List of numerical analysis topics

 

This is a list of numerical analysis topics, by Wikipedia page.

General

• Iterative method
• Sequence transformations — methods to accelerate the speed of convergence
  • Richardson extrapolation
• Level set method
  • Level set (data structures) — data structures for representing level sets
• Abramowitz and Stegun
• Curse of dimensionality
• Superconvergence
• Termination
• Discretization
  • Collocation method — discretizes a continuous equation by requiring it only to hold at certain points
• Difference quotient
• Computational complexity of mathematical operations

Error

Error analysis

• Approximation
• Approximation error
• Arithmetic precision
• Condition number
• Discretization error
• Floating point number
  • Guard digit — extra precision introduced during a computation to reduce round-off error
  • Arbitrary-precision arithmetic
• Loss of significance
• Numerical error
• Numerical stability
• Error propagation:
  • Propagation of uncertainty
  • Significance arithmetic
• Residual (mathematics)
• Round-off error
  • Stochastic rounding
• Significant figures
• Truncation
• Well-posed problem
• Affine arithmetic

Elementary and special functions

• Summation:
  • Kahan summation algorithm
  • Binary splitting
• 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 fastest
• Exponentiation:
  • Exponentiation by squaring
  • Addition-chain exponentiation
• Polynomials:
  • Horner scheme
  • Clenshaw algorithm
  • De Casteljau's algorithm
• Square roots and other roots:
  • Integer square root
  • Methods of computing square roots
  • nth root algorithm
  • Alpha max plus beta min algorithm — approximates
• Elementary functions (exponential, logarithm, trigonometric functions):
  • Generating trigonometric tables
  • 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
• Computation of π:
  • Gauss-Legendre algorithm — iteration which converges quadratically to π, based on arithmetic-geometric mean
  • Bailey-Borwein-Plouffe formula — can be used to compute individual hexadecimal digits of π
  • Borwein's algorithm — iteration which converges quartically to 1/π, and other algorithms

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
    • Tridiagonal matrix
    • Pentadiagonal matrix
    • Skyline matrix
  • Circulant matrix
  • Triangular matrix
  • Diagonally dominant matrix
  • Hilbert matrix — example of a matrix which is extremely ill-conditioned (and thus difficult to handle)
• Algorithms for matrix multiplication:
  • Strassen algorithm
  • Coppersmith–Winograd algorithm
  • Freivald's algorithm — a randomized algorithm for checking the result of a multiplication

Solving systems of linear equations

• Gaussian elimination
  • Row echelon form — matrix in which all entries below a nonzero entry are zero
  • Reduced row echelon form — matrix in row echelon form, in which the leading coefficients are one and have only zeros above them
  • Gauss–Jordan elimination — variant in which the entries below the pivot are also zeroed
  • Montante's method — 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
  • Crout matrix decomposition
• Block LU decomposition
• Cholesky decomposition — for solving a system with a positive definite matrix
  • Minimum degree algorithm
  • Symbolic Cholesky decomposition
• Frontal solver — for sparse matrices; used in finite element methods
• Levinson recursion — for Toeplitz matrices
• Iterative methods:
  • Jacobi method
  • Gauss-Seidel method
    • Successive over-relaxation (SOR) — a technique to accelerate the Gauss-Seidel method
  • Modified Richardson iteration
  • Conjugate gradient method — assumes that the matrix is positive definite
    • Preconditioned conjugate gradient method
    • Nonlinear conjugate gradient method — generalization for nonlinear optimization problems
  • Biconjugate gradient method (BiCG)
  • Generalized minimal residual method (GMRES) — based on the Arnoldi iteration
  • Stone method — uses an incomplete LU decomposition
  • Preconditioner

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
• 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
• Divide-and-conquer eigenvalue algorithm

Other concepts and algorithms

• Orthogonalization algorithms:
  • Gram–Schmidt process
  • Householder transformation
  • Givens rotation
• QR decomposition
• Krylov subspace
• Block matrix pseudoinverse
• Bidiagonalization
• Pivot element — entry in a matrix on which the algorithm concentrates

Interpolation

• 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 constant (interpolation)
• 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
• Extensions to multiple dimensions:
  • Bilinear interpolation
  • Trilinear interpolation
  • Bicubic interpolation
• Hermite interpolation
• Birkhoff 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
• Monotone cubic interpolation
• Hermite spline
• Cardinal spline
• Kochanek-Bartels spline
• Catmull-Rom spline
• Bézier spline
  • Bézier curve
  • De Casteljau's algorithm
• B-spline
  • Truncated power function
  • De Boor's algorithm — generalizes De Casteljau's algorithm
• Nonuniform rational B-spline (NURBS)
• Blossom (mathematics) — 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 — 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
  • Butterfly diagram
  • Twiddle factor — the trigonometric constant coefficients that are multiplied by the data
  • Overlap-add method — computes discrete convulations with finite impulse response filters using the FFT
• Sigma approximation
• Dirichlet kernel — convolving any function with the Dirichlet kernel yields its trigonometric interpolant

Other interpolants

• Simple rational approximation
• Wavelet
  • Continuous wavelet
    • Continuous wavelet transform
  • Transfer matrix
• Inverse distance weighting
  • Shepard's method
  • Cascade algorithm — iterative algorithm to compute wavelets
• Radial basis function
  • Polyharmonic spline — a commonly used radial basis function
  • Thin plate spline — a specific polyharmonic spline: r² log r
  • Radial basis function network — neural network using radial basis functions as activation functions
• Subdivision surface — constructed by recursively subdividing a piecewise linear interpolant
  • Catmull-Clark subdivision surface
  • Doo-Sabin subdivision surface
• Slerp
• Irrational base discrete weighted transform
• Pick matrix — appears when interpolating by analytic functions which must be bounded in the unit disc
• Pareto interpolation
• Multivariate interpolation — the function being interpolated depends on more than one variable
  • Lanczos resampling — based on convolution with a sinc function
  • Natural neighbor interpolation
  • PDE surface
  • Method based on polynomials are listed under Polynomial interpolation

Approximation theory

Approximation theory

• Orders of approximation
• Lebesgue's lemma
• Curve fitting
• Modulus of continuity — measures smoothness of a function
• Remez algorithm — for constructing the best polynomial approximation
• Jackson's inequality — upper bound for best approximation by a trigonometric polynomial
• Surrogate model — application: replacing a function that is hard to evaluate by a simpler function
• Different approximations:
  • Linear approximation
  • Moving least squares
  • Padé approximant
    • Padé table — table of Padé approximants

Miscellaneous

• Extrapolation
  • Linear predictive analysis — linear extrapolation
• Regression analysis
  • Isotonic regression
• Curve-fitting compaction
• Interpolation (computer programming) — interpolation in the context of 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
    • Newton fractal
    • Quasi-Newton method — use an approximation of the Jacobian:
      • Broyden's method — uses a rank-one update for the Jacobian
      • SR1 formula — 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
      • BFGS method — rank-two update of the Jacobian in which the matrix remains positive definite
  • Secant method — based on linear interpolation at last two iterates
  • False position method — secant method with ideas from the bisection method
  • Müller's method — based on quadratic interpolation at last three iterates
  • Inverse quadratic interpolation — similar to Müller's method, but interpolates the inverse
  • Halley's method — uses f, f' and f''; achieves the cubic convergence
  • Brent's method — combines bisection method, secant method and inverse quadratic interpolation
• 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
• Methods for other special cases:
  • Shifting nth-root algorithm
• Analysis:
  • Wilkinson's polynomial
• Numerical continuation — tracking a root as one parameters in the equation changes

Optimization

Optimization (mathematics) — algorithm for finding maxima or minima of a given function

Basic concepts

• Active set
• Candidate solution
• Constraint (mathematics)
• Corner solution
• Global optimum and Local optimum
• Maxima and minima
• Slack variable
• Surplus 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
  • Interior point method
    • Karmarkar's algorithm
    • Mehrotra predictor-corrector method
  • Delayed column generation
• Algorithms for integer programming:
  • Branch and bound
  • Cutting-plane method
• Linear complementarity problem
• Fourier–Motzkin elimination

Nonlinear programming

Nonlinear programming — the most general optimization problem in the usual framework

• Special cases of nonlinear programming:
  • Quadratic programming
    • Linear least squares
    • Frank–Wolfe algorithm
    • Bilinear program
  • Convex optimization
    • Linear matrix inequality
    • Conic optimization
      • Semidefinite programming
      • Second order cone programming
      • Quadratic programming (see above)
    • Subgradient method — extension of steepest descent for problems with a nondifferentiable objective function
  • Geometric programming — problems involving posynomials
    • Posynomial — similar to polynomials, but coefficients have to be positive while exponents need not be integers
  • Quadratically constrained quadratic program
• General algorithms:
  • Concepts:
    • Descent direction
    • Linesearch
      • Backtracking linesearch
      • Wolfe conditions
  • Gradient descent
    • Stochastic gradient descent
  • Successive linear programming (SLP) — replace problem by a linear 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
  • Nelder-Mead method
  • Golden section search
  • Ternary search
  • Tabu search
  • Least absolute deviations
  • Nonlinear least squares
    • Gauss-Newton algorithm
    • Levenberg-Marquardt algorithm
    • Expectation-maximization algorithm
      • Osem (mathematics) — ordered subset expectation maximization
  • Nearest neighbor search
• Mixed complementarity problem

Uncertainty and randomness

• Approaches to deal with uncertainty:
  • Robust optimization
  • Stochastic approximation
  • Stochastic optimization
  • Stochastic programming
  • Stochastic gradient descent
• Random optimization algorithms:
  • Simulated annealing
    • Adaptive simulated annealing — variant in which the algorithm parameters are adjusted during the computation.
    • Great Deluge algorithm
  • Evolutionary algorithm, Evolution strategy
    • Differential evolution
    • Evolutionary programming
    • Evolution window
    • Genetic algorithm, Genetic programming
      • Genetic algorithm in economics
      • Speciation (genetic algorithm)
    • Genetic representation
    • Gaussian adaptation
  • Memetic algorithm
  • Particle swarm optimization
  • Cooperative optimization
    • Repulsive particle swarm optimization
  • Stochastic tunneling
  • Harmony search - mimicks the improvisation process of musicians
  • see also the section Monte Carlo method

Theoretical aspects

• Convex analysis
  • Quasiconvex function
  • Subderivative
• Duality:
  • Dual problem, Shadow price
  • Dual cone and polar cone
  • Lagrange duality
• Farkas' lemma
• Karush-Kuhn-Tucker conditions
• Lagrange multipliers
  • Lagrange multipliers on Banach spaces
• Danskin's theorem — used in the analysis of minimax problems
• No free lunch in search and optimization
• Relaxation technique (mathematics)
  • Lagrangian relaxation
  • LP relaxation

Applications

• Automatic label placement
• Cutting stock problem
• Demand Optimization
• Entropy maximization
• Expenditure minimization problem
• Inventory control problem
  • Newsvendor
  • Extended Newsvendor models
• Job-shop problem
• Multidisciplinary design optimization
• Paper bag problem
• Process optimization
• Stigler diet
• Stress majorization
• Trajectory optimization
• Utility maximization problem
• Wing shape optimization

Miscellaneous

• Combinatorial optimization
• Dynamic programming
  • Bellman equation
• Global optimization:
  • BRST algorithm
  • MCS algorithm
• Infinite-dimensional optimization
  • Optimal control
    • Pontryagin's minimum principle — infinite-dimensional version of Lagrange multipliers
  • Shape optimization — optimization over a set of regions
    • Topological derivative — derivative with respect to changing in the shape
• Optimal substructure
• Algorithmic concepts:
  • Barrier function
  • Penalty method
  • Trust region
• Famous test functions for optimization:
  • Rosenbrock function — two-dimensional function with a banana-shaped valley
  • Shekel function — multimodal and multidimensional
• Mathematical Programming Society

Numerical quadrature

Numerical integration — the numerical evaluation of an integral

• Rectangle method
• Trapezium rule
• Simpson's rule
  • Adaptive Simpson's method
• Newton-Cotes formulas
• Romberg's method - Richardson extrapolation applied to Trapezium rule
• Gaussian quadrature - highest possible degree with given number of points
• 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
• T-integration — a non-standard method
• Lebedev grid — grid on a sphere with octahedral symmetry
• Sparse grid
• Numerical differentiation
• Euler-Maclaurin formula

Numerical ordinary differential equations

Numerical 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
• 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
  • Cash-Karp — a fifth-order method with six stages and an embedded fourth-order method
  • Dormand-Prince — another fifth-order method with six stages and an embedded fourth-order method
  • Runge–Kutta–Fehlberg method — another fifth-order method with six stages and an embedded fourth-order method
• Linear multistep method — the other main class of methods for initial-value problems
• 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
  • Beeman's algorithm — a two-step method extending the Verlet method
• 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
    • Euler–Cromer algorithm — variant of Euler method which is symplectic when applied to separable Hamiltonians
• Adaptive stepsize — automatically changing the step size when that seems advantageous
• Stiff equation — roughly, an ODE for which the unstable methods needs a very short step size, but stable methods do not.
• Methods for solving two-point boundary value problems (BVPs):
  • Shooting method
• Methods for solving differential-algebraic equations (DAEs), i.e., ODEs with constraints:
  • Constraint algorithm — for solving Newton's equations with constraints
• 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
• Bi-directional delay line
• History of numerical solution of differential equations using computers

Numerical partial differential equations

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

• Methods for the solution of PDEs:
  • Finite difference method — based on approximating differential operators with difference operators
    • Finite difference — the discrete analogue of a differential operator
      • Difference operator — the numerator of a finite difference
      • Discrete Laplace operator — finite-difference approximation of the Laplace operator
      • Discrete Poisson equation — discrete analogue of the Poisson equation using the discrete Laplace operator
      • Five-point stencil — standard finite-difference approximation of the Laplace operator in two dimensions
    • Crank-Nicolson method — second-order method for heat and related PDEs
    • Lax-Wendroff method — second-order method for hyperbolic PDEs
    • Finite-difference time-domain method — a finite-difference method for electrodynamics
  • Finite element method, finite element analysis — based on a discretization of the space of solutions
    • 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
    • Rayleigh-Ritz method — a finite element method based on variational principles
    • Spectral element method — high-order finite element methods
    • Direct stiffness method — a particular implementation of the finite element method, often used in structural analysis
    • Trefftz method
    • 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
  • 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 — based on transforming the PDE to an integral equation on the boundary of the domain
  • 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
  • Discrete element method — a method in which the elements can move freely relative to each other
  • Meshfree methods — does not use a mesh, but uses a particle view of the field
    • Diffuse element method —
  • Uniform theory of diffraction — specifically designed for scattering problems in electromagnetics
  • Particle-in-cell — used especially in fluid dynamics
  • High-resolution scheme
  • Split-step method
  • Fast marching method
• Techniques for improving these methods:
  • Multigrid, multigrid method — uses a hierarchy of nested meshes to speed up the methods
  • Domain decomposition method — divides the domain in a few subdomains and solves the PDE on these subdomains
  • Adaptive mesh refinement — uses the computed solution to refine the mesh only where necessary
  • Fast Multipole Method — hierarchical method for evaluating particle-particle interactions
• 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
• 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
  • Numerical diffusion — diffusion introduced by the numerical method, above to that which is naturally present
  • 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
• Boundary conditions:
  • Perfectly matched layer
  • Uniaxial perfectly matched layer
• Meshing and gridding techniques for finite differences, finite elements, finite volumes, and boundary elements
  • Geodesic grid — forming isotropic grids/meshes on a sphere

Monte Carlo method

• Variants of the Monte Carlo method:
  • Direct simulation Monte Carlo
  • Quasi-Monte Carlo method
  • Markov chain Monte Carlo
    • Metropolis-Hastings algorithm
    • Gibbs sampling
  • Dynamic Monte Carlo method
    • Kinetic Monte Carlo
    • Gillespie algorithm
  • Particle filter
  • Reverse Monte Carlo
• Sampling methods:
  • Inverse transform sampling — general and straightforward method but computationally expensive
  • Rejection sampling — sample from a simpler distribution but reject some of the samples
  • For sampling from a normal distribution:
    • Box-Muller transform
    • Marsaglia polar method
• Low-discrepancy sequence
  • Constructions of low-discrepancy sequences
  • Illustration of a low-discrepancy sequence
• Event generator
• Parallel tempering
• Umbrella sampling — improves sampling in physical systems with significant energy barriers
• Variance reduction techniques:
  • Control variate
  • Importance sampling
  • Stratified sampling
  • VEGAS algorithm
• Applications:
  • Metropolis light transport
  • Monte Carlo methods in finance
    • Monte Carlo option model
  • Monte Carlo project — research project modelling of human exposure to food chemicals and nutrients
  • 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
  • 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

• Climate model
• Numerical weather prediction
  • Geodesic grid
• Computational chemistry
  • Cell lists
  • Coupled cluster
  • Density functional theory
  • Self-consistent field method
• Computational electromagnetics
• Computational fluid dynamics
  • Large eddy simulation
  • Smoothed particle hydrodynamics
  • Acoustic analogy — used in numerical aeroacoustics to reduce sound sources to simple emitter types
• Computational physics
• Celestial Mechanics
  • Numerical model of solar system

Software

For software, see the list of numerical analysis software.

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