Numerical method - Revision history

Jdbtwo: “cross-product” —> “Cartesian product”

2025-04-14T17:06:45Z

“cross-product” —> “Cartesian product”

← Previous revision		Revision as of 17:06, 14 April 2025
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	==Mathematical definition==		==Mathematical definition==
	Let <math>F(x,y)=0</math> be a [[well-posed problem (numerical analysis)\|well-posed problem]], i.e. <math>F:X \times Y \rightarrow \mathbb{R}</math> is a [[Real number\|real]] or [[Complex Numbers\|complex]] functional relationship, defined on the ~~cross-~~product of an input data set <math>X</math> and an output data set <math>Y</math>, such that exists a [[Lipschitz continuity\|locally lipschitz]] function <math>g:X \rightarrow Y</math> called [[Resolvent (direct problem)\|resolvent]], which has the property that for every root <math>(x,y)</math> of <math>F</math>, <math>y=g(x)</math>. We define '''numerical method''' for the approximation of <math>F(x,y)=0</math>, the [[sequence]] of problems		Let <math>F(x,y)=0</math> be a [[well-posed problem (numerical analysis)\|well-posed problem]], i.e. <math>F:X \times Y \rightarrow \mathbb{R}</math> is a [[Real number\|real]] or [[Complex Numbers\|complex]] functional relationship, defined on the [[Cartesian product]] of an input data set <math>X</math> and an output data set <math>Y</math>, such that exists a [[Lipschitz continuity\|locally lipschitz]] function <math>g:X \rightarrow Y</math> called [[Resolvent (direct problem)\|resolvent]], which has the property that for every root <math>(x,y)</math> of <math>F</math>, <math>y=g(x)</math>. We define '''numerical method''' for the approximation of <math>F(x,y)=0</math>, the [[sequence]] of problems

	: <math>\left \{ M_n \right \}_{n \in \mathbb{N}} = \left \{ F_n(x_n,y_n)=0 \right \}_{n \in \mathbb{N}},</math>		: <math>\left \{ M_n \right \}_{n \in \mathbb{N}} = \left \{ F_n(x_n,y_n)=0 \right \}_{n \in \mathbb{N}},</math>

MWAK: /* Convergence */

2024-12-17T15:49:22Z

Convergence

← Previous revision		Revision as of 15:49, 17 December 2024
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	</math>		</math>

	One can easily prove that the point-wise convergence of <math> \{y_n\} _{n \in \mathbb{N}}</math> to <math>y</math> implies the convergence of the associated method ~~is function~~.<ref name="quartsaccsal" />		One can easily prove that the point-wise convergence of <math> \{y_n\} _{n \in \mathbb{N}}</math> to <math>y</math> implies the convergence of the associated method.<ref name="quartsaccsal" />

	==See also==		==See also==

Charhead: Undid revision 1238263759 by 2601:C2:1580:CA10:8C04:54A7:8855:3849 (talk)

2024-08-10T22:27:52Z

Undid revision 1238263759 by 2601:C2:1580:CA10:8C04:54A7:8855:3849 (talk)

← Previous revision		Revision as of 22:27, 10 August 2024
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	{{Short description\|Mathematical tool to algorithmically solve equations}}		{{Short description\|Mathematical tool to algorithmically solve equations}}
		⚫	{{more footnotes needed\|date=September 2016}}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.
	{{more footnotes needed\|date=September 2016}}
⚫	It is ~~estimation (oversimplified).~~ 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==		==Mathematical definition==

2601:C2:1580:CA10:8C04:54A7:8855:3849: Added one sentence in the beginning that ACTUALLY tells me what it is in layman's terms.

2024-08-03T00:30:07Z

Added one sentence in the beginning that ACTUALLY tells me what it is in layman's terms.

← Previous revision		Revision as of 00:30, 3 August 2024
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	{{Short description\|Mathematical tool to algorithmically solve equations}}		{{Short description\|Mathematical tool to algorithmically solve equations}}
	{{more footnotes needed\|date=September 2016}}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.		{{more footnotes needed\|date=September 2016}}
			It is estimation (oversimplified). 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==		==Mathematical definition==

Tom.Reding: +{{Authority control}} (1 ID from Wikidata); WP:GenFixes & cleanup on

2023-10-20T22:17:32Z

+{{Authority control}} (1 ID from Wikidata); WP:GenFixes & cleanup on

← Previous revision		Revision as of 22:17, 20 October 2023
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	{{Short description\|Mathematical tool to algorithmically solve equations}}		{{Short description\|Mathematical tool to algorithmically solve equations}}
	{{more footnotes\|date=September 2016}}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.		{{more footnotes needed\|date=September 2016}}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==		==Mathematical definition==
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	==References==		==References==
	{{Reflist}}		{{Reflist}}

			{{Authority control}}

	[[Category:Numerical analysis]]		[[Category:Numerical analysis]]

200.58.135.162: /* See also */

2023-01-30T16:57:38Z

Dunelm86 at 05:15, 14 August 2022

2022-08-14T05:15:50Z

← Previous revision		Revision as of 05:15, 14 August 2022
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			{{Short description\|Mathematical tool to algorithmically solve equations}}
	{{more footnotes\|date=September 2016}}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.		{{more footnotes\|date=September 2016}}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.

ComicsAreJustAllRight at 23:19, 12 February 2021

2021-02-12T23:19:08Z

← Previous revision		Revision as of 23:19, 12 February 2021
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	{{more footnotes\|date=September 2016}}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		{{more footnotes\|date=September 2016}}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==		==Mathematical definition==

GeordieMcBain: /* Convergence */ delete spurious markup "math"

2020-06-29T00:46:04Z

Convergence: delete spurious markup "math"

← Previous revision		Revision as of 00:46, 29 June 2020
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	==Convergence==		==Convergence==
	Denote by <math>\ell_n</math> a sequence of ''admissible perturbations'' of <math>x \in X</math> for some numerical method <math>M</math> (i.e. <math>x+\ell_n \in X_n \forall n \in \mathbb{N}</math>) and with <math>y_n(x+\ell_n) \in Y_n</math> the value such that <math>F_n(x+\ell_n,y_n(x+\ell_n)) = 0</math>. A condition which the method has to satisfy to be a meaningful tool for solving the problem <math>F(x,y)=0</math> is ''convergence''~~math~~:		Denote by <math>\ell_n</math> a sequence of ''admissible perturbations'' of <math>x \in X</math> for some numerical method <math>M</math> (i.e. <math>x+\ell_n \in X_n \forall n \in \mathbb{N}</math>) and with <math>y_n(x+\ell_n) \in Y_n</math> the value such that <math>F_n(x+\ell_n,y_n(x+\ell_n)) = 0</math>. A condition which the method has to satisfy to be a meaningful tool for solving the problem <math>F(x,y)=0</math> is ''convergence'':

	: <math>		: <math>

2409:4072:6080:9178:E9FC:3EF0:1BCF:EC82 at 07:40, 16 June 2020

2020-06-16T07:40:52Z

← Previous revision		Revision as of 07:40, 16 June 2020
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			{{more footnotes\|date=September 2016}}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
	{{more footnotes\|date=September 2016}}
	'''Numerical method'''

	==Mathematical definition==		==Mathematical definition==

← Previous revision		Revision as of 16:57, 30 January 2023
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	One can easily prove that the point-wise convergence of <math> \{y_n\} _{n \in \mathbb{N}}</math> to <math>y</math> implies the convergence of the associated method is function.<ref name="quartsaccsal" />		One can easily prove that the point-wise convergence of <math> \{y_n\} _{n \in \mathbb{N}}</math> to <math>y</math> implies the convergence of the associated method is function.<ref name="quartsaccsal" />

			==See also==
			* [[Numerical methods for ordinary differential equations]]
			* [[Numerical methods for partial differential equations]]

	==References==		==References==