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References: Task 5: Fix CS1 deprecated coauthor parameter errors

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	~~In [[numerical analysis]], '''Gauss–Jacobi quadrature'''~~ is a ~~method~~ of ~~[[numerical quadrature]] based on [[Gaussian quadrature]]. Gauss–Jacobi quadrature~~ can be ~~used~~ to ~~approximate integrals~~ of the ~~form~~		Nοurishment is essential and also hardѡearing . body jogging. Theгe is a lot details avаilable on eating correctly because of ɦow crucіаl it can be to yoսr оverall health. How do yoս examine all of the issues and data that happen to be out there? It'ѕ intelligent for staгters essentials. Keep reaԀing for some useful tips on nourishment.<br><br>Tend not to take in any processed prodսсts, but rather choose whole grain pгoductѕ. Dietary fiber and protein-սnique cereals will alԝays be a greater choice than processed products. Not only do whole ɡrain foods make you feel more full, nevertheless they lessen youг cholesterol levelѕ at the same time. 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	:<~~math~~> ~~\int_{-1}^1 f(x) (1-x)^\alpha (1+x)^\beta \,\mathrm{d}x~~ <~~/math~~>

	~~where ƒ is a smooth function on [−1~~, 1] and ~~''α''~~, ''β'' > −1. The ~~interval [−1, 1] can be replaced by any other interval by a linear transformation~~. ~~Thus~~, ~~Gauss–Jacobi quadrature can~~ be ~~used to approximate integrals~~ with ~~singularities at the end points~~. [~~[Gauss–Legendre quadrature]~~] is a ~~special case~~ of ~~Gauss–Jacobi quadrature with ''α'' = ''β'' = 0~~. ~~Similarly~~, ~~[[Chebyshev–Gauss quadrature]] arises when one takes ''α'' = ''β'' = ±½. More generally~~, ~~the special case ''α'' = ''β'' turns Jacobi polynomials into [[Gegenbauer polynomials]]~~, in ~~which case~~ the ~~technique is sometimes called '''Gauss–Gegenbauer quadrature''~~'.

	~~Gauss–Jacobi quadrature uses ''ω''(''x'') = (1 − ''x'')~~<~~sup~~>~~''α''~~</~~sup> (1 + ''x'')~~<~~sup~~>~~''β''~~<~~/sup~~> as the ~~weight function~~. ~~The corresponding sequence of~~ [~~[orthogonal polynomials~~]~~] consist~~ of ~~[[Jacobi polynomials]]~~. ~~Thus~~, ~~the Gauss–Jacobi quadrature rule on ''n'' points has the form~~

	:<~~math~~> ~~\int_{-1}^1 f(x) (1-x)^\alpha (1+x)^\beta \,\mathrm{d}x \approx \lambda_1 f(x_1) + \lambda_2 f(x_2) + \cdots + \lambda_n f(x_n),~~ <~~/math~~>

	~~where ''x''~~<~~sub~~>1<~~/sub~~>, …, '~~'x''~~<~~sub~~>~~''n''~~<~~/sub~~> ~~are~~ the ~~roots~~ of the ~~Jacobi polynomial of degree ''n''~~. ~~The weights ''λ''~~<~~sub~~>1<~~/sub~~>, …, '~~'λ''~~<~~sub~~>~~''n''~~<~~/sub~~> ~~are given~~ by ~~the formula~~

	:<~~math~~> ~~\lambda_i = - \frac{2n+\alpha+\beta+2}{n+\alpha+\beta+1} \frac{\Gamma(n+\alpha+1)\Gamma(n+\beta+1)}{\Gamma(n+\alpha+\beta+1)(n+1)!} \frac{2^{\alpha+\beta}}{P'_n(x_i)P_{n+1}(x_i)}~~, <~~/math~~>

	~~where Γ denotes the [[Gamma function]] and ''P''~~<~~sub~~>'~~'n''<~~/~~sub> the Jacobi polynomial of degree ''n''~~.

	~~==References==~~
	* {{Citation \| last1=Rabinowitz \| first1=Philip \| author1-~~link=Philip Rabinowitz (mathematician) \| title=A First Course in Numerical Analysis \| publisher=[[Dover Publications]] \| location=New York \| edition=2nd \| isbn=978~~-0-~~486~~-~~41454~~-~~6 \| year=2001 \| chapter=§4~~.~~8-1: Gauss–Jacobi quadrature}}~~.

	~~==External links==~~
	* [http://people.sc.~~fsu~~.~~edu/~jburkardt/m_src/jacobi_rule/jacobi_rule~~.~~html Jacobi rule] - [[free software]] (Matlab, C++~~, ~~and Fortran)~~ to ~~evaluate integrals by Gauss–Jacobi quadrature rules.~~
	* [http://~~people~~.sc.~~fsu.edu/~jburkardt/cpp_src~~/~~gegenbauer_rule~~/~~gegenbauer_rule~~.~~html Gegenbauer rule~~] - ~~free software (Matlab~~, ~~C++~~, ~~and Fortran) for Gauss–Gegenbauer quadrature~~

	~~{{DEFAULTSORT:Gauss~~-~~Jacobi quadrature}}~~
	~~[[Category:Numerical integration (quadrature)]]~~

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2013-03-19T07:22:31Z

Bot: Migrating 1 interwiki links, now provided by Wikidata on d:q7586536

← Older revision		Revision as of 09:22, 19 March 2013
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	~~Based~~ on the ~~Search Engine Marketing Professionals Organization, SEM practices include paid position, paid addition, and search engine optimization (or SEO). <br><br>~~		In [[numerical analysis]], '''Gauss–Jacobi quadrature''' is a method of [[numerical quadrature]] based on [[Gaussian quadrature]]. Gauss–Jacobi quadrature can be used to approximate integrals of the form

	~~Search~~-~~engines began appearing~~ to ~~aid people find information fast, as~~ the ~~amount~~ of ~~sites online increased within the mid-to-late~~. ~~Search engine marketing experts expanded their offerings to help firms understand and utilize~~ the ~~advertising opportunities offered by search engines~~, ~~and new organizations focusing mainly upon advertising and advertising through search engines appeared. Visit~~ the ~~Search Engine Optimisation Philippines for more information about this~~. <br><br>~~Search engine optimization~~ (~~SEO~~) is the ~~process~~ of ~~improving the capacity and quality~~ of ~~traffic to your website from search engines via natural search results for focused keywords. To check up additional info, consider glancing at:~~ [~~http://smallbusinessinternetmarketinginfo.blogspot.com/2013/08/marketing-for-small-business.html marketing for small business~~]. ~~Generally~~, the ~~earlier a site is presented in~~ the ~~research results or even the larger it "ranks", the more hunters can visit that site. Search Engine Optimisation Delhi organization e~~-~~fuzion may also target different types of search~~, ~~including picture search~~, ~~local search, and industry-specific vertical search engines.~~ <br><br>~~As a marketing strategy for increasing a site"s significance, SEO Delhi organization e-fuzion views how search algorithms work and what people search for. Visit [http:~~/~~/medium.com/@enfusen this site] to read when to deal with this belief. Search Engine Optimization Delhi company e-fuzion efforts may possibly include a site"s coding~~, ~~presentation~~, ~~and design, along with solving issues that can stop se indexing plans from entirely spidering a site. Look at~~ the ~~e-fuzion.com to find out more relating to this~~. <br><br>~~Other~~, ~~more visible efforts can include putting unique content to a site~~, ~~ensuring that content is simply found~~ by ~~search engine robots~~, ~~and making~~ the ~~site more appealing to consumers. Another class of practices, referred to as black hat SEO o-r spamdexing use techniques such as for example link farms~~ and ~~keyword stuffing that have a tendency to harm search engine user experience.~~ <br><br>~~The initialism SEO can also make reference to search engine optimizers, terms adopted by a market of professionals who by workers who conduct SEO Delhi Company~~ of ~~your home, and execute marketing projects on behalf of clients~~. ~~The Search Engine Optimization Delhi company e~~-~~fuzion could possibly offer additional information about that~~.. ~~If you think anything, you will maybe want to read about~~ [~~https~~://~~medium~~.~~com~~/~~online-marketing~~/~~8f36c7b7f5f8 your internet marketing].Enfusen Marketing Services<br>http:~~//~~www~~.~~enfunse.com<br><br>In the event you loved this post~~ and ~~you would love to receive more details relating~~ to [http://~~rhetoricalfurvo37~~.~~shutterfly~~.~~com state health insurance~~] ~~assure visit the internet site.~~		:<math> \int_{-1}^1 f(x) (1-x)^\alpha (1+x)^\beta \,\mathrm{d}x </math>

			where ƒ is a smooth function on [−1, 1] and ''α'', ''β'' > −1. The interval [−1, 1] can be replaced by any other interval by a linear transformation. Thus, Gauss–Jacobi quadrature can be used to approximate integrals with singularities at the end points. [[Gauss–Legendre quadrature]] is a special case of Gauss–Jacobi quadrature with ''α'' = ''β'' = 0. Similarly, [[Chebyshev–Gauss quadrature]] arises when one takes ''α'' = ''β'' = ±½. More generally, the special case ''α'' = ''β'' turns Jacobi polynomials into [[Gegenbauer polynomials]], in which case the technique is sometimes called '''Gauss–Gegenbauer quadrature'''.

			Gauss–Jacobi quadrature uses ''ω''(''x'') = (1 − ''x'')<sup>''α''</sup> (1 + ''x'')<sup>''β''</sup> as the weight function. The corresponding sequence of [[orthogonal polynomials]] consist of [[Jacobi polynomials]]. Thus, the Gauss–Jacobi quadrature rule on ''n'' points has the form

			:<math> \int_{-1}^1 f(x) (1-x)^\alpha (1+x)^\beta \,\mathrm{d}x \approx \lambda_1 f(x_1) + \lambda_2 f(x_2) + \cdots + \lambda_n f(x_n), </math>

			where ''x''<sub>1</sub>, …, ''x''<sub>''n''</sub> are the roots of the Jacobi polynomial of degree ''n''. The weights ''λ''<sub>1</sub>, …, ''λ''<sub>''n''</sub> are given by the formula

			:<math> \lambda_i = - \frac{2n+\alpha+\beta+2}{n+\alpha+\beta+1} \frac{\Gamma(n+\alpha+1)\Gamma(n+\beta+1)}{\Gamma(n+\alpha+\beta+1)(n+1)!} \frac{2^{\alpha+\beta}}{P'_n(x_i)P_{n+1}(x_i)}, </math>

			where Γ denotes the [[Gamma function]] and ''P''<sub>''n''</sub> the Jacobi polynomial of degree ''n''.

			==References==
			* {{Citation \| last1=Rabinowitz \| first1=Philip \| author1-link=Philip Rabinowitz (mathematician) \| title=A First Course in Numerical Analysis \| publisher=[[Dover Publications]] \| location=New York \| edition=2nd \| isbn=978-0-486-41454-6 \| year=2001 \| chapter=§4.8-1: Gauss–Jacobi quadrature}}.

			==External links==
			* [http://people.sc.fsu.edu/~jburkardt/m_src/jacobi_rule/jacobi_rule.html Jacobi rule] - [[free software]] (Matlab, C++, and Fortran) to evaluate integrals by Gauss–Jacobi quadrature rules.
			* [http://people.sc.fsu.edu/~jburkardt/cpp_src/gegenbauer_rule/gegenbauer_rule.html Gegenbauer rule] - free software (Matlab, C++, and Fortran) for Gauss–Gegenbauer quadrature

			{{DEFAULTSORT:Gauss-Jacobi quadrature}}
			[[Category:Numerical integration (quadrature)]]

en>Luizpuodzius at 01:51, 3 June 2012

2012-06-03T01:51:54Z

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