Numerical method

Last updated October 21, 2023

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 language is called a numerical algorithm.

Mathematical definition

Let $F(x,y)=0$ be a well-posed problem, i.e. $F:X\times Y\rightarrow \mathbb {R}$ is a real or complex functional relationship, defined on the cross-product of an input data set $X$ and an output data set $Y$ , such that exists a locally lipschitz function $g:X\rightarrow Y$ called resolvent, which has the property that for every root $(x,y)$ of $F$ , $y=g(x)$ . We define numerical method for the approximation of $F(x,y)=0$ , the sequence of problems

\left\{M_{n}\right\}_{n\in \mathbb {N} }=\left\{F_{n}(x_{n},y_{n})=0\right\}_{n\in \mathbb {N} },

with $F_{n}:X_{n}\times Y_{n}\rightarrow \mathbb {R}$ , $x_{n}\in X_{n}$ and $y_{n}\in Y_{n}$ for every $n\in \mathbb {N}$ . The problems of which the method consists need not be well-posed. If they are, the method is said to be stable or well-posed.^[1]

Consistency

Necessary conditions for a numerical method to effectively approximate $F(x,y)=0$ are that $x_{n}\rightarrow x$ and that $F_{n}$ behaves like $F$ when $n\rightarrow \infty$ . So, a numerical method is called consistent if and only if the sequence of functions $\left\{F_{n}\right\}_{n\in \mathbb {N} }$ pointwise converges to $F$ on the set $S$ of its solutions:

\lim F_{n}(x,y+t)=F(x,y,t)=0,\quad \quad \forall (x,y,t)\in S.

When $F_{n}=F,\forall n\in \mathbb {N}$ on $S$ the method is said to be strictly consistent.^[1]

Convergence

Denote by $\ell _{n}$ a sequence of admissible perturbations of $x\in X$ for some numerical method $M$ (i.e. $x+\ell _{n}\in X_{n}\forall n\in \mathbb {N}$ ) and with $y_{n}(x+\ell _{n})\in Y_{n}$ the value such that $F_{n}(x+\ell _{n},y_{n}(x+\ell _{n}))=0$ . A condition which the method has to satisfy to be a meaningful tool for solving the problem $F(x,y)=0$ is convergence:

{\begin{aligned}&\forall \varepsilon >0,\exists n_{0}(\varepsilon )>0,\exists \delta _{\varepsilon ,n_{0}}{\text{ such that}}\\&\forall n>n_{0},\forall \ell _{n}:\|\ell _{n}\|<\delta _{\varepsilon ,n_{0}}\Rightarrow \|y_{n}(x+\ell _{n})-y\|\leq \varepsilon .\end{aligned}}

One can easily prove that the point-wise convergence of $\{y_{n}\}_{n\in \mathbb {N} }$ to $y$ implies the convergence of the associated method is function.^[1]

Related Research Articles

In mathematics, a continuous function is a function such that a continuous variation of the argument induces a continuous variation of the value of the function. This means there are no abrupt changes in value, known as discontinuities. More precisely, a function is continuous if arbitrarily small changes in its value can be assured by restricting to sufficiently small changes of its argument. A discontinuous function is a function that is not continuous. Until the 19th century, mathematicians largely relied on intuitive notions of continuity and considered only continuous functions. The epsilon–delta definition of a limit was introduced to formalize the definition of continuity.

In mathematics, the branch of real analysis studies the behavior of real numbers, sequences and series of real numbers, and real functions. Some particular properties of real-valued sequences and functions that real analysis studies include convergence, limits, continuity, smoothness, differentiability and integrability.

<span class="mw-page-title-main">Uniform continuity</span> Uniform restraint of the change in functions

In mathematics, a real function $of real numbers is said to be uniformly continuous if there is a positive real number such that function values over any function domain interval of the size are as close to each other as we want. In other words, for a uniformly continuous real function of real numbers, if we want function value differences to be less than any positive real number, then there is a positive real number such that at any and in any function interval of the size .$

In mathematical physics, the Dirac delta distribution, also known as the unit impulse, is a generalized function or distribution over the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire real line is equal to one.

In probability theory, there exist several different notions of convergence of random variables. The convergence of sequences of random variables to some limit random variable is an important concept in probability theory, and its applications to statistics and stochastic processes. The same concepts are known in more general mathematics as stochastic convergence and they formalize the idea that a sequence of essentially random or unpredictable events can sometimes be expected to settle down into a behavior that is essentially unchanging when items far enough into the sequence are studied. The different possible notions of convergence relate to how such a behavior can be characterized: two readily understood behaviors are that the sequence eventually takes a constant value, and that values in the sequence continue to change but can be described by an unchanging probability distribution.

In mathematics, the limit of a function is a fundamental concept in calculus and analysis concerning the behavior of that function near a particular input.

In mathematics, the limit of a sequence is the value that the terms of a sequence "tend to", and is often denoted using the $symbol. If such a limit exists, the sequence is called convergent . A sequence that does not converge is said to be divergent . The limit of a sequence is said to be the fundamental notion on which the whole of mathematical analysis ultimately rests.$

<span class="mw-page-title-main">Squeeze theorem</span> Method for finding limits in calculus

In calculus, the squeeze theorem is a theorem regarding the limit of a function that is trapped between two other functions.

Various types of stability may be discussed for the solutions of differential equations or difference equations describing dynamical systems. The most important type is that concerning the stability of solutions near to a point of equilibrium. This may be discussed by the theory of Aleksandr Lyapunov. In simple terms, if the solutions that start out near an equilibrium point $stay near forever, then is Lyapunov stable . More strongly, if is Lyapunov stable and all solutions that start out near converge to, then is said to be asymptotically stable . The notion of exponential stability guarantees a minimal rate of decay, i.e., an estimate of how quickly the solutions converge. The idea of Lyapunov stability can be extended to infinite-dimensional manifolds, where it is known as structural stability, which concerns the behavior of different but "nearby" solutions to differential equations. Input-to-state stability (ISS) applies Lyapunov notions to systems with inputs.$

<span class="mw-page-title-main">One-sided limit</span> Limit of a function approaching a value point from values below or above the value point

In calculus, a one-sided limit refers to either one of the two limits of a function $of a real variable as approaches a specified point either from the left or from the right.$

The Arzelà–Ascoli theorem is a fundamental result of mathematical analysis giving necessary and sufficient conditions to decide whether every sequence of a given family of real-valued continuous functions defined on a closed and bounded interval has a uniformly convergent subsequence. The main condition is the equicontinuity of the family of functions. The theorem is the basis of many proofs in mathematics, including that of the Peano existence theorem in the theory of ordinary differential equations, Montel's theorem in complex analysis, and the Peter–Weyl theorem in harmonic analysis and various results concerning compactness of integral operators.

In mathematics, more specifically measure theory, there are various notions of the convergence of measures. For an intuitive general sense of what is meant by convergence of measures, consider a sequence of measures μ_n on a space, sharing a common collection of measurable sets. Such a sequence might represent an attempt to construct 'better and better' approximations to a desired measure μ that is difficult to obtain directly. The meaning of 'better and better' is subject to all the usual caveats for taking limits; for any error tolerance ε > 0 we require there be N sufficiently large for n ≥ N to ensure the 'difference' between μ_n and μ is smaller than ε. Various notions of convergence specify precisely what the word 'difference' should mean in that description; these notions are not equivalent to one another, and vary in strength.

In mathematical analysis, the Russo–Vallois integral is an extension to stochastic processes of the classical Riemann–Stieltjes integral

Stochastic approximation methods are a family of iterative methods typically used for root-finding problems or for optimization problems. The recursive update rules of stochastic approximation methods can be used, among other things, for solving linear systems when the collected data is corrupted by noise, or for approximating extreme values of functions which cannot be computed directly, but only estimated via noisy observations.

In applied mathematics, discontinuous Galerkin methods (DG methods) form a class of numerical methods for solving differential equations. They combine features of the finite element and the finite volume framework and have been successfully applied to hyperbolic, elliptic, parabolic and mixed form problems arising from a wide range of applications. DG methods have in particular received considerable interest for problems with a dominant first-order part, e.g. in electrodynamics, fluid mechanics and plasma physics.

In real analysis and measure theory, the Vitali convergence theorem, named after the Italian mathematician Giuseppe Vitali, is a generalization of the better-known dominated convergence theorem of Henri Lebesgue. It is a characterization of the convergence in L^p in terms of convergence in measure and a condition related to uniform integrability.

In mathematics, the Möbius energy of a knot is a particular knot energy, i.e., a functional on the space of knots. It was discovered by Jun O'Hara, who demonstrated that the energy blows up as the knot's strands get close to one another. This is a useful property because it prevents self-intersection and ensures the result under gradient descent is of the same knot type.

In probability theory, the continuous mapping theorem states that continuous functions preserve limits even if their arguments are sequences of random variables. A continuous function, in Heine’s definition, is such a function that maps convergent sequences into convergent sequences: if x_n → x then g(x_n) → g(x). The continuous mapping theorem states that this will also be true if we replace the deterministic sequence {x_n} with a sequence of random variables {X_n}, and replace the standard notion of convergence of real numbers “→” with one of the types of convergence of random variables.

This article is supplemental for “Convergence of random variables” and provides proofs for selected results.

In mathematics, a limit is the value that a function approaches as the input approaches some value. Limits are essential to calculus and mathematical analysis, and are used to define continuity, derivatives, and integrals.

References

1 2 3 Quarteroni, Sacco, Saleri (2000). Numerical Mathematics (PDF). Milano: Springer. p. 33. Archived from the original (PDF) on 2017-11-14. Retrieved 2016-09-27.{{cite book}}: CS1 maint: multiple names: authors list (link)

This page is based on this Wikipedia article
Text is available under the CC BY-SA 4.0 license; additional terms may apply.
Images, videos and audio are available under their respective licenses.

[quartsaccsal-1] 1 2 3 Quarteroni, Sacco, Saleri (2000). Numerical Mathematics (PDF). Milano: Springer. p. 33. Archived from the original (PDF) on 2017-11-14. Retrieved 2016-09-27.{{cite book}}: CS1 maint: multiple names: authors list (link)

[1]