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numerical method wikipedia jump to content main menu main menu move to sidebar hide navigation main page contents current events random article about wikipedia contact us contribute help learn to edit community portal recent changes upload file special pages search search appearance donate create account log in personal tools donate create account log in contents move to sidebar hide top 1 mathematical definition 2 consistency 3 convergence 4 see also 5 references toggle the table of contents numerical method 4 languages čeština فارسی galego slovenčina edit links article talk english read edit view history tools tools move to sidebar hide actions read edit view history general what links here related changes upload file permanent link page information cite this page get shortened url print export download as pdf printable version in other projects wikibooks wikidata item appearance move to sidebar hide from wikipedia the free encyclopedia mathematical tool to algorithmically solve equations this article includes a list of general references but it lacks sufficient corresponding inline citations please help to improve this article by introducing more precise citations september 2016 learn how and when to remove this message 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 edit let f x y 0 displaystyle f x y 0 be a well posed problem i e f x y r displaystyle f x times y rightarrow mathbb r is a real or complex functional relationship defined on the cartesian product of an input data set x displaystyle x and an output data set y displaystyle y such that exists a locally lipschitz function g x y displaystyle g x rightarrow y called resolvent which has the property that for every root x y displaystyle x y of f displaystyle f y g x displaystyle y g x we define numerical method for the approximation of f x y 0 displaystyle f x y 0 the sequence of problems m n n n f n x n y n 0 n n displaystyle 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 y n r displaystyle f_ n x_ n times y_ n rightarrow mathbb r x n x n displaystyle x_ n in x_ n and y n y n displaystyle y_ n in y_ n for every n n displaystyle 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 edit necessary conditions for a numerical method to effectively approximate f x y 0 displaystyle f x y 0 are that x n x displaystyle x_ n rightarrow x and that f n displaystyle f_ n behaves like f displaystyle f when n displaystyle n rightarrow infty so a numerical method is called consistent if and only if the sequence of functions f n n n displaystyle left f_ n right _ n in mathbb n pointwise converges to f displaystyle f on the set s displaystyle s of its solutions lim f n x y t f x y t 0 x y t s displaystyle lim f_ n x y t f x y t 0 quad quad forall x y t in s when f n f n n displaystyle f_ n f forall n in mathbb n on s displaystyle s the method is said to be strictly consistent 1 convergence edit denote by ℓ n displaystyle ell _ n a sequence of admissible perturbations of x x displaystyle x in x for some numerical method m displaystyle m i e x ℓ n x n n n displaystyle x ell _ n in x_ n forall n in mathbb n and with y n x ℓ n y n displaystyle y_ n x ell _ n in y_ n the value such that f n x ℓ n y n x ℓ n 0 displaystyle 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 displaystyle f x y 0 is convergence ε 0 n 0 ε 0 δ ε n 0 such that n n 0 ℓ n ℓ n δ ε n 0 y n x ℓ n y ε displaystyle 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 n displaystyle y_ n _ n in mathbb n to y displaystyle y implies the convergence of the associated method 1 see also edit numerical methods for ordinary differential equations numerical methods for partial differential equations references edit a b c 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 authority control databases national czech republic retrieved from https en wikipedia org w index php title numerical_method oldid 1304942310 category numerical analysis hidden categories cs1 maint multiple names authors list articles with short description short description is different from wikidata articles lacking in text citations from september 2016 all articles lacking in text citations this page was last edited on 9 august 2025 at 02 48 utc text is available under the creative commons attribution sharealike 4 0 license additional terms may apply by using this site you agree to the terms of use and privacy policy wikipedia is a registered trademark of the wikimedia foundation inc a non profit organization privacy policy about wikipedia disclaimers contact wikipedia legal safety contacts code of conduct developers statistics cookie statement mobile view search search toggle the table of contents numerical method 4 languages add topic
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