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===Numerical algorithms===
{{further|List of numerical analysis topics|Numerical analysis}}

====Differential equation solving====
{{further|Differential equation}}
* [[Backward Euler method]]
* [[Euler method]]
* [[Linear multistep method]]s
* [[Multigrid method]]s (MG methods), a group of algorithms for solving differential equations using a hierarchy of discretizations
* [[Partial differential equation]]:
** [[Crank–Nicolson method]] for diffusion equations
** [[Finite difference method]]
** [[Lax–Wendroff method|Lax–Wendroff]] for wave equations
* [[Runge–Kutta methods]]
** [[Euler integration]]
* [[Trapezoidal rule (differential equations)]]
* [[Verlet integration]] ({{IPA|fr|v???l?}}): integrate Newton's equations of motion

====Elementary and special functions====
{{further|Special functions}}
* [[Computing π|Computation of π]]:
** [[Bailey–Borwein–Plouffe formula]]: (BBP formula) a spigot algorithm for the computation of the nth binary digit of π
** [[Borwein's algorithm]]: an algorithm to calculate the value of 1/π
** [[Chudnovsky algorithm]]: a fast method for calculating the digits of π
** [[Gauss–Legendre algorithm]]: computes the digits of [[pi]]
* [[Division algorithm]]s: for computing quotient and/or remainder of two numbers
** [[Goldschmidt division]]
** [[Long division]]
** [[Newton–Raphson division]]: uses [[Newton's method]] to find the [[Multiplicative inverse|reciprocal]] of D, and multiply that reciprocal by N to find the final quotient Q.
** [[Non-restoring division]]
** [[Restoring division]]
** [[SRT division]]
* Exponentiation:
** [[Addition-chain exponentiation]]: exponentiation by positive integer powers that requires a minimal number of multiplications
** [[Exponentiating by squaring]]: an algorithm used for the fast computation of [[Arbitrary-precision arithmetic|large integer]] powers of a number
* Hyperbolic and Trigonometric Functions:
** [[BKM algorithm]]: computes [[Elementary function (differential algebra)|elementary functions]] using a table of logarithms
** [[CORDIC]]: computes hyperbolic and trigonometric functions using a table of arctangents
* [[Montgomery reduction]]: an algorithm that allows [[modular arithmetic]] to be performed efficiently when the modulus is large
* [[Multiplication algorithm]]s: fast multiplication of two numbers
** [[Booth's multiplication algorithm]]: a multiplication algorithm that multiplies two signed binary numbers in two's complement notation
** [[Fürer's algorithm]]: an integer multiplication algorithm for very large numbers possessing a very low [[Computational complexity theory|asymptotic complexity]]
** [[Karatsuba algorithm]]: an efficient procedure for multiplying large numbers
** [[Sch?nhage–Strassen algorithm]]: an asymptotically fast multiplication algorithm for large integers
** [[Toom–Cook multiplication]]: (Toom3) a multiplication algorithm for large integers
* [[Multiplicative inverse#Algorithms|Multiplicative inverse Algorithms]]: for computing a number's multiplicative inverse (reciprocal).
** [[Newton's method#Multiplicative inverses of numbers and power series|Newton's method]]
* [[Rounding functions]]: the classic ways to round numbers
* [[Spigot algorithm]]: a way to compute the value of a [[mathematical constant]] without knowing preceding digits
* Square and Nth root of a number:
** [[Alpha max plus beta min algorithm]]: an approximation of the square-root of the sum of two squares
** [[Methods of computing square roots]]
** [[Nth root algorithm|''n''th root algorithm]]
* Summation:
** [[Binary splitting]]: a [[Divide and conquer algorithm|divide and conquer]] technique which speeds up the numerical evaluation of many types of series with rational terms
** [[Kahan summation algorithm]]: a more accurate method of summing floating-point numbers
* [[Unrestricted algorithm]]

====Geometric====
* [[Radon transform#Radon inversion formula|Filtered back-projection]]: efficiently computes the inverse 2-dimensional [[Radon transform]].
* [[Level set method]] (LSM): a numerical technique for tracking interfaces and shapes

====Interpolation and extrapolation====
{{further|Extrapolation|Interpolation}}
* [[Birkhoff interpolation]]: an extension of polynomial interpolation
* [[Cubic interpolation]]
* [[Hermite interpolation]]
* [[Lagrange interpolation]]: interpolation using [[Lagrange polynomial]]s
* [[Linear interpolation]]: a method of curve fitting using linear polynomials
* [[Monotone cubic interpolation]]: a variant of cubic interpolation that preserves monotonicity of the data set being interpolated.
* [[Multivariate interpolation]]
** [[Bicubic interpolation]]: a generalization of [[cubic interpolation]] to two dimensions
** [[Bilinear interpolation]]: an extension of [[linear interpolation]] for interpolating functions of two variables on a regular grid
** [[Lanczos resampling]] ("Lanzosh"): a multivariate interpolation method used to compute new values for any digitally sampled data
** [[Nearest-neighbor interpolation]]
** [[Tricubic interpolation]]: a generalization of [[cubic interpolation]] to three dimensions
* [[Pareto interpolation]]: a method of estimating the median and other properties of a population that follows a [[Pareto distribution]].
* [[Polynomial interpolation]]
** [[Neville's algorithm]]
* [[Spline interpolation]]: Reduces error with [[Runge's phenomenon]].
** [[De Boor algorithm]]: [[B-spline]]s
** [[De Casteljau's algorithm]]: [[Bézier curve]]s
* [[Trigonometric interpolation]]

====Linear algebra====
{{further|Numerical linear algebra}}
* [[Eigenvalue algorithm]]s
** [[Arnoldi iteration]]
** [[Inverse iteration]]
** [[Jacobi eigenvalue algorithm|Jacobi method]]
** [[Lanczos iteration]]
** [[Power iteration]]
** [[QR algorithm]]
** [[Rayleigh quotient iteration]]
* [[Gram–Schmidt process]]: orthogonalizes a set of vectors
* Krylov methods (for large sparse matrix problems; third most-important numerical method class of the 20th century as ranked by SISC; after fast-fourier and fast-multipole)
* [[Matrix multiplication algorithm]]s
** [[Cannon's algorithm]]: a [[distributed algorithm]] for [[matrix multiplication]] especially suitable for computers laid out in an N × N mesh
** [[Coppersmith–Winograd algorithm]]: square [[matrix multiplication]]
** [[Freivalds' algorithm]]: a randomized algorithm used to verify matrix multiplication
** [[Strassen algorithm]]: faster [[matrix multiplication]]
{{anchor|Solving systems of linear equations}}
* Solving [[system of linear equations|systems of linear equations]]
** [[Biconjugate gradient method]]: solves systems of linear equations
** [[Conjugate gradient]]: an algorithm for the numerical solution of particular systems of linear equations
** [[Gauss–Jordan elimination]]: solves systems of linear equations
** [[Gauss–Seidel method]]: solves systems of linear equations iteratively
** [[Gaussian elimination]]
** [[Levinson recursion]]: solves equation involving a [[Toeplitz matrix]]
** [[Stone's method]]: also known as the strongly implicit procedure or SIP, is an algorithm for solving a sparse linear system of equations
** [[Successive over-relaxation]] (SOR): method used to speed up convergence of the [[Gauss–Seidel method]]
** [[Tridiagonal matrix algorithm]] (Thomas algorithm): solves systems of tridiagonal equations
* [[Sparse matrix]] algorithms
** [[Cuthill–McKee algorithm]]: reduce the [[bandwidth (matrix theory)|bandwidth]] of a [[symmetric sparse matrix]]
** [[Minimum degree algorithm]]: permute the rows and columns of a [[symmetric sparse matrix]] before applying the [[Cholesky decomposition]]
** [[Symbolic Cholesky decomposition]]: Efficient way of storing [[sparse matrix]]

====Monte Carlo====
{{further|Monte Carlo method}}
* [[Gibbs sampling]]: generates a sequence of samples from the joint probability distribution of two or more random variables
* [[Hybrid Monte Carlo]]: generates a sequence of samples using [[Hamiltonian dynamics|Hamiltonian]] weighted [[Markov chain Monte Carlo]], from a probability distribution which is difficult to sample directly.
* [[Metropolis–Hastings algorithm]]: used to generate a sequence of samples from the [[probability distribution]] of one or more variables
* [[Wang and Landau algorithm]]: an extension of [[Metropolis–Hastings algorithm]] sampling

====Numerical integration====
{{further|Numerical integration}}
* [[MISER algorithm]]: Monte Carlo simulation, [[numerical integration]]

====Root finding====
{{main|Root-finding algorithm}}
* [[Bisection method]]
* [[False position method]]: and Illinois method: 2-point, bracketing
* [[Halley's method]]: uses first and second derivatives
* [[ITP Method|ITP method]]: minmax optimal and superlinear convergence simultaneously
* [[Muller's method]]: 3-point, quadratic interpolation
* [[Newton's method]]: finds zeros of functions with [[calculus]]
* [[Ridder's method]]: 3-point, exponential scaling
* [[Secant method]]: 2-point, 1-sided