Weakened weak form
Weakened weak form (or W2 form) is used in the formulation of general numerical methods based on meshfree methods and/or finite element method settings. These numerical methods are applicable to solid
Multigrid method
In numerical analysis, a multigrid method (MG method) is an algorithm for solving differential equations using a hierarchy of discretizations. They are an example of a class of techniques called multi
Wilkinson's polynomial
In numerical analysis, Wilkinson's polynomial is a specific polynomial which was used by James H. Wilkinson in 1963 to illustrate a difficulty when finding the root of a polynomial: the location of th
Local linearization method
In numerical analysis, the local linearization (LL) method is a general strategy for designing numerical integrators for differential equations based on a local (piecewise) linearization of the given
Numerical error
In software engineering and mathematics, numerical error is the error in the numerical computations.
Numerical methods in fluid mechanics
Fluid motion is governed by the Navier–Stokes equations, a set of coupled and nonlinearpartial differential equations derived from the basic laws of conservation of mass, momentumand energy. The unkno
Order of approximation
In science, engineering, and other quantitative disciplines, order of approximation refers to formal or informal expressions for how accurate an approximation is.
Generalized Gauss–Newton method
The generalized Gauss–Newton method is a generalization of the least-squares method originally described by Carl Friedrich Gauss and of Newton's method due to Isaac Newton to the case of constrained n
Unisolvent functions
In mathematics, a set of n functions f1, f2, ..., fn is unisolvent (meaning "uniquely solvable") on a domain Ω if the vectors are linearly independent for any choice of n distinct points x1, x2 ... xn
Hundred-dollar, Hundred-digit Challenge problems
The Hundred-dollar, Hundred-digit Challenge problems are 10 problems in numerical mathematics published in 2002 by Nick Trefethen. A $100 prize was offered to whoever produced the most accurate soluti
De Boor's algorithm
In the mathematical subfield of numerical analysis de Boor's algorithm is a polynomial-time and numerically stable algorithm for evaluating spline curves in B-spline form. It is a generalization of de
Multilevel Monte Carlo method
Multilevel Monte Carlo (MLMC) methods in numerical analysis are algorithms for computing expectations that arise in stochastic simulations. Just as Monte Carlo methods, they rely on repeated random sa
Closest point method
The closest point method (CPM) is an embedding method for solving partial differential equations on surfaces. The closest point method uses standard numerical approaches such as finite differences, fi
Generalized-strain mesh-free formulation
The generalized-strain mesh-free (GSMF) formulation is a local meshfree method in the field of numerical analysis, completely integration free, working as a weighted-residual weak-form collocation. Th
Regge calculus
In general relativity, Regge calculus is a formalism for producing simplicial approximations of spacetimes that are solutions to the Einstein field equation. The calculus was introduced by the Italian
Discrete Fourier transform
In mathematics, the discrete Fourier transform (DFT) converts a finite sequence of equally-spaced samples of a function into a same-length sequence of equally-spaced samples of the discrete-time Fouri
Butcher group
In mathematics, the Butcher group, named after the New Zealand mathematician John C. Butcher by , is an infinite-dimensional Lie group first introduced in numerical analysis to study solutions of non-
Curse of dimensionality
The curse of dimensionality refers to various phenomena that arise when analyzing and organizing data in high-dimensional spaces that do not occur in low-dimensional settings such as the three-dimensi
Stiffness matrix
In the finite element method for the numerical solution of elliptic partial differential equations, the stiffness matrix is a matrix that represents the system of linear equations that must be solved
Vector field reconstruction
Vector field reconstruction is a method of creating a vector field from experimental or computer generated data, usually with the goal of finding a differential equation model of the system. A differe
Radial basis function
A radial basis function (RBF) is a real-valued function whose value depends only on the distance between the input and some fixed point, either the origin, so that , or some other fixed point , called
Newton–Krylov method
Newton–Krylov methods are numerical methods for solving non-linear problems using Krylov subspace linear solvers. The Newton method, when generalised to systems of multiple variables, includes the inv
Von Neumann stability analysis
In numerical analysis, von Neumann stability analysis (also known as Fourier stability analysis) is a procedure used to check the stability of finite difference schemes as applied to linear partial di
Condition number
In numerical analysis, the condition number of a function measures how much the output value of the function can change for a small change in the input argument. This is used to measure how sensitive
Aitken's delta-squared process
In numerical analysis, Aitken's delta-squared process or Aitken extrapolation is a series acceleration method, used for accelerating the rate of convergence of a sequence. It is named after Alexander
Simpson's rule
In numerical integration, Simpson's rules are several approximations for definite integrals, named after Thomas Simpson (1710–1761). The most basic of these rules, called Simpson's 1/3 rule, or just S
Numerical stability
In the mathematical subfield of numerical analysis, numerical stability is a generally desirable property of numerical algorithms. The precise definition of stability depends on the context. One is nu
Approximation
An approximation is anything that is intentionally similar but not exactly equal to something else.
Discrete wavelet transform
In numerical analysis and functional analysis, a discrete wavelet transform (DWT) is any wavelet transform for which the wavelets are discretely sampled. As with other wavelet transforms, a key advant
GetFEM++
GetFEM++ is a generic finite element C++ library with interfaces for Python, Matlab and Scilab. It aims at providing finite element methods and elementary matrix computations for solving linear and no
Composite methods for structural dynamics
Composite methods are an approach applied in structural dynamics and related fields. They combine various methods in each time step, in order to acquire the advantages of different methods. The existi
Multiphysics simulation
In computational modelling, multiphysics simulation (often shortened to simply "multiphysics") is defined as the simultaneous simulation of different aspects of a physical system or systems and the in
Newton fractal
The Newton fractal is a boundary set in the complex plane which is characterized by Newton's method applied to a fixed polynomial p(Z) ∈ ℂ[Z] or transcendental function. It is the Julia set of the mer
Mesh generation
Mesh generation is the practice of creating a mesh, a subdivision of a continuous geometric space into discrete geometric and topological cells.Often these cells form a simplicial complex.Usually the
Richardson extrapolation
In numerical analysis, Richardson extrapolation is a sequence acceleration method used to improve the rate of convergence of a sequence of estimates of some value . In essence, given the value of for
Uzawa iteration
In numerical mathematics, the Uzawa iteration is an algorithm for solving saddle point problems. It is named after Hirofumi Uzawa and was originally introduced in the context of concave programming.
Remez algorithm
The Remez algorithm or Remez exchange algorithm, published by Evgeny Yakovlevich Remez in 1934, is an iterative algorithm used to find simple approximations to functions, specifically, approximations
Sinc numerical methods
In numerical analysis and applied mathematics, sinc numerical methods are numerical techniques for finding approximate solutions of partial differential equations and integral equations based on the t
Meshfree methods
In the field of numerical analysis, meshfree methods are those that do not require connection between nodes of the simulation domain, i.e. a mesh, but are rather based on interaction of each node with
Discretization
In applied mathematics, discretization is the process of transferring continuous functions, models, variables, and equations into discrete counterparts. This process is usually carried out as a first
Artificial precision
In numerical mathematics, artificial precision is a source of error that occurs when a numerical value or semantic is expressed with more precision than was initially provided from measurement or user
Overlap–add method
In signal processing, the overlap–add method is an efficient way to evaluate the discrete convolution of a very long signal with a finite impulse response (FIR) filter : where h[m] = 0 for m outside t
Nyström method
In mathematics numerical analysis, the Nyström method or quadrature method seeks the numerical solution of an integral equation by replacing the integral with a representative weighted sum. The contin
Estrin's scheme
In numerical analysis, Estrin's scheme (after Gerald Estrin), also known as Estrin's method, is an algorithm for numerical evaluation of polynomials. Horner's method for evaluation of polynomials is o
Van Wijngaarden transformation
In mathematics and numerical analysis, the van Wijngaarden transformation is a variant on the Euler transform used to accelerate the convergence of an alternating series. One algorithm to compute Eule
Partial differential algebraic equation
In mathematics a partial differential algebraic equation (PDAE) set is an incomplete system of partial differential equations that is closed with a set of algebraic equations.
Piecewise linear continuation
Simplicial continuation, or piecewise linear continuation (Allgower and Georg), is a one-parameter continuation method which is well suited to small to medium embedding spaces. The algorithm has been
INTLAB
INTLAB (INTerval LABoratory) is an interval arithmetic library using MATLAB and GNU Octave, available in Windows and Linux, macOS. It was developed by S.M. Rump from Hamburg University of Technology.
Runge–Kutta–Fehlberg method
In mathematics, the Runge–Kutta–Fehlberg method (or Fehlberg method) is an algorithm in numerical analysis for the numerical solution of ordinary differential equations. It was developed by the German
Padé approximant
In mathematics, a Padé approximant is the "best" approximation of a function near a specific point by a rational function of given order. Under this technique, the approximant's power series agrees wi
Bi-directional delay line
In mathematics, a bi-directional delay line is a numerical analysis technique used in computer simulation for solving ordinary differential equations by converting them to hyperbolic equations. In thi
Truncation
In mathematics and computer science, truncation is limiting the number of digits right of the decimal point.
Iterative rational Krylov algorithm
The iterative rational Krylov algorithm (IRKA), is an iterative algorithm, useful for model order reduction (MOR) of single-input single-output (SISO) linear time-invariant dynamical systems. At each
Affine arithmetic
Affine arithmetic (AA) is a model for self-validated numerical analysis. In AA, the quantities of interest are represented as affine combinations (affine forms) of certain primitive variables, which s
Unrestricted algorithm
An unrestricted algorithm is an algorithm for the computation of a mathematical function that puts no restrictions on the range of the argument or on the precision that may be demanded in the result.
Anderson acceleration
In mathematics, Anderson acceleration, also called Anderson mixing, is a method for the acceleration of the convergence rate of fixed-point iterations. Introduced by Donald G. Anderson, this technique
Scarborough criterion
The Scarborough criterion is used for satisfying convergence of a solution while solving linear equations using an iterative method.
Peano kernel theorem
In numerical analysis, the Peano kernel theorem is a general result on error bounds for a wide class of numerical approximations (such as numerical quadratures), defined in terms of linear functionals
Guard digit
In numerical analysis, one or more guard digits can be used to reduce the amount of roundoff error. For example, suppose that the final result of a long, multi-step calculation can be safely rounded o
Monte Carlo method
Monte Carlo methods, or Monte Carlo experiments, are a broad class of computational algorithms that rely on repeated random sampling to obtain numerical results. The underlying concept is to use rando
FEE method
In mathematics, the FEE method, or fast E-function evaluation method, is the method of fast summation of series of a special form. It was constructed in 1990 by and is so-named because it makes fast c
Legendre pseudospectral method
The Legendre pseudospectral method for optimal control problems is based on Legendre polynomials. It is part of the larger theory of pseudospectral optimal control, a term coined by Ross. A basic vers
Difference quotient
In single-variable calculus, the difference quotient is usually the name for the expression which when taken to the limit as h approaches 0 gives the derivative of the function f. The name of the expr
Well-posed problem
The mathematical term well-posed problem stems from a definition given by 20th-century French mathematician Jacques Hadamard. He believed that mathematical models of physical phenomena should have the
Numerical model of the Solar System
A numerical model of the Solar System is a set of mathematical equations, which, when solved, give the approximate positions of the planets as a function of time. Attempts to create such a model estab
Numerical differentiation
In numerical analysis, numerical differentiation algorithms estimate the derivative of a mathematical function or function subroutine using values of the function and perhaps other knowledge about the
Horner's method
In mathematics and computer science, Horner's method (or Horner's scheme) is an algorithm for polynomial evaluation. Although named after William George Horner, this method is much older, as it has be
Transfer matrix
In applied mathematics, the transfer matrix is a formulation in terms of a block-Toeplitz matrix of the two-scale equation, which characterizes refinable functions. Refinable functions play an importa
Cell-based models
Cell-based models are mathematical models that represent biological cells as a discrete entities. Within the field of computational biology they are often simply called agent-based models of which the
Hermes Project
Hermes2D (Higher-order modular finite element system) is a C++/Python library of algorithms for rapid development of adaptive hp-FEM solvers. hp-FEM is a modern version of the finite element method (F
Bernstein polynomial
In the mathematical field of numerical analysis, a Bernstein polynomial is a polynomial that is a linear combination of Bernstein basis polynomials. The idea is named after Sergei Natanovich Bernstein
Applied element method
The applied element method (AEM) is a numerical analysis used in predicting the continuum and discrete behavior of structures. The modeling method in AEM adopts the concept of discrete cracking allowi
Minimum polynomial extrapolation
In mathematics, minimum polynomial extrapolation is a sequence transformation used for convergence acceleration of vector sequences, due to Cabay and Jackson. While Aitken's method is the most famous,
CORDIC
CORDIC (for "coordinate rotation digital computer"), also known as Volder's algorithm, or: Digit-by-digit method Circular CORDIC (Jack E. Volder), Linear CORDIC, Hyperbolic CORDIC (John Stephen Walthe
Exponential integrator
Exponential integrators are a class of numerical methods for the solution of ordinary differential equations, specifically initial value problems. This large class of methods from numerical analysis i
Adaptive step size
In mathematics and numerical analysis, an adaptive step size is used in some methods for the numerical solution of ordinary differential equations (including the special case of numerical integration)
Approximation error
The approximation error in a data value is the discrepancy between an exact value and some approximation to it. This error can be expressed as an absolute error (the numerical amount of the discrepanc
Numeric precision in Microsoft Excel
As with other spreadsheets, Microsoft Excel works only to limited accuracy because it retains only a certain number of figures to describe numbers (it has limited precision). With some exceptions rega
Order of accuracy
In numerical analysis, order of accuracy quantifies the rate of convergence of a numerical approximation of a differential equation to the exact solution.Consider , the exact solution to a differentia
Identifiability analysis
Identifiability analysis is a group of methods found in mathematical statistics that are used to determine how well the parameters of a model are estimated by the quantity and quality of experimental
Bellman pseudospectral method
The Bellman pseudospectral method is a pseudospectral method for optimal control based on Bellman's principle of optimality. It is part of the larger theory of pseudospectral optimal control, a term c
Iterative method
In computational mathematics, an iterative method is a mathematical procedure that uses an initial value to generate a sequence of improving approximate solutions for a class of problems, in which the
Comparison of linear algebra libraries
The following tables provide a comparison of linear algebra software libraries, either specialized or general purpose libraries with significant linear algebra coverage.
Equioscillation theorem
In mathematics, the equioscillation theorem concerns the approximation of continuous functions using polynomials when the merit function is the maximum difference (uniform norm). Its discovery is attr
Padé table
In complex analysis, a Padé table is an array, possibly of infinite extent, of the rational Padé approximants Rm, n to a given complex formal power series. Certain sequences of approximants lying with
Ross–Fahroo lemma
Named after I. Michael Ross and F. Fahroo, the Ross–Fahroo lemma is a fundamental result in optimal control theory. It states that dualization and discretization are, in general, non-commutative opera
Method of fundamental solutions
In scientific computation and simulation, the method of fundamental solutions (MFS) is a technique for solving partial differential equations based on using the fundamental solution as a basis functio
Sigma approximation
In mathematics, σ-approximation adjusts a Fourier summation to greatly reduce the Gibbs phenomenon, which would otherwise occur at discontinuities. A σ-approximated summation for a series of period T
Finite Legendre transform
The finite Legendre transform (fLT) transforms a mathematical function defined on the finite interval into its Legendre spectrum.Conversely, the inverse fLT (ifLT) reconstructs the original function f
False precision
False precision (also called overprecision, fake precision, misplaced precision and spurious precision) occurs when numerical data are presented in a manner that implies better precision than is justi
Computational statistics
Computational statistics, or statistical computing, is the bond between statistics and computer science. It means statistical methods that are enabled by using computational methods. It is the area of
Flat pseudospectral method
The flat pseudospectral method is part of the family of the Ross–Fahroo pseudospectral methods introduced by Ross and Fahroo. The method combines the concept of differential flatness with pseudospectr
Predictor–corrector method
In numerical analysis, predictor–corrector methods belong to a class of algorithms designed to integrate ordinary differential equations – to find an unknown function that satisfies a given differenti
Pairwise summation
In numerical analysis, pairwise summation, also called cascade summation, is a technique to sum a sequence of finite-precision floating-point numbers that substantially reduces the accumulated round-o
Round-off error
A roundoff error, also called rounding error, is the difference between the result produced by a given algorithm using exact arithmetic and the result produced by the same algorithm using finite-preci
Validated numerics
Validated numerics, or rigorous computation, verified computation, reliable computation, numerical verification (German: Zuverlässiges Rechnen) is numerics including mathematically strict error (round
Numerical analysis
Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathemat
Spectral method
Spectral methods are a class of techniques used in applied mathematics and scientific computing to numerically solve certain differential equations. The idea is to write the solution of the differenti
Successive parabolic interpolation
Successive parabolic interpolation is a technique for finding the extremum (minimum or maximum) of a continuous unimodal function by successively fitting parabolas (polynomials of degree two) to a fun
Isotonic regression
In statistics and numerical analysis, isotonic regression or monotonic regression is the technique of fitting a free-form line to a sequence of observations such that the fitted line is non-decreasing
Rate of convergence
In numerical analysis, the order of convergence and the rate of convergence of a convergent sequence are quantities that represent how quickly the sequence approaches its limit. A sequence that conver
Probability box
A probability box (or p-box) is a characterization of uncertain numbers consisting of both aleatoric and epistemic uncertainties that is often used in risk analysis or quantitative uncertainty modelin
Coopmans approximation
The Coopmans approximation is a method for approximating a fractional-order integrator in a continuous process with constant space complexity. The most correct and accurate methods for calculating the
Explicit algebraic stress model
The algebraic stress model arises in computational fluid dynamics. Two main approaches can be undertaken. In the first, the transport of the turbulent stresses is assumed proportional to the turbulent
Linear multistep method
Linear multistep methods are used for the numerical solution of ordinary differential equations. Conceptually, a numerical method starts from an initial point and then takes a short step forward in ti
Interval contractor
In mathematics, an interval contractor (or contractor for short) associated to a set X is an operator C which associates to a box [x] in Rn another box C([x]) of Rn such that the two following propert
Residual (numerical analysis)
Loosely speaking, a residual is the error in a result. To be precise, suppose we want to find x such that Given an approximation x0 of x, the residual is that is, "what is left of the right hand side"
Abramowitz and Stegun
Abramowitz and Stegun (AS) is the informal name of a 1964 mathematical reference work edited by Milton Abramowitz and Irene Stegun of the United States National Bureau of Standards (NBS), now the Nati
Numerical integration
In analysis, numerical integration comprises a broad family of algorithms for calculating the numerical value of a definite integral, and by extension, the term is also sometimes used to describe the
Runge–Kutta methods
In numerical analysis, the Runge–Kutta methods (English: /ˈrʊŋəˈkʊtɑː/ RUUNG-ə-KUUT-tah) are a family of implicit and explicit iterative methods, which include the Euler method, used in temporal discr
Trajectory (fluid mechanics)
In fluid mechanics, meteorology and oceanography, a trajectory traces the motion of a single point, often called a parcel, in the flow. Trajectories are useful for tracking atmospheric contaminants, s
Linear algebra
Linear algebra is the branch of mathematics concerning linear equations such as: linear maps such as: and their representations in vector spaces and through matrices. Linear algebra is central to almo
Forward problem of electrocardiology
The forward problem of electrocardiology is a computational and mathematical approach to study the electrical activity of the heart through the body surface. The principal aim of this study is to comp
Series acceleration
In mathematics, series acceleration is one of a collection of sequence transformations for improving the rate of convergence of a series. Techniques for series acceleration are often applied in numeri
Discretization error
In numerical analysis, computational physics, and simulation, discretization error is the error resulting from the fact that a function of a continuous variable is represented in the computer by a fin
Levinson recursion
Levinson recursion or Levinson–Durbin recursion is a procedure in linear algebra to recursively calculate the solution to an equation involving a Toeplitz matrix. The algorithm runs in Θ(n2) time, whi
Singular boundary method
In numerical analysis, the singular boundary method (SBM) belongs to a family of meshless boundary collocation techniques which include the method of fundamental solutions (MFS), boundary knot method
Sparse grid
Sparse grids are numerical techniques to represent, integrate or interpolate high dimensional functions. They were originally developed by the Russian mathematician , a student of Lazar Lyusternik, an
Material point method
The material point method (MPM) is a numerical technique used to simulate the behavior of solids, liquids, gases, and any other continuum material. Especially, it is a robust spatial discretization me
Stechkin's lemma
In mathematics – more specifically, in functional analysis and numerical analysis – Stechkin's lemma is a result about the ℓq norm of the tail of a sequence, when the whole sequence is known to have f
Finite difference
A finite difference is a mathematical expression of the form f (x + b) − f (x + a). If a finite difference is divided by b − a, one gets a difference quotient. The approximation of derivatives by fini
Numerical method
In numerical analysis, a numerical method is a mathematical tool designed to solve numerical problems. The implementation of a numerical method with an appropriate convergence check in a programming l
Minimax approximation algorithm
A minimax approximation algorithm (or L∞ approximation or uniform approximation) is a method to find an approximation of a mathematical function that minimizes maximum error. For example, given a func
Level set (data structures)
In computer science a level set data structure is designed to represent discretely sampled dynamic level sets functions. A common use of this form of data structure is in efficient image rendering. Th
Whitney inequality
In mathematics, the Whitney inequality gives an upper bound for the error of best approximation of a function by polynomials in terms of the moduli of smoothness. It was first proved by Hassler Whitne
Clenshaw algorithm
In numerical analysis, the Clenshaw algorithm, also called Clenshaw summation, is a recursive method to evaluate a linear combination of Chebyshev polynomials. The method was published by Charles Will
Computer-assisted proof
A computer-assisted proof is a mathematical proof that has been at least partially generated by computer. Most computer-aided proofs to date have been implementations of large proofs-by-exhaustion of
Error analysis (mathematics)
In mathematics, error analysis is the study of kind and quantity of error, or uncertainty, that may be present in the solution to a problem. This issue is particularly prominent in applied areas such
Nonstandard finite difference scheme
Nonstandard finite difference schemes is a general set of methods in numerical analysis that gives numerical solutions to differential equations by constructing a discrete model. The general rules for
Kempner series
The Kempner series is a modification of the harmonic series, formed by omitting all terms whose denominator expressed in base 10 contains the digit 9. That is, it is the sum where the prime indicates
Dynamic relaxation
Dynamic relaxation is a numerical method, which, among other things, can be used to do "" for cable and fabric structures. The aim is to find a geometry where all forces are in equilibrium. In the pas
Ross–Fahroo pseudospectral method
Introduced by I. Michael Ross and F. Fahroo, the Ross–Fahroo pseudospectral methods are a broad collection of pseudospectral methods for optimal control. Examples of the Ross–Fahroo pseudospectral met
Truncation error
In numerical analysis and scientific computing, truncation error is an error caused by approximating a mathematical process.
Quantification of margins and uncertainties
Quantification of Margins and Uncertainty (QMU) is a decision support methodology for complex technical decisions. QMU focuses on the identification, characterization, and analysis of performance thre
Least-squares spectral analysis
Least-squares spectral analysis (LSSA) is a method of estimating a frequency spectrum, based on a least squares fit of sinusoids to data samples, similar to Fourier analysis. Fourier analysis, the mos
Numerical continuation
Numerical continuation is a method of computing approximate solutions of a system of parameterized nonlinear equations, The parameter is usually a real scalar, and the solution an n-vector. For a fixe
Radial basis function interpolation
Radial basis function (RBF) interpolation is an advanced method in approximation theory for constructing high-order accurate interpolants of unstructured data, possibly in high-dimensional spaces. The
Overlap–save method
In signal processing, overlap–save is the traditional name for an efficient way to evaluate the discrete convolution between a very long signal and a finite impulse response (FIR) filter : where h[m]
Interval propagation
In numerical mathematics, interval propagation or interval constraint propagation is the problem of contracting interval domains associated to variables of R without removing any value that is consist
Semi-infinite programming
In optimization theory, semi-infinite programming (SIP) is an optimization problem with a finite number of variables and an infinite number of constraints, or an infinite number of variables and a fin
Trigonometric tables
In mathematics, tables of trigonometric functions are useful in a number of areas. Before the existence of pocket calculators, trigonometric tables were essential for navigation, science and engineeri
Superconvergence
In numerical analysis, a superconvergent or supraconvergent method is one which converges faster than generally expected (superconvergence or supraconvergence). For example, in the Finite Element Meth
Karlsruhe Accurate Arithmetic
Karlsruhe Accurate Arithmetic (KAA) or Karlsruhe Accurate Arithmetic Approach (KAAA), augments conventional floating-point arithmetic with good error behaviour with new operations to calculate scalar
Timeline of numerical analysis after 1945
The following is a timeline of numerical analysis after 1945, and deals with developments after the invention of the modern electronic computer, which began during Second World War. For a fuller histo
Kantorovich theorem
The Kantorovich theorem, or Newton–Kantorovich theorem, is a mathematical statement on the semi-local convergence of Newton's method. It was first stated by Leonid Kantorovich in 1948. It is similar t
Discrete calculus
Discrete calculus or the calculus of discrete functions, is the mathematical study of incremental change, in the same way that geometry is the study of shape and algebra is the study of generalization
Natural element method
The natural element method (NEM) is a meshless method to solve partial differential equation, where the elements do not have a predefined shape as in the finite element method, but depend on the geome
Bueno-Orovio–Cherry–Fenton model
The Bueno-Orovio–Cherry–Fenton model, also simply called Bueno-Orovio model, is a minimal ionic model for human ventricular cells. It belongs to the category of phenomenological models, because of its
Matrix analysis
In mathematics, particularly in linear algebra and applications, matrix analysis is the study of matrices and their algebraic properties. Some particular topics out of many include; operations defined
Boundary knot method
In numerical mathematics, the boundary knot method (BKM) is proposed as an alternative boundary-type meshfree distance function collocation scheme. Recent decades have witnessed a research boom on the
Catastrophic cancellation
In numerical analysis, catastrophic cancellation is the phenomenon that subtracting good approximations to two nearby numbers may yield a very bad approximation to the difference of the original numbe
Multi-time-step integration
In numerical analysis, multi-time-step integration, also referred to as multiple-step or asynchronous time integration, is a numerical time-integration method that uses different time-steps or time-in
Approximation theory
In mathematics, approximation theory is concerned with how functions can best be approximated with simpler functions, and with quantitatively characterizing the errors introduced thereby. Note that wh
Calderón projector
In applied mathematics, the Calderón projector is a pseudo-differential operator used widely in boundary element methods. It is named after Alberto Calderón.
Multilevel fast multipole method
The multilevel fast multipole method (MLFMM) is used along with method of moments (MoM) a numerical computational method of solving linear partial differential equations which have been formulated as
Differential-algebraic system of equations
In mathematics, a differential-algebraic system of equations (DAEs) is a system of equations that either contains differential equations and algebraic equations, or is equivalent to such a system. Suc
Gal's accurate tables
Gal's accurate tables is a method devised by Shmuel Gal to provide accurate values of special functions using a lookup table and interpolation. It is a fast and efficient method for generating values
De Casteljau's algorithm
In the mathematical field of numerical analysis, De Casteljau's algorithm is a recursive method to evaluate polynomials in Bernstein form or Bézier curves, named after its inventor Paul de Casteljau.
Miller's recurrence algorithm
Miller's recurrence algorithm is a procedure for calculating a rapidly decreasing solution of a linear recurrence relation developed by J. C. P. Miller. It was originally developed to compute tables o
Parareal
Parareal is a parallel algorithm from numerical analysis and used for the solution of initial value problems.It was introduced in 2001 by Lions, Maday and Turinici. Since then, it has become one of th
Continuous wavelet
In numerical analysis, continuous wavelets are functions used by the continuous wavelet transform. These functions are defined as analytical expressions, as functions either of time or of frequency.Mo
Interval arithmetic
Interval arithmetic (also known as interval mathematics, interval analysis, or interval computation) is a mathematical technique used to put bounds on rounding errors and measurement errors in mathema
Linear approximation
In mathematics, a linear approximation is an approximation of a general function using a linear function (more precisely, an affine function). They are widely used in the method of finite differences
Sterbenz lemma
In floating-point arithmetic, the Sterbenz lemma or Sterbenz's lemma is a theorem giving conditions under which floating-point differences are computed exactly.It is named after Pat H. Sterbenz, who p
Dormand–Prince method
In numerical analysis, the Dormand–Prince (RKDP) method or DOPRI method, is an embedded method for solving ordinary differential equations. The method is a member of the Runge–Kutta family of ODE solv
Galerkin method
In mathematics, in the area of numerical analysis, Galerkin methods, named after the Russian mathematician Boris Galerkin, convert a continuous operator problem, such as a differential equation, commo
Adjoint state method
The adjoint state method is a numerical method for efficiently computing the gradient of a function or operator in a numerical optimization problem. It has applications in geophysics, seismic imaging,
Bidomain model
The bidomain model is a mathematical model to define the electrical activity of the heart. It consists in a continuum (volume-average) approach in which the cardiac mictrostructure is defined in terms
Pseudospectral knotting method
In applied mathematics, the pseudospectral knotting method is a generalization and enhancement of a standard pseudospectral method for optimal control. The concept was introduced by I. Michael Ross an
Ross' π lemma
Ross' π lemma, named after I. Michael Ross, is a result in computational optimal control. Based on generating Carathéodory-π solutions for feedback control, Ross' π-lemma states that there is fundamen
Kahan summation algorithm
In numerical analysis, the Kahan summation algorithm, also known as compensated summation, significantly reduces the numerical error in the total obtained by adding a sequence of finite-precision floa
Blossom (functional)
In numerical analysis, a blossom is a functional that can be applied to any polynomial, but is mostly used for Bézier and spline curves and surfaces. The blossom of a polynomial ƒ, often denoted is co
Finite volume method
The finite volume method (FVM) is a method for representing and evaluating partial differential equations in the form of algebraic equations.In the finite volume method, volume integrals in a partial
Integrable algorithm
Integrable algorithms are numerical algorithms that rely on basic ideas from the mathematical theory of integrable systems.
Pseudo-spectral method
Pseudo-spectral methods, also known as discrete variable representation (DVR) methods, are a class of numerical methods used in applied mathematics and scientific computing for the solution of partial
Shanks transformation
In numerical analysis, the Shanks transformation is a non-linear series acceleration method to increase the rate of convergence of a sequence. This method is named after Daniel Shanks, who rediscovere
Movable cellular automaton
The movable cellular automaton (MCA) method is a method in computational solid mechanics based on the discrete concept. It provides advantages both of classical cellular automaton and discrete element
Gradient discretisation method
In numerical mathematics, the gradient discretisation method (GDM) is a framework which contains classical and recent numerical schemes for diffusion problems of various kinds: linear or non-linear, s
Particle method
In the field of numerical analysis, particle methods discretize fluid into particles. Particle methods enable the simulation of some otherwise difficult types of problems, at the cost of extra computi
Surrogate model
A surrogate model is an engineering method used when an outcome of interest cannot be easily measured or computed, so a model of the outcome is used instead. Most engineering design problems require e
Proper generalized decomposition
The proper generalized decomposition (PGD) is an iterative numerical method for solving boundary value problems (BVPs), that is, partial differential equations constrained by a set of boundary conditi
Riemann solver
A Riemann solver is a numerical method used to solve a Riemann problem. They are heavily used in computational fluid dynamics and computational magnetohydrodynamics.
Kummer's transformation of series
In mathematics, specifically in the field of numerical analysis, Kummer's transformation of series is a method used to accelerate the convergence of an infinite series. The method was first suggested
2Sum
2Sum is a floating-point algorithm for computing the exact round-off error in a floating-point addition operation. 2Sum and its variant Fast2Sum were first published by Møller in 1965.Fast2Sum is ofte
Boundary particle method
In applied mathematics, the boundary particle method (BPM) is a boundary-only meshless (meshfree) collocation technique, in the sense that none of inner nodes are required in the numerical solution of
Fast multipole method
The fast multipole method (FMM) is a numerical technique that was developed to speed up the calculation of long-ranged forces in the n-body problem. It does this by expanding the system Green's functi
Lie group integrator
A Lie group integrator is a numerical integration method for differential equations built from such as Lie group actions on a manifold. They have been used for the animation and control of vehicles in
Scale co-occurrence matrix
Scale co-occurrence matrix (SCM) is a method for image feature extraction within scale space after wavelet transformation, proposed by Wu Jun and Zhao Zhongming (Institute of Remote Sensing Applicatio
Bernstein's constant
Bernstein's constant, usually denoted by the Greek letter β (beta), is a mathematical constant named after Sergei Natanovich Bernstein and is equal to 0.2801694990... .
Relative change and difference
In any quantitative science, the terms relative change and relative difference are used to compare two quantities while taking into account the "sizes" of the things being compared, i.e. dividing by a
Chebyshev nodes
In numerical analysis, Chebyshev nodes are specific real algebraic numbers, namely the roots of the Chebyshev polynomials of the first kind. They are often used as nodes in polynomial interpolation be
Modulus of smoothness
In mathematics, moduli of smoothness are used to quantitatively measure smoothness of functions. Moduli of smoothness generalise modulus of continuity and are used in approximation theory and numerica
Lady Windermere's Fan (mathematics)
In mathematics, Lady Windermere's Fan is a telescopic identity employed to relate global and local error of a numerical algorithm. The name is derived from Oscar Wilde's 1892 play Lady Windermere's Fa
Lanczos approximation
In mathematics, the Lanczos approximation is a method for computing the gamma function numerically, published by Cornelius Lanczos in 1964. It is a practical alternative to the more popular Stirling's
Model order reduction
Model order reduction (MOR) is a technique for reducing the computational complexity of mathematical models in numerical simulations. As such it is closely related to the concept of metamodeling, with
Pythagorean addition
In mathematics, Pythagorean addition is a binary operation on the real numbers that computes the length of the hypotenuse of a right triangle, given its two sides. According to the Pythagorean theorem
Local convergence
In numerical analysis, an iterative method is called locally convergent if the successive approximations produced by the method are guaranteed to converge to a solution when the initial approximation
Low-discrepancy sequence
In mathematics, a low-discrepancy sequence is a sequence with the property that for all values of N, its subsequence x1, ..., xN has a low discrepancy. Roughly speaking, the discrepancy of a sequence
Chebyshev pseudospectral method
The Chebyshev pseudospectral method for optimal control problems is based on Chebyshev polynomials of the first kind. It is part of the larger theory of pseudospectral optimal control, a term coined b
Variational multiscale method
The variational multiscale method (VMS) is a technique used for deriving models and numerical methods for multiscale phenomena. The VMS framework has been mainly applied to design stabilized finite el
Regularized meshless method
In numerical mathematics, the regularized meshless method (RMM), also known as the singular meshless method or desingularized meshless method, is a meshless boundary collocation method designed to sol
Jenkins–Traub algorithm
The Jenkins–Traub algorithm for polynomial zeros is a fast globally convergent iterative polynomial root-finding method published in 1970 by and Joseph F. Traub. They gave two variants, one for genera
Significant figures
Significant figures (also known as the significant digits, precision or resolution) of a number in positional notation are digits in the number that are reliable and necessary to indicate the quantity
Propagation of uncertainty
In statistics, propagation of uncertainty (or propagation of error) is the effect of variables' uncertainties (or errors, more specifically random errors) on the uncertainty of a function based on the
Significance arithmetic
Significance arithmetic is a set of rules (sometimes called significant figure rules) for approximating the propagation of uncertainty in scientific or statistical calculations. These rules can be use
Basis function
In mathematics, a basis function is an element of a particular basis for a function space. Every function in the function space can be represented as a linear combination of basis functions, just as e
Curve fitting
Curve fitting is the process of constructing a curve, or mathematical function, that has the best fit to a series of data points, possibly subject to constraints. Curve fitting can involve either inte
Digital Library of Mathematical Functions
The Digital Library of Mathematical Functions (DLMF) is an online project at the National Institute of Standards and Technology (NIST) to develop a database of mathematical reference data for special
Symbolic-numeric computation
In mathematics and computer science, symbolic-numeric computation is the use of software that combines symbolic and numeric methods to solve problems.
Truncated power function
In mathematics, the truncated power function with exponent is defined as In particular, and interpret the exponent as conventional power.