cone
** Duality gap — difference between primal and dual solution
** Fenchel's duality theorem — relates minimization problems with maximization problems of convex conjugates
** Perturbation function — any function which relates to primal and dual problems
** Slater's condition — sufficient condition for strong duality to hold in a convex optimization problem
** Total dual integrality — concept of duality for integer linear programming
** Wolfe duality — for when objective function and constraints are differentiable
* Farkas' lemma
* Karush–Kuhn–Tucker conditions (KKT) — sufficient conditions for a solution to be optimal
** Fritz John conditions — variant of KKT conditions
* Lagrange multiplier
** Lagrange multipliers on Banach spaces
* Semi-continuity
* Complementarity theory — study of problems with constraints of the form ⟨u, v⟩ = 0
** Mixed complementarity problem
*** Mixed linear complementarity problem
*** Lemke's algorithm — method for solving (mixed) linear complementarity problems
* Danskin's theorem — used in the analysis of minimax problems
* Maximum theorem — the maximum and maximizer are continuous as function of parameters, under some conditions
* No free lunch in search and optimization
* Relaxation (approximation) — approximating a given problem by an easier problem by relaxing some constraints
** Lagrangian relaxation
** Linear programming relaxation — ignoring the integrality constraints in a linear programming problem
* Self-concordant function
* Reduced cost — cost for increasing a variable by a small amount
* Hardness of approximation — computational complexity of getting an approximate solution

### Applications

* In geometry:
** Geometric median — the point minimizing the sum of distances to a given set of points
** Chebyshev center — the centre of the smallest ball containing a given set of points
* In statistics:
** Iterated conditional modes — maximizing joint probability of Markov random field
** Response surface methodology — used in the design of experiments
* Automatic label placement
* Compressed sensing — reconstruct a signal from knowledge that it is sparse or compressible
* Cutting stock problem
* Demand optimization
* Destination dispatch — an optimization technique for dispatching elevators
* Energy minimization
* Entropy maximization
* Highly optimized tolerance
* Hyperparameter optimization
* Inventory control problem
** Newsvendor model
** Extended newsvendor model
** Assemble-to-order system
* Linear programming decoding
* Linear search problem — find a point on a line by moving along the line
* Low-rank approximation — find best approximation, constraint is that rank of some matrix is smaller than a given number
* Meta-optimization — optimization of the parameters in an optimization method
* Multidisciplinary design optimization
* Optimal computing budget allocation — maximize the overall simulation efficiency for finding an optimal decision
* Paper bag problem
* Process optimization
* Recursive economics — individuals make a series of two-period optimization decisions over time.
* Stigler diet
* Space allocation problem
* Stress majorization
* Trajectory optimization
* Transportation theory
* Wing-shape optimization

### Miscellaneous

* Combinatorial optimization
* Dynamic programming
** Bellman equation
** Hamilton–Jacobi–Bellman equation — continuous-time analogue of Bellman equation
** Backward induction — solving dynamic programming problems by reasoning backwards in time
** Optimal stopping — choosing the optimal time to take a particular action
*** Odds algorithm
*** Robbins' problem
* Global optimization:
** BRST algorithm
** MCS algorithm
* Multi-objective optimization — there are multiple conflicting objectives
** Benson's algorithm — for linear vector optimization problems
* Bilevel optimization — studies problems in which one problem is embedded in another
* Optimal substructure
* Dykstra's projection algorithm — finds a point in intersection of two convex sets
* Algorithmic concepts:
** Barrier function
** Penalty method
** Trust region
* Test functions for optimization:
** Rosenbrock function — two-dimensional function with a banana-shaped valley
** Himmelblau's function — two-dimensional with four local minima, defined by $f(x,y)=(x^{2}+y-11)^{2}+(x+y^{2}-7)^{2}$
** Rastrigin function — two-dimensional function with many local minima
** Shekel function — multimodal and multidimensional
* Mathematical Optimization Society

## Numerical quadrature (integration)

Numerical integration — the numerical evaluation of an integral
* Rectangle method — first-order method, based on (piecewise) constant approximation
* Trapezoidal rule — second-order method, based on (piecewise) linear approximation
* Simpson's rule — fourth-order method, based on (piecewise) quadratic approximation
** Adaptive Simpson's method
* Boole's rule — sixth-order method, based on the values at five equidistant points
* Newton–Cotes formulas — generalizes the above methods
* Romberg's method — Richardson extrapolation applied to trapezium rule
* Gaussian quadrature — highest possible degree with given number of points
** Chebyshev–Gauss quadrature — extension of Gaussian quadrature for integrals with weight (1 − x$^2$)$^{±1/2}$ on [−1, 1]
** Gauss–Hermite quadrature — extension of Gaussian quadrature for integrals with weight exp(−x$^2$) on [−∞, ∞]
** Gauss–Jacobi quadrature — extension of Gaussian quadrature for integrals with weight (1 − x)$^α$ (1 + x)$^β$ on [−1, 1]
** Gauss–Laguerre quadrature — extension of Gaussian quadrature for integrals with weight exp(−x) on [0, ∞]
** Gauss–Kronrod quadrature formula — nested rule based on Gaussian quadrature
** Gauss–Kronrod rules
* Tanh-sinh quadrature — variant of Gaussian quadrature which works well with singularities at the end points
* Clenshaw–Curtis quadrature — based on expanding the integrand in terms of Chebyshev polynomials
* Adaptive quadrature — adapting the subintervals in which the integration interval is divided depending on the integrand
* Monte Carlo integration — takes random samples of the integrand
** See also #Monte Carlo method
* Quantized state systems method (QSS) — based on the idea of state quantization
* Lebedev quadrature — uses a grid on a sphere with octahedral symmetry
* Sparse grid
* Coopmans approximation
* Numerical differentiation — for fractional-order integrals
** Numerical smoothing and differentiation
** Adjoint state method — approximates gradient of a function in an optimization problem
* Euler–Maclaurin formula

## Numerical methods for ordinary differential equations

Numerical methods for ordinary differential equations — the numerical solution of ordinary differential equations (ODEs)
* Euler method — the most basic method for solving an ODE
* Explicit and implicit methods — implicit methods need to solve an equation at every step
* Backward Euler method — implicit variant of the Euler method
* Trapezoidal rule — second-order implicit method
* Runge–Kutta methods — one of the two main classes of methods for initial-value problems
** Midpoint method — a second-order method with two stages
** Heun's method — either a second-order method with two stages, or a third-order method with three stages
** Bogacki–Shampine method — a third-order method with four stages (FSAL) and an embedded fourth-order method
** Cash–Karp method — a fifth-order method with six stages and an embedded fourth-order method
** Dormand–Prince method — a fifth-order method with seven stages (FSAL) and an embedded fourth-order method
** Runge–Kutta–Fehlberg method — a fifth-order method with six stages and an embedded fourth-order method
** Gauss–Legendre method — family of A-stable method with optimal order based on Gaussian quadrature
** Butcher group — algebraic formalism involving rooted trees for analysing Runge–Kutta methods
** List of Runge–Kutta methods
* Linear multistep method — the other main class of methods for initial-value problems
** Backward differentiation formula — implicit methods of order 2 to 6; especially suitable for stiff equations
** Numerov's method — fourth-order method for equations of the form $y''=f(t,y)$
** Predictor–corrector method — uses one method to approximate solution and another one to increase accuracy
* General linear methods — a class of methods encapsulating linear multistep and Runge-Kutta methods
* Bulirsch–Stoer algorithm — combines the midpoint method with Richardson extrapolation to attain arbitrary order
* Exponential integrator — based on splitting ODE in a linear part, which is solved exactly, and a nonlinear part
* Methods designed for the solution of ODEs from classical physics:
** Newmark-beta method — based on the extended mean-value theorem
** Verlet integration — a popular second-order method
** Leapfrog integration — another name for Verlet integration
** Beeman's algorithm — a two-step method extending the Verlet method
** Dynamic relaxation
* Geometric integrator — a method that preserves some geometric structure of the equation
** Symplectic integrator — a method for the solution of Hamilton's equations that preserves the symplectic structure
*** Variational integrator — symplectic integrators derived using the underlying variational principle
*** Semi-implicit Euler method — variant of Euler method which is symplectic when applied to separable Hamiltonians
** Energy drift — phenomenon that energy, which should be conserved, drifts away due to numerical errors
* Other methods for initial value problems (IVPs):
** Bi-directional delay line
** Partial element equivalent circuit
* Methods for solving two-point boundary value problems (BVPs):
** Shooting method
** Direct multiple shooting method — divides interval in several subintervals and applies the shooting method on each subinterval
* Methods for solving differential-algebraic equations (DAEs), i.e., ODEs with constraints:
** Constraint algorithm — for solving Newton's equations with constraints
** Pantelides algorithm — for reducing the index of a DEA
* Methods for solving stochastic differential equations (SDEs):
** Euler–Maruyama method — generalization of the Euler method for SDEs
** Milstein method — a method with strong order one
** Runge–Kutta method (SDE) — generalization of the family of Runge–Kutta methods for SDEs
* Methods for solving integral equations:
** Nyström method — replaces the integral with a quadrature rule
* Analysis:
** Truncation error (numerical integration) — local and global truncation errors, and their relationships
*** Lady Windermere's Fan (mathematics) — telescopic identity relating local and global truncation errors
* Stiff equation — roughly, an ODE for which unstable methods need a very short step size, but stable methods do not
** L-stability — method is A-stable and stability function vanishes at infinity
* Adaptive stepsize — automatically changing the step size when that seems advantageous
* Parareal -- a parallel-in-time integration algorithm

## Numerical methods for partial differential equations

Numerical partial differential equations — the numerical solution of partial differential equations (PDEs)

### Finite difference methods

Finite difference method — based on approximating differential operators with difference operators
* Finite difference — the discrete analogue of a differential operator
** Finite difference coefficient — table of coefficients of finite-difference approximations to derivatives
** Discrete Laplace operator — finite-difference approximation of the Laplace operator
*** Eigenvalues and eigenvectors of the second derivative — includes eigenvalues of discrete Laplace operator
*** Kronecker sum of discrete Laplacians — used for Laplace operator in multiple dimensions
** Discrete Poisson equation — discrete analogue of the Poisson equation using the discrete Laplace operator
* Stencil (numerical analysis) — the geometric arrangements of grid points affected by a basic step of the algorithm
** Compact stencil — stencil which only uses a few grid points, usually only the immediate and diagonal neighbours
*** Higher-order compact finite difference scheme
** Non-compact stencil — any stencil that is not compact
** Five-point stencil — two-dimensional stencil consisting of a point and its four immediate neighbours on a rectangular grid
* Finite difference methods for heat equation and related PDEs:
** FTCS scheme (forward-time central-space) — first-order explicit
** Crank–Nicolson method — second-order implicit
* Finite difference methods for hyperbolic PDEs like the wave equation:
** Lax–Friedrichs method — first-order explicit
** Lax–Wendroff method — second-order explicit
** MacCormack method — second-order explicit
** Upwind scheme
*** Upwind differencing scheme for convection — first-order scheme for convection–diffusion problems
** Lax–Wendroff theorem — conservative scheme for hyperbolic system of conservation laws converges to the weak solution
* Alternating direction implicit method (ADI) — update using the flow in x-direction and then using flow in y-direction
* Nonstandard finite difference scheme
* Specific applications:
** Finite difference methods for option pricing
** Finite-difference time-domain method — a finite-difference method for electrodynamics

### Finite element methods, gradient discretisation methods

Finite element method — based on a discretization of the space of solutions gradient discretisation method — based on both the discretization of the solution and of its gradient
* Finite element method in structural mechanics — a physical approach to finite element methods
* Galerkin method — a finite element method in which the residual is orthogonal to the finite element space
** Discontinuous Galerkin method — a Galerkin method in which the approximate solution is not continuous
* Rayleigh–Ritz method — a finite element method based on variational principles
* Spectral element method — high-order finite element methods
* hp-FEM — variant in which both the size and the order of the elements are automatically adapted
* Examples of finite elements:
** Bilinear quadrilateral element — also known as the Q4 element
** Constant strain triangle element (CST) — also known as the T3 element
** Quadratic quadrilateral element — also known as the Q8 element
** Barsoum elements
* Direct stiffness method — a particular implementation of the finite element method, often used in structural analysis
* Trefftz method
* Finite element updating
* Extended finite element method — puts functions tailored to the problem in the approximation space
* Functionally graded elements — elements for describing functionally graded materials
* Superelement — particular grouping of finite elements, employed as a single element
* Interval finite element method — combination of finite elements with interval arithmetic
* Discrete exterior calculus — discrete form of the exterior calculus of differential geometry
* Modal analysis using FEM — solution of eigenvalue problems to find natural vibrations
* Céa's lemma — solution in the finite-element space is an almost best approximation in that space of the true solution
* Patch test (finite elements) — simple test for the quality of a finite element
* MAFELAP (MAthematics of Finite ELements and APplications) — international conference held at Brunel University
* NAFEMS — not-for-profit organisation that sets and maintains standards in computer-aided engineering analysis
* Multiphase topology optimisation — technique based on finite elements for determining optimal composition of a mixture
* Interval finite element
* Applied element method — for simulation of cracks and structural collapse
* Wood–Armer method — structural analysis method based on finite elements used to design reinforcement for concrete slabs
* Isogeometric analysis — integrates finite elements into conventional NURBS-based CAD design tools
* Loubignac iteration
* Stiffness matrix — finite-dimensional analogue of differential operator
* Combination with meshfree methods:
** Weakened weak form — form of a PDE that is weaker than the standard weak form
** G space — functional space used in formulating the weakened weak form
** Smoothed finite element method
* Variational multiscale method
* List of finite element software packages

### Other methods

* Spectral method — based on the Fourier transformation
** Pseudo-spectral method
* Method of lines — reduces the PDE to a large system of ordinary differential equations
* Boundary element method (BEM) — based on transforming the PDE to an integral equation on the boundary of the domain
** Interval boundary element method — a version using interval arithmetics
* Analytic element method — similar to the boundary element method, but the integral equation is evaluated analytically
* Finite volume method — based on dividing the domain in many small domains; popular in computational fluid dynamics
** Godunov's scheme — first-order conservative scheme for fluid flow, based on piecewise constant approximation
** MUSCL scheme — second-order variant of Godunov's scheme
** AUSM — advection upstream splitting method
** Flux limiter — limits spatial derivatives (fluxes) in order to avoid spurious oscillations
** Riemann solver — a solver for Riemann problems (a conservation law with piecewise constant data)
** Properties of discretization schemes — finite volume methods can be conservative, bounded, etc.
* Discrete element method — a method in which the elements can move freely relative to each other
** Extended discrete element method — adds properties such as strain to each particle
** Movable cellular automaton — combination of cellular automata with discrete elements
* Meshfree methods — does not use a mesh, but uses a particle view of the field
** Discrete least squares meshless method — based on minimization of weighted summation of the squared residual
** Diffuse element method
** Finite pointset method — represent continuum by a point cloud
** Moving Particle Semi-implicit Method
** Method of fundamental solutions (MFS) — represents solution as linear combination of fundamental solutions
** Variants of MFS with source points on the physical boundary:
*** Boundary knot method (BKM)
*** Boundary particle method (BPM)
*** Regularized meshless method (RMM)
*** Singular boundary method (SBM)
* Methods designed for problems from electromagnetics:
** Finite-difference time-domain method — a finite-difference method
** Rigorous coupled-wave analysis — semi-analytical Fourier-space method based on Floquet's theorem
** Transmission-line matrix method (TLM) — based on analogy between electromagnetic field and mesh of transmission lines
** Uniform theory of diffraction — specifically designed for scattering problems
* Particle-in-cell — used especially in fluid dynamics
** Multiphase particle-in-cell method — considers solid particles as both numerical particles and fluid
* High-resolution scheme
* Shock capturing method
* Vorticity confinement — for vortex-dominated flows in fluid dynamics, similar to shock capturing
* Split-step method
* Fast marching method
* Orthogonal collocation
* Lattice Boltzmann methods — for the solution of the Navier-Stokes equations
* Roe solver — for the solution of the Euler equation
* Relaxation (iterative method) — a method for solving elliptic PDEs by converting them to evolution equations
* Broad classes of methods:
** Mimetic methods — methods that respect in some sense the structure of the original problem
** Multiphysics — models consisting of various submodels with different physics
** Immersed boundary method — for simulating elastic structures immersed within fluids
* Multisymplectic integrator — extension of symplectic integrators, which are for ODEs
* Stretched grid method — for problems solution that can be related to an elastic grid behavior.

### Techniques for improving these methods

* Multigrid method — uses a hierarchy of nested meshes to speed up the methods
* Domain decomposition methods — divides the domain in a few subdomains and solves the PDE on these subdomains
** Additive Schwarz method
** Abstract additive Schwarz method — abstract version of additive Schwarz without reference to geometric information
** Balancing domain decomposition method (BDD) — preconditioner for symmetric positive definite matrices
** Balancing domain decomposition by constraints (BDDC) — further development of BDD
** Finite element tearing and interconnect (FETI)
** FETI-DP — further development of FETI
** Fictitious domain method — preconditioner constructed with a structured mesh on a fictitious domain of simple shape
** Mortar methods — meshes on subdomain do not mesh
** Neumann–Dirichlet method — combines Neumann problem on one subdomain with Dirichlet problem on other subdomain
** Neumann–Neumann methods — domain decomposition methods that use Neumann problems on the subdomains
** Poincaré–Steklov operator — maps tangential electric field onto the equivalent electric current
** Schur complement method — early and basic method on subdomains that do not overlap
** Schwarz alternating method — early and basic method on subdomains that overlap
* Coarse space — variant of the problem which uses a discretization with fewer degrees of freedom
* Adaptive mesh refinement — uses the computed solution to refine the mesh only where necessary
* Fast multipole method — hierarchical method for evaluating particle-particle interactions
* Perfectly matched layer — artificial absorbing layer for wave equations, used to implement absorbing boundary conditions

### Grids and meshes

* Grid classification / Types of mesh:
** Polygon mesh — consists of polygons in 2D or 3D
** Triangle mesh — consists of triangles in 2D or 3D
*** Triangulation (geometry) — subdivision of given region in triangles, or higher-dimensional analogue
*** Nonobtuse mesh — mesh in which all angles are less than or equal to 90°
*** Point-set triangulation — triangle mesh such that given set of point are all a vertex of a triangle
*** Polygon triangulation — triangle mesh inside a polygon
*** Delaunay triangulation — triangulation such that no vertex is inside the circumcentre of a triangle
*** Constrained Delaunay triangulation — generalization of the Delaunay triangulation that forces certain required segments into the triangulation
*** Pitteway triangulation — for any point, triangle containing it has nearest neighbour of the point as a vertex
*** Minimum-weight triangulation — triangulation of minimum total edge length
*** Kinetic triangulation — a triangulation that moves over time
*** Triangulated irregular network
*** Quasi-triangulation — subdivision into simplices, where vertices are not points but arbitrary sloped line segments
** Volume mesh — consists of three-dimensional shapes
** Regular grid — consists of congruent parallelograms, or higher-dimensional analogue
** Unstructured grid
** Geodesic grid — isotropic grid on a sphere
* Mesh generation
** Image-based meshing — automatic procedure of generating meshes from 3D image data
** Marching cubes — extracts a polygon mesh from a scalar field
** Parallel mesh generation
** Ruppert's algorithm — creates quality Delauney triangularization from piecewise linear data
* Subdivisions:
* Apollonian network — undirected graph formed by recursively subdividing a triangle
* Barycentric subdivision — standard way of dividing arbitrary convex polygons into triangles, or the higher-dimensional analogue
* Improving an existing mesh:
** Chew's second algorithm — improves Delauney triangularization by refining poor-quality triangles
** Laplacian smoothing — improves polynomial meshes by moving the vertices
* Jump-and-Walk algorithm — for finding triangle in a mesh containing a given point
* Spatial twist continuum — dual representation of a mesh consisting of hexahedra
* Pseudotriangle — simply connected region between any three mutually tangent convex sets
* Simplicial complex — all vertices, line segments, triangles, tetrahedra, ..., making up a mesh

### Analysis

* Lax equivalence theorem — a consistent method is convergent if and only if it is stable
* Courant–Friedrichs–Lewy condition — stability condition for hyperbolic PDEs
* Von Neumann stability analysis — all Fourier components of the error should be stable
* Numerical diffusion — diffusion introduced by the numerical method, above to that which is naturally present
** False diffusion
* Numerical dispersion
* Numerical resistivity — the same, with resistivity instead of diffusion
* Weak formulation — a functional-analytic reformulation of the PDE necessary for some methods
* Total variation diminishing — property of schemes that do not introduce spurious oscillations
* Godunov's theorem — linear monotone schemes can only be of first order
* Motz's problem — benchmark problem for singularity problems

## Monte Carlo method

* Variants of the Monte Carlo method:
** Direct simulation Monte Carlo
** Quasi-Monte Carlo method
** Markov chain Monte Carlo
*** Metropolis–Hastings algorithm
**** Multiple-try Metropolis — modification which allows larger step sizes
**** Wang and Landau algorithm — extension of Metropolis Monte Carlo
**** Equation of State Calculations by Fast Computing Machines — 1953 article proposing the Metropolis Monte Carlo algorithm
**** Multicanonical ensemble — sampling technique that uses Metropolis–Hastings to compute integrals
*** Gibbs sampling
*** Coupling from the past
*** Reversible-jump Markov chain Monte Carlo
** Dynamic Monte Carlo method
*** Kinetic Monte Carlo
*** Gillespie algorithm
** Particle filter
*** Auxiliary particle filter
** Reverse Monte Carlo
** Demon algorithm
* Pseudo-random number sampling
** Inverse transform sampling — general and straightforward method but computationally expensive
** Rejection sampling — sample from a simpler distribution but reject some of the samples
*** Ziggurat algorithm — uses a pre-computed table covering the probability distribution with rectangular segments
** For sampling from a normal distribution:
*** Box–Muller transform
*** Marsaglia polar method
** Convolution random number generator — generates a random variable as a sum of other random variables
** Indexed search
* Variance reduction techniques:
** Antithetic variates
** Control variates
** Importance sampling
** Stratified sampling
** VEGAS algorithm
* Low-discrepancy sequence
** Constructions of low-discrepancy sequences
* Event generator
* Parallel tempering
* Umbrella sampling — improves sampling in physical systems with significant energy barriers
* Hybrid Monte Carlo
* Ensemble Kalman filter — recursive filter suitable for problems with a large number of variables
* Transition path sampling
* Walk-on-spheres method — to generate exit-points of Brownian motion from bounded domains
* Applications:
** Ensemble forecasting — produce multiple numerical predictions from slightly initial conditions or parameters
** Bond fluctuation model — for simulating the conformation and dynamics of polymer systems
** Iterated filtering
** Metropolis light transport
** Monte Carlo localization — estimates the position and orientation of a robot
** Monte Carlo methods for electron transport
** Monte Carlo method for photon transport
** Monte Carlo methods in finance
*** Monte Carlo methods for option pricing
*** Quasi-Monte Carlo methods in finance
** Monte Carlo molecular modeling
*** Path integral molecular dynamics — incorporates Feynman path integrals
** Quantum Monte Carlo
*** Diffusion Monte Carlo — uses a Green function to solve the Schrödinger equation
*** Gaussian quantum Monte Carlo
*** Path integral Monte Carlo
*** Reptation Monte Carlo
*** Variational Monte Carlo
** Methods for simulating the Ising model:
*** Swendsen–Wang algorithm — entire sample is divided into equal-spin clusters
*** Wolff algorithm — improvement of the Swendsen–Wang algorithm
*** Metropolis–Hastings algorithm
** Auxiliary field Monte Carlo — computes averages of operators in many-body quantum mechanical problems
** Cross-entropy method — for multi-extremal optimization and importance sampling
* Also see the list of statistics topics

## Applications

* Computational physics
** Computational electromagnetics
** Computational fluid dynamics (CFD)
*** Numerical methods in fluid mechanics
*** Large eddy simulation
*** Smoothed-particle hydrodynamics
*** Aeroacoustic analogy — used in numerical aeroacoustics to reduce sound sources to simple emitter types
*** Stochastic Eulerian Lagrangian method — uses Eulerian description for fluids and Lagrangian for structures
*** Explicit algebraic stress model
** Computational magnetohydrodynamics (CMHD) — studies electrically conducting fluids
** Climate model
** Numerical weather prediction
*** Geodesic grid
** Celestial mechanics
*** Numerical model of the Solar System
** Quantum jump method — used for simulating open quantum systems, operates on wave function
** Dynamic design analysis method (DDAM) — for evaluating effect of underwater explosions on equipment
* Computational chemistry
** Cell lists
** Coupled cluster
** Density functional theory
** DIIS — direct inversion in (or of) the iterative subspace
* Computational sociology
* Computational statistics

## Software

For a large list of software, see the list of numerical-analysis software.

## Journals

* Acta Numerica
* Mathematics of Computation (published by the American Mathematical Society)
* Journal of Computational and Applied Mathematics
* BIT Numerical Mathematics
* Numerische Mathematik
* Journals from the Society for Industrial and Applied Mathematics
** SIAM Journal on Numerical Analysis
** SIAM Journal on Scientific Computing

## Researchers

* Cleve Moler
* Gene H. Golub
* James H. Wilkinson
* Margaret H. Wright
* Nicholas J. Higham
* Nick Trefethen
* Peter Lax
* Richard S. Varga
* Ulrich W. Kulisch
* Vladik Kreinovich
* Chi-Wang Shu