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| This is a '''list of [[numerical analysis]] topics'''.
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| ==General==
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| *[[Iterative method]]
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| *[[Rate of convergence]] — the speed at which a convergent sequence approaches its limit
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| **[[Order of accuracy]] — rate at which numerical solution of differential equation converges to exact solution
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| *[[Series acceleration]] — methods to accelerate the speed of convergence of a series
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| **[[Aitken's delta-squared process]] — most useful for linearly converging sequences
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| **[[Minimum polynomial extrapolation]] — for vector sequences
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| **[[Richardson extrapolation]]
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| **[[Shanks transformation]] — similar to Aitken's delta-squared process, but applied to the partial sums
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| **[[Van Wijngaarden transformation]] — for accelerating the convergence of an alternating series
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| *[[Abramowitz and Stegun]] — book containing formulas and tables of many special functions
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| **[[Digital Library of Mathematical Functions]] — successor of book by Abramowitz and Stegun
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| *[[Curse of dimensionality]]
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| *[[Local convergence]] and global convergence — whether you need a good initial guess to get convergence
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| *[[Superconvergence]]
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| *[[Discretization]]
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| *[[Difference quotient]]
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| *Complexity:
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| **[[Computational complexity of mathematical operations]]
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| **[[Smoothed analysis]] — measuring the expected performance of algorithms under slight random perturbations of worst-case inputs
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| *[[Symbolic-numeric computation]] — combination of symbolic and numeric methods
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| *Cultural and historical aspects:
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| **[[History of numerical solution of differential equations using computers]]
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| **[[Hundred-dollar, Hundred-digit Challenge problems]] — list of ten problems proposed by Nick Trefethen in 2002
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| **[[International Workshops on Lattice QCD and Numerical Analysis]]
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| **[[Timeline of numerical analysis after 1945]]
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| *General classes of methods:
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| **[[Collocation method]] — discretizes a continuous equation by requiring it only to hold at certain points
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| **[[Level set method]]
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| ***[[Level set (data structures)]] — data structures for representing level sets
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| **[[Sinc numerical methods]] — methods based on the sinc function, sinc(''x'') = sin(''x'') / ''x''
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| **[[ABS methods]]
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| ==Error==
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| [[Error analysis]]
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| *[[Approximation]]
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| *[[Approximation error]]
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| *[[Condition number]]
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| *[[Discretization error]]
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| *[[Floating point]] number
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| **[[Guard digit]] — extra precision introduced during a computation to reduce round-off error
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| **[[Truncation]] — rounding a floating-point number by discarding all digits after a certain digit
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| **[[Round-off error]]
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| ***[[Numeric precision in Microsoft Excel]]
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| **[[Arbitrary-precision arithmetic]]
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| *[[Interval arithmetic]] — represent every number by two floating-point numbers guaranteed to have the unknown number between them
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| **[[Interval contractor]] — maps interval to subinterval which still contains the unknown exact answer
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| **[[Interval propagation]] — contracting interval domains without removing any value consistent with the constraints
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| ***See also: [[Interval boundary element method]], [[Interval finite element]]
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| *[[Loss of significance]]
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| *[[Numerical error]]
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| *[[Numerical stability]]
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| *Error propagation:
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| **[[Propagation of uncertainty]]
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| ***[[List of uncertainty propagation software]]
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| **[[Significance arithmetic]]
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| ** [[Residual (numerical analysis)]]
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| *[[Relative change and difference]] — the relative difference between ''x'' and ''y'' is |''x'' − ''y''| / max(|''x''|, |''y''|)
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| *[[Significant figures]]
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| **[[False precision]] — giving more significant figures than appropriate
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| *[[Truncation error]] — error committed by doing only a finite numbers of steps
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| *[[Well-posed problem]]
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| *[[Affine arithmetic]]
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| ==Elementary and special functions==
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| *Summation:
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| **[[Kahan summation algorithm]]
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| **[[Pairwise summation]] — slightly worse than Kahan summation but cheaper
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| **[[Binary splitting]]
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| *Multiplication:
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| **[[Multiplication algorithm]] — general discussion, simple methods
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| **[[Karatsuba algorithm]] — the first algorithm which is faster than straightforward multiplication
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| **[[Toom–Cook multiplication]] — generalization of Karatsuba multiplication
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| **[[Schönhage–Strassen algorithm]] — based on Fourier transform, asymptotically very fast
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| **[[Fürer's algorithm]] — asymptotically slightly faster than Schönhage–Strassen
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| *[[Division algorithm]] — for computing quotient and remainder of two numbers
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| *Exponentiation:
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| **[[Exponentiation by squaring]]
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| **[[Addition-chain exponentiation]]
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| *Polynomials:
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| **[[Horner's method]]
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| **[[Estrin's scheme]] — modification of the Horner scheme with more possibilities for parallellization
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| **[[Clenshaw algorithm]]
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| **[[De Casteljau's algorithm]]
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| *Square roots and other roots:
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| **[[Integer square root]]
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| **[[Methods of computing square roots]]
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| **[[Nth root algorithm|''n''th root algorithm]]
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| **[[Shifting nth root algorithm|Shifting ''n''th root algorithm]] — similar to long division
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| **[[hypot]] — the function (''x''<sup>2</sup> + ''y''<sup>2</sup>)<sup>1/2</sup>
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| **[[Alpha max plus beta min algorithm]] — approximates hypot(x,y)
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| **[[Fast inverse square root]] — calculates 1 / √''x'' using details of the IEEE floating-point system
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| *Elementary functions (exponential, logarithm, trigonometric functions):
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| **[[Trigonometric tables]] — different methods for generating them
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| **[[CORDIC]] — shift-and-add algorithm using a table of arc tangents
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| **[[BKM algorithm]] — shift-and-add algorithm using a table of logarithms and complex numbers
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| *Gamma function:
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| **[[Lanczos approximation]]
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| **[[Spouge's approximation]] — modification of Stirling's approximation; easier to apply than Lanczos
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| *[[AGM method]] — computes arithmetic–geometric mean; related methods compute special functions
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| *[[FEE method]] (Fast E-function Evaluation) — fast summation of series like the power series for e<sup>''x''</sup>
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| *[[Gal's accurate tables]] — table of function values with unequal spacing to reduce round-off error
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| *[[Spigot algorithm]] — algorithms that can compute individual digits of a real number
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| *[[Approximations of π|Approximations of {{pi}}]]:
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| **[[Liu Hui's π algorithm]] — first algorithm that can compute π to arbitrary precision
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| **[[Leibniz formula for π]] — alternating series with very slow convergence
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| **[[Wallis product]] — infinite product converging slowly to π/2
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| **[[Viète's formula]] — more complicated infinite product which converges faster
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| **[[Gauss–Legendre algorithm]] — iteration which converges quadratically to π, based on arithmetic–geometric mean
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| **[[Borwein's algorithm]] — iteration which converges quartically to 1/π, and other algorithms
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| **[[Chudnovsky algorithm]] — fast algorithm that calculates a hypergeometric series
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| **[[Bailey–Borwein–Plouffe formula]] — can be used to compute individual hexadecimal digits of π
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| **[[Bellard's formula]] — faster version of Bailey–Borwein–Plouffe formula
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| **[[List of formulae involving π]]
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| ==Numerical linear algebra==
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| [[Numerical linear algebra]] — study of numerical algorithms for linear algebra problems
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| ===Basic concepts===
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| *Types of matrices appearing in numerical analysis:
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| **[[Sparse matrix]]
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| ***[[Band matrix]]
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| ***[[Bidiagonal matrix]]
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| ***[[Tridiagonal matrix]]
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| ***[[Pentadiagonal matrix]]
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| ***[[Skyline matrix]]
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| **[[Circulant matrix]]
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| **[[Triangular matrix]]
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| **[[Diagonally dominant matrix]]
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| **[[Block matrix]] — matrix composed of smaller matrices
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| **[[Stieltjes matrix]] — symmetric positive definite with non-positive off-diagonal entries
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| **[[Hilbert matrix]] — example of a matrix which is extremely ill-conditioned (and thus difficult to handle)
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| **[[Wilkinson matrix]] — example of a symmetric tridiagonal matrix with pairs of nearly, but not exactly, equal eigenvalues
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| **[[Convergent matrix]] – square matrix whose successive powers approach the zero matrix
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| *Algorithms for matrix multiplication:
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| **[[Strassen algorithm]]
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| **[[Coppersmith–Winograd algorithm]]
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| **[[Cannon's algorithm]] — a distributed algorithm, especially suitable for processors laid out in a 2d grid
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| **[[Freivalds' algorithm]] — a randomized algorithm for checking the result of a multiplication
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| *[[Matrix decomposition]]s:
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| **[[LU decomposition]] — lower triangular times upper triangular
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| **[[QR decomposition]] — orthogonal matrix times triangular matrix
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| ***[[RRQR factorization]] — rank-revealing QR factorization, can be used to compute rank of a matrix
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| **[[Polar decomposition]] — unitary matrix times positive-semidefinite Hermitian matrix
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| **Decompositions by similarity:
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| ***[[Eigendecomposition of a matrix|Eigendecomposition]] — decomposition in terms of eigenvectors and eigenvalues
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| ***[[Jordan normal form]] — bidiagonal matrix of a certain form; generalizes the eigendecomposition
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| ****[[Weyr canonical form]] — permutation of Jordan normal form
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| ***[[Jordan–Chevalley decomposition]] — sum of commuting nilpotent matrix and diagonalizable matrix
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| ***[[Schur decomposition]] — similarity transform bringing the matrix to a triangular matrix
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| **[[Singular value decomposition]] — unitary matrix times diagonal matrix times unitary matrix
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| *[[Matrix splitting]] – expressing a given matrix as a sum or difference of matrices
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| ===Solving systems of linear equations===
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| *[[Gaussian elimination]]
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| **[[Row echelon form]] — matrix in which all entries below a nonzero entry are zero
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| **[[Bareiss algorithm]] — variant which ensures that all entries remain integers if the initial matrix has integer entries
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| **[[Tridiagonal matrix algorithm]] — simplified form of Gaussian elimination for tridiagonal matrices
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| *[[LU decomposition]] — write a matrix as a product of an upper- and a lower-triangular matrix
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| **[[Crout matrix decomposition]]
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| **[[LU reduction]] — a special parallelized version of a LU decomposition algorithm
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| *[[Block LU decomposition]]
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| *[[Cholesky decomposition]] — for solving a system with a positive definite matrix
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| **[[Minimum degree algorithm]]
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| **[[Symbolic Cholesky decomposition]]
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| *[[Iterative refinement]] — procedure to turn an inaccurate solution in a more accurate one
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| *Direct methods for sparse matrices:
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| **[[Frontal solver]] — used in finite element methods
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| **[[Nested dissection]] — for symmetric matrices, based on graph partitioning
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| *[[Levinson recursion]] — for Toeplitz matrices
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| *[[SPIKE algorithm]] — hybrid parallel solver for narrow-banded matrices
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| *[[Cyclic reduction]] — eliminate even or odd rows or columns, repeat
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| *Iterative methods:
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| **[[Jacobi method]]
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| **[[Gauss–Seidel method]]
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| ***[[Successive over-relaxation]] (SOR) — a technique to accelerate the Gauss–Seidel method
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| ****[[Symmetric successive overrelaxation]] (SSOR) — variant of SOR for symmetric matrices
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| ***[[Backfitting algorithm]] — iterative procedure used to fit a generalized additive model, often equivalent to Gauss–Seidel
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| **[[Modified Richardson iteration]]
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| **[[Conjugate gradient method]] (CG) — assumes that the matrix is positive definite
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| ***[[Derivation of the conjugate gradient method]]
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| ***[[Nonlinear conjugate gradient method]] — generalization for nonlinear optimization problems
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| **[[Biconjugate gradient method]] (BiCG)
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| ***[[Biconjugate gradient stabilized method]] (BiCGSTAB) — variant of BiCG with better convergence
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| **[[Conjugate residual method]] — similar to CG but only assumed that the matrix is symmetric
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| **[[Generalized minimal residual method]] (GMRES) — based on the Arnoldi iteration
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| **[[Chebyshev iteration]] — avoids inner products but needs bounds on the spectrum
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| **[[Stone method|Stone's method]] (SIP – Srongly Implicit Procedure) — uses an incomplete LU decomposition
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| **[[Kaczmarz method]]
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| **[[Preconditioner]]
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| ***[[Incomplete Cholesky factorization]] — sparse approximation to the Cholesky factorization
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| ***[[Incomplete LU factorization]] — sparse approximation to the LU factorization
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| **[[Uzawa iteration]] — for saddle node problems
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| *Underdetermined and overdetermined systems (systems that have no or more than one solution):
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| **[[Kernel (matrix)#Numerical computation of null space|Numerical computation of null space]] — find all solutions of an underdetermined system
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| **[[Moore–Penrose pseudoinverse]] — for finding solution with smallest 2-norm (for underdetermined systems) or smallest residual
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| **[[Sparse approximation]] — for finding the sparsest solution (i.e., the solution with as many zeros as possible)
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| ===Eigenvalue algorithms===
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| [[Eigenvalue algorithm]] — a numerical algorithm for locating the eigenvalues of a matrix
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| *[[Power iteration]]
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| *[[Inverse iteration]]
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| *[[Rayleigh quotient iteration]]
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| *[[Arnoldi iteration]] — based on Krylov subspaces
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| *[[Lanczos algorithm]] — Arnoldi, specialized for positive-definite matrices
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| **[[Block Lanczos algorithm]] — for when matrix is over a finite field
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| *[[QR algorithm]]
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| *[[Jacobi eigenvalue algorithm]] — select a small submatrix which can be diagonalized exactly, and repeat
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| **[[Jacobi rotation]] — the building block, almost a Givens rotation
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| **[[Jacobi method for complex Hermitian matrices]]
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| *[[Divide-and-conquer eigenvalue algorithm]]
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| *[[Folded spectrum method]]
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| *[[LOBPCG]] — Locally Optimal Block Preconditioned Conjugate Gradient Method
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| *[[Eigenvalue perturbation]] — stability of eigenvalues under perturbations of the matrix
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| ===Other concepts and algorithms===
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| *[[Orthogonalization]] algorithms:
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| **[[Gram–Schmidt process]]
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| **[[Householder transformation]]
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| ***[[Householder operator]] — analogue of Householder transformation for general inner product spaces
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| **[[Givens rotation]]
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| *[[Krylov subspace]]
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| *[[Block matrix pseudoinverse]]
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| *[[Bidiagonalization]]
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| *[[Cuthill–McKee algorithm]] — permutes rows/columns in sparse matrix to yield a narrow band matrix
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| *[[In-place matrix transposition]] — computing the transpose of a matrix without using much additional storage
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| *[[Pivot element]] — entry in a matrix on which the algorithm concentrates
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| *[[Matrix-free methods]] — methods that only access the matrix by evaluating matrix-vector products
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| ==Interpolation and approximation==
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| [[Interpolation]] — construct a function going through some given data points
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| *[[Nearest-neighbor interpolation]] — takes the value of the nearest neighbor
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| ===Polynomial interpolation===
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| [[Polynomial interpolation]] — interpolation by polynomials
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| *[[Linear interpolation]]
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| *[[Runge's phenomenon]]
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| *[[Vandermonde matrix]]
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| *[[Chebyshev polynomials]]
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| *[[Chebyshev nodes]]
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| *[[Lebesgue constant (interpolation)]]
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| *Different forms for the interpolant:
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| **[[Newton polynomial]]
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| ***[[Divided differences]]
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| ***[[Neville's algorithm]] — for evaluating the interpolant; based on the Newton form
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| **[[Lagrange polynomial]]
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| **[[Bernstein polynomial]] — especially useful for approximation
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| **[[Brahmagupta's interpolation formula]] — seventh-century formula for quadratic interpolation
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| *Extensions to multiple dimensions:
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| **[[Bilinear interpolation]]
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| **[[Trilinear interpolation]]
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| **[[Bicubic interpolation]]
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| **[[Tricubic interpolation]]
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| **[[Padua points]] — set of points in '''R'''<sup>2</sup> with unique polynomial interpolant and minimal growth of Lebesgue constant
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| *[[Hermite interpolation]]
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| *[[Birkhoff interpolation]]
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| *[[Abel–Goncharov interpolation]]
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| ===Spline interpolation===
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| [[Spline interpolation]] — interpolation by piecewise polynomials
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| *[[Spline (mathematics)]] — the piecewise polynomials used as interpolants
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| *[[Perfect spline]] — polynomial spline of degree ''m'' whose ''m''th derivate is ±1
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| *[[Cubic Hermite spline]]
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| **[[Centripetal Catmull–Rom spline]] — special case of cubic Hermite splines without self-intersections or cusps
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| *[[Monotone cubic interpolation]]
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| *[[Hermite spline]]
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| *[[Bézier spline]]
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| **[[Bézier curve]]
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| **[[De Casteljau's algorithm]]
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| **Generalizations to more dimensions:
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| ***[[Bézier triangle]] — maps a triangle to '''R'''<sup>3</sup>
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| ***[[Bézier surface]] — maps a square to '''R'''<sup>3</sup>
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| *[[B-spline]]
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| **[[Box spline]] — multivariate generalization of B-splines
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| **[[Truncated power function]]
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| **[[De Boor's algorithm]] — generalizes De Casteljau's algorithm
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| *[[Non-uniform rational B-spline]] (NURBS)
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| **[[T-spline]] — can be thought of as a NURBS surface for which a row of control points is allowed to terminate
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| *[[Kochanek–Bartels spline]]
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| *[[Coons patch]] — type of manifold parametrization used to smoothly join other surfaces together
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| *[[M-spline]] — a non-negative spline
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| *[[I-spline]] — a monotone spline, defined in terms of M-splines
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| *[[Smoothing spline]] — a spline fitted smoothly to noisy data
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| *[[Blossom (functional)]] — a unique, affine, symmetric map associated to a polynomial or spline
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| *See also: [[List of numerical computational geometry topics]]
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| ===Trigonometric interpolation===
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| [[Trigonometric interpolation]] — interpolation by trigonometric polynomials
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| *[[Discrete Fourier transform]] — can be viewed as trigonometric interpolation at equidistant points
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| **[[Relations between Fourier transforms and Fourier series]]
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| *[[Fast Fourier transform]] (FFT) — a fast method for computing the discrete Fourier transform
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| **[[Bluestein's FFT algorithm]]
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| **[[Bruun's FFT algorithm]]
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| **[[Cooley–Tukey FFT algorithm]]
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| **[[Split-radix FFT algorithm]] — variant of Cooley–Tukey that uses a blend of radices 2 and 4
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| **[[Goertzel algorithm]]
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| **[[Prime-factor FFT algorithm]]
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| **[[Rader's FFT algorithm]]
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| **[[Bit-reversal permutation]] — particular permutation of vectors with 2<sup>''m''</sup> entries used in many FFTs.
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| **[[Butterfly diagram]]
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| **[[Twiddle factor]] — the trigonometric constant coefficients that are multiplied by the data
| |
| **[[Cyclotomic fast Fourier transform]] — for FFT over finite fields
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| **Methods for computing discrete convolutions with finite impulse response filters using the FFT:
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| ***[[Overlap–add method]]
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| ***[[Overlap–save method]]
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| *[[Sigma approximation]]
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| *[[Dirichlet kernel]] — convolving any function with the Dirichlet kernel yields its trigonometric interpolant
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| *[[Gibbs phenomenon]]
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| ===Other interpolants===
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| *[[Simple rational approximation]]
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| **[[Polynomial and rational function modeling]] — comparison of polynomial and rational interpolation
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| *[[Wavelet]]
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| **[[Continuous wavelet]]
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| **[[Transfer matrix]]
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| **See also: [[List of functional analysis topics]], [[List of wavelet-related transforms]]
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| *[[Inverse distance weighting]]
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| *[[Radial basis function]] (RBF) — a function of the form ƒ(''x'') = ''φ''(|''x''−''x''<sub>0</sub>|)
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| **[[Polyharmonic spline]] — a commonly used radial basis function
| |
| **[[Thin plate spline]] — a specific polyharmonic spline: ''r''<sup>2</sup> log ''r''
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| **[[Hierarchical RBF]]
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| *[[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
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| **[[Nearest neighbor value interpolation]]
| |
| **[[PDE surface]]
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| **[[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''
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| | |
| ===Approximation theory=== | |
| [[Approximation theory]]
| |
| *[[Orders of approximation]]
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| *[[Lebesgue's lemma]]
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| *[[Curve fitting]]
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| **[[Vector field reconstruction]]
| |
| *[[Modulus of continuity]] — measures smoothness of a function
| |
| *[[Least squares (function approximation)]] — minimizes the error in the L<sup>2</sup>-norm
| |
| *[[Minimax approximation algorithm]] — minimizes the maximum error over an interval (the L<sup>∞</sup>-norm)
| |
| **[[Equioscillation theorem]] — characterizes the best approximation in the L<sup>∞</sup>-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<sup>∞</sup>-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<sup>p</sup> 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<sup>−''n''</sup> ''x''<sup>''k''</sup> 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<sup>2</sup> 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''<nowiki>'</nowiki> and ''f''<nowiki>''</nowiki>; 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 parameters 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<sub>1</sub>-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 Methods]] — 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''<sup>T</sup> ''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 weigted 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 [[#Finding roots of nonlinear equations|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 contrained 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]]
| |
| **[[Adaptive projected subgradient method]]
| |
| **[[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]]
| |
| **[[Probabilistic-based design optimization]]
| |
| **[[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 (mathematics)|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 <math>f(x, y) = (x^2+y-11)^2 + (x+y^2-7)^2</math>
| |
| **[[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 {{nowrap|(1 − ''x''<sup>2</sub>)<sup>±1/2</sup>}} on [−1, 1]
| |
| **[[Gauss–Hermite quadrature]] — extension of Gaussian quadrature for integrals with weight exp(−''x''<sup>2</sub>) on [−∞, ∞]
| |
| **[[Gauss–Jacobi quadrature]] — extension of Gaussian quadrature for integrals with weight (1 − ''x'')<sup>''α''</sup> (1 + ''x'')<sup>''β''</sup> 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
| |
| **[[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 (differential equations)|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 <math>y'' = f(t,y)</math>
| |
| **[[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
| |
| **[[Dynamic errors of numerical methods of ODE discretization]] — logarithm of stability function
| |
| *[[Adaptive stepsize]] — automatically changing the step size when that seems advantageous
| |
| | |
| ==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===
| |
| [[Finite element method]] — 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
| |
| **[[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 elemets:
| |
| **[[Bilinear quadrilateral element]] — also known as the Q4 element
| |
| **[[Constant strain triangle element]] (CST) — also known as the T3 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 element]]s — 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
| |
| *[[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]]
| |
| *[[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:
| |
| **[[Mimesis (mathematics)|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
| |
| **[[BDDC|Balancing domain decomposition by constraints]] (BDDC) — further development of BDD
| |
| **[[FETI|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 (numerical analysis)|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 vertiсes 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 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]]
| |
| *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 software, see the [[list of numerical analysis software]].
| |
| | |
| [[Category:Numerical analysis|*Topics]]
| |
| [[Category:Mathematics-related lists|Numerical analysis topics]]
| |
| [[Category:Outlines]]
| |