Template:Numerical analysis topics

v t e Numerical analysis
List of numerical analysis topics • Scientific computing • Applied mathematics • Computational science
General	Validated numerics • Iterative method • Rate of convergence • Order of accuracy • Series acceleration (Aitken Δ² • Shanks transformation • Richardson extrapolation • Van Wijngaarden transformation) • Curse of dimensionality • Local convergence / Global convergence • Superconvergence • Discretization • Difference quotient • Complexity • Smoothed analysis • Symbolic-numeric computation
Error	Error analysis • Approximation error • Catastrophic cancellation • Condition number • Discretization error • Floating point • Guard digit • Truncation • Round-off error • Arbitrary-precision arithmetic • Interval arithmetic / Affine arithmetic • Loss of significance • Numerical stability • Propagation of uncertainty • Residual • Significant figures • Sterbenz lemma • Truncation error • Well-posed problem
Elementary & special functions	Summation: Kahan summation algorithm • Pairwise summation • Binary splitting • 2Sum Multiplication & division: Karatsuba algorithm • Toom–Cook multiplication • Schönhage–Strassen algorithm • Fürer's algorithm • Division algorithm • SRT division • Newton–Raphson division • Goldschmidt division Exponentiation & inverse: Exponentiation by squaring • Addition-chain exponentiation • Newton's method Polynomials: Horner's method • Estrin's scheme • Clenshaw algorithm • De Casteljau's algorithm Roots: Square roots • Fast inverse square root Elementary functions: CORDIC • BKM algorithm Gamma & AGM: Lanczos approximation • Spouge's approximation • AGM π algorithms: Gauss–Legendre algorithm • Borwein's algorithm • Chudnovsky algorithm • Bailey–Borwein–Plouffe formula • Bellard's formula
Numerical linear algebra	Matrix types: Sparse matrix • Band matrix • Tridiagonal matrix • Hilbert matrix • Wilkinson matrix Matrix mult.: Strassen algorithm • Coppersmith–Winograd algorithm • Freivalds' algorithm Decompositions: LU decomposition • QR decomposition • RRQR factorization • Polar decomposition • Eigendecomposition • Jordan normal form • Schur decomposition • Singular value decomposition Linear systems: Gaussian elimination • Tridiagonal matrix algorithm • Cholesky decomposition • Iterative refinement • Frontal solver • Nested dissection Iterative methods: Jacobi method • Gauss–Seidel method • SOR • GMRES • Conjugate gradient method • BiCGSTAB • Chebyshev iteration • Kaczmarz method • Preconditioner (ILU • Incomplete Cholesky factorization) Eigenvalues: Power iteration • QR algorithm • Arnoldi iteration • Lanczos algorithm • Divide-and-conquer
Interpolation & approximation	Interpolation: Polynomial interpolation (Newton polynomial • Divided differences • Lagrange polynomial) • Runge's phenomenon • Chebyshev nodes • Neville's algorithm • Spline interpolation (Cubic Hermite spline • Bézier curve • B-spline • NURBS • De Boor's algorithm) • Trigonometric interpolation / Discrete Fourier transform / Fast Fourier transform Other: Radial basis function • Lanczos resampling • Transfinite interpolation Approximation theory: Least squares • Minimax approximation / Remez algorithm • Stone–Weierstrass theorem • Padé approximant • Wavelet
Root finding (nonlinear)	Bisection method • Fixed-point iteration • Newton's method • Secant method • False position method • Muller's method • Brent's method • Halley's method • Householder's method • Polynomials: Jenkins–Traub algorithm • Laguerre's method • Durand–Kerner method • Aberth method
Optimization	Linear & convex: Linear programming (Simplex algorithm • Interior-point method) • Quadratic programming • Semidefinite programming • Second-order cone programming • Convex optimization Nonlinear: Nonlinear programming • Sequential quadratic programming • Trust region • Levenberg–Marquardt algorithm Gradient-based: Gradient descent • Stochastic gradient descent • Nonlinear CG • Proximal gradient Derivative-free: Nelder–Mead method • Pattern search • Coordinate descent Global & stochastic: Simulated annealing • Genetic algorithm • Bayesian optimization • Particle swarm optimization • Tabu search Theory: Duality • KKT • Farkas' lemma
Numerical quadrature (integration)	Numerical integration • Rectangle method • Trapezoidal rule • Simpson's rule • Newton–Cotes formulas • Romberg's method • Gaussian quadrature (Legendre • Hermite • Laguerre • Jacobi • Chebyshev) • Clenshaw–Curtis quadrature • Gauss–Kronrod • Tanh-sinh quadrature • Adaptive quadrature • Monte Carlo integration
ODE methods	Numerical methods for ordinary differential equations • Euler method • Backward Euler method • Trapezoidal rule • Runge–Kutta methods (Dormand–Prince method • Runge–Kutta–Fehlberg method) • Linear multistep method / Backward differentiation formula • Predictor–corrector method • Exponential integrator • SDEs: Euler–Maruyama method • Milstein method • DAEs: Pantelides algorithm • Analysis: Stiff equation • Adaptive stepsize • L-stability
PDE methods	Numerical partial differential equations • Finite difference: Finite difference method • Crank–Nicolson method • Lax–Wendroff method • Upwind scheme • ADI method Finite element: Finite element method • Galerkin method • Discontinuous Galerkin method • Rayleigh–Ritz method • Spectral element method • Other: Spectral method • Method of lines • Boundary element method • Finite volume method (Godunov's scheme • MUSCL scheme) • Lattice Boltzmann methods Acceleration & stability: Multigrid method • Domain decomposition methods • Adaptive mesh refinement • Perfectly matched layer • Courant–Friedrichs–Lewy condition • Von Neumann stability analysis • Lax equivalence theorem
Monte Carlo method	Monte Carlo method • Quasi-Monte Carlo method • Markov chain Monte Carlo (Metropolis–Hastings algorithm • Gibbs sampling • Reversible-jump MCMC) • Kinetic Monte Carlo • Particle filter • Variance reduction: Antithetic variates • Control variates • Importance sampling • Stratified sampling • Low-discrepancy sequence
Applications • Software • Journals • Researchers	Applications: Computational physics • Computational fluid dynamics • Numerical weather prediction • Computational chemistry • Computational statistics Software: List of numerical analysis software Journals: Acta Numerica • Mathematics of Computation • Journal of Computational and Applied Mathematics • BIT • Numerische Mathematik • SIAM Journal on Numerical Analysis • SIAM Journal on Scientific Computing Researchers: Cleve Moler • Gene H. Golub • James H. Wilkinson • Margaret H. Wright • Nicholas Higham • Nick Trefethen • Peter Lax • Richard S. Varga • Ulrich W. Kulisch • Vladik Kreinovich • Chi-Wang Shu
Numerical linear algebra • Numerical differential equations • Numerical optimization • Scientific visualization

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