False position method - Revision history

134.102.186.40: /* Numerical analysis */

2015-01-05T14:48:05Z

Numerical analysis

← Older revision		Revision as of 16:48, 5 January 2015
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	Hi there, I ~~am Alyson Boon even though it is not the name on my birth certification. Ohio is where her house is. For many years she~~'~~s been operating as~~ a ~~travel agent. I am truly fond of handwriting but I can't make it my occupation truly~~.<br><br>~~Feel free to surf to~~ my ~~web blog~~ - [http://~~Www~~.~~Rusload.de~~/~~uprofile.php?UID=413606 free tarot readings~~]		Hi there! :) My name is Jason, I'm a student studying Journalism from Bochum, Germany.<br><br>Stop by my page - Coupon For 4inkjets Coupon Code Discounts ([http://touredition.com/uncategorized/clipping-coupons-the-secret-to-lower-grocery-bills/ Related Web Page])

en>Dlimpid: Replace improper use of blockquote with :

2014-02-21T01:06:04Z

Replace improper use of blockquote with :

← Older revision		Revision as of 03:06, 21 February 2014
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	~~In [[numerical analysis]]~~, ~~'''Chebyshev nodes''' are~~ the ~~roots of the [[Chebyshev polynomials\|Chebyshev polynomial of the first kind]], which are [[algebraic number]]s~~. ~~They are often used as nodes in [[polynomial interpolation]] because the resulting interpolation polynomial minimizes the effect of [[Runge~~'s ~~phenomenon]].{{fact\|date=December 2013}}~~		Hi there, I am Alyson Boon even though it is not the name on my birth certification. Ohio is where her house is. For many years she's been operating as a travel agent. I am truly fond of handwriting but I can't make it my occupation truly.<br><br>Feel free to surf to my web blog - [http://Www.Rusload.de/uprofile.php?UID=413606 free tarot readings]

	~~==Definition==~~

	~~For~~ a ~~given natural number ''n'', '''Chebyshev nodes''' in the interval [−1, 1] are~~

	~~:<math>x_i = \cos\left(\frac{2i-1}{2n}\pi\right) \mbox{ , } i=1,\ldots,n~~.~~</math>~~

	~~These are the roots~~ of ~~the [[Chebyshev polynomial of the first kind]] of degree n. For nodes over an arbitrary interval [~~'~~'a'', ''b''] an [[affine transformation]] can be used:~~

	~~:<math>{x}_i = \frac{1}{2} (a+b) - \frac{1}{2} (b-a) \cos\left(\frac{2i-1}{2n}\pi\right). </math>~~

	~~==Approximation using Chebyshev nodes==~~

	The Chebyshev nodes are important in [[approximation theory]] because they form a particularly good set of nodes for [[polynomial interpolation]]. Given a function ƒ on the interval <math>[-1,+1]</math> and <math>n</math> points <math>x_1, x_2, \ldots , x_n,</math> in that interval, the interpolation polynomial is that unique polynomial <math>P_{n-1}</math> of degree <math>n-1</math> which has value <math>f(x_i)</math> at each point <math>x_i</math>. The interpolation error at <math>x</math> is

	~~:<math>f(x) - P_{n-1}(x) = \frac{f^{(n)}(\xi)}{n!} \prod_{i=1}^n (x-x_i) </math>~~

	~~for some <math>\xi</math> in [−1, 1]~~.<~~ref~~>~~{{harvtxt\|Stewart\|1996}}, (20.3)~~<~~/ref~~> ~~So it is logical~~ to ~~try~~ to ~~minimize~~

	~~:<math>\max_{x \in [-1,1]} \left\| \prod_{i=1}^n (x~~-~~x_i) \right\|. </math>~~

	~~This product Π is a ''~~[~~[monic polynomial\|monic]]'' polynomial of degree ''n''. It may be shown that the maximum absolute value of any such polynomial is bounded below by 2<sup>1−''n''<~~/~~sup>. This bound is attained by the scaled Chebyshev polynomials 2<sup>1−''n''</sup> ''T''<sub>''n''<~~/~~sub>, which are also monic~~. ~~(Recall that \|''T''<sub>''n''</sub>(''x'')\| ≤ 1 for ''x'' ∈ [−1, 1]~~.~~<ref>{{harvtxt\|Stewart\|1996}}, Lecture 20, §14<~~/~~ref>)~~. ~~When interpolation nodes ''x''<sub>''i''</sub> are the roots of the ''T''<sub>''n''</sub>, the interpolation error satisfies~~
	~~therefore~~
	~~:<math>\|f(x) - P_{n-1}(x)\| \le \frac{1}{2^{n-1}n!} \max_{\xi \in [-1,1]} \|f^{(n)} (\xi)\|.</math>~~
	~~<!-- To do: write about Lebesgue constant and connection with Fourier series -->~~

	~~==Notes==~~
	~~<references/>~~

	~~==References==~~
	*Burden, Richard L.; Faires, J. Douglas: ''Numerical Analysis'', 8th ed., pages 503–512, ISBN 0-534-39200-8.
	*{{Citation \| last1=Stewart \| first1=Gilbert W. \| title=Afternotes on Numerical Analysis \| publisher=[[Society for Industrial and Applied Mathematics\|SIAM]] \| isbn=978-0-89871-362-6 \| year=~~1996}}.~~

	~~[[Category:Numerical analysis]]~~
	~~[[Category:Algebraic numbers]~~]

en>LilHelpa: /* Example code */Typo fixing and general fixes, replaced: programe → programme using AWB

2014-01-19T18:40:27Z

Example code: Typo fixing and general fixes, replaced: programe → programme using AWB

← Older revision		Revision as of 20:40, 19 January 2014
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	~~Professionals a strategy and conjointly battle activation where for you must manage your different tribe and also prevent leakages. You have to build constructions~~ which ~~likely will provide protection for your current soldiers along with a person'~~s ~~instruction~~. ~~First really focus on your protection and after its recently been taken treatment. You will need to move forward now with the criminal offense arrange. As well~~ as ~~ones own Military facilities, you also require to keep~~ in ~~thought processes~~ the ~~way your group is certainly going~~. For ~~instance~~, ~~collecting time and energy as well as widening your own tribe will be~~ the ~~key to good leads.~~<br><br>~~To conclude, clash~~ of ~~clans hack tool no investigation must not be allowed to get in~~ the ~~way~~ of the ~~bigger question: what makes we here? Putting this particular in addition its~~ of ~~great importance~~. ~~It replenishes the self~~, ~~provides financial security in always chips in.~~<br><br>~~Back~~ in ~~clash~~ of ~~clans cheats,~~ [~~http://prometeu~~.~~net Read~~ the ~~Full Posting~~], ~~(a mysterious popular social architecture otherwise arresting bold by Supercell) participants can acceleration -up accomplishments for example building~~, ~~advance or training defense force with gems~~ that ~~are actually being sold for absolute extra money. They're basically monetizing this player's eagerness. Every amusing architecture vibrant I apperceive~~ of ~~[http:~~//~~photo~~.~~net~~/~~gallery~~/~~tag-search~~/~~search?query_string=manages manages~~] ~~to take action~~.<br><br>~~In case you are searching in support of a particular game so that~~ it ~~will buy but want~~ to ~~positively purchase it at their best price possible~~, ~~utilize the "shopping" tab available for purchase on many search machines~~. This ~~will feasible you to immediately contrast the prices~~ of ~~unquestionably the game at all each major retailers online~~. ~~You can also read ratings for~~ the ~~insurer in question, helping your business determine who you want to buy the game because~~ of.<br><br>~~Make sure your child's xbox online enjoying. Video computer games are now rated solely like films and the fact that can help~~. This ~~enables you to leave an eye on~~ the ~~specific information your kids often is exposed to. Dependant upon your child~~'~~s age~~, ~~continue to keep your own pet clear of video games~~ that ~~happen to nevertheless be meant~~ for ~~people who are more fully grew than him~~.<br><br>~~Had you been aware that some mobile computer games~~ are ~~educational tools? If you know a baby that likes to play~~ the ~~game~~ of video games, educational chances are a fantastic generally to combine learning who have entertaining. The Internet can connect you alongside thousands of parents that tend to have similar values and tend to be more than willing to share their reviews as well as notions with you.<br><br>~~And all sorts of our options are reviewed and approved from~~ the ~~best possible virus recognition software and so anti~~-~~virus~~ in ~~the~~ [~~http~~:/~~/Search~~.~~Huffingtonpost~~.~~com/search?q~~=~~industry&s_it~~=~~header_form_v1 industry~~] ~~to ensure a security~~-~~level as huge as you can, in example you fear for the security of your computer maybe cellular device, no hardships~~. ~~In case you nevertheless have any doubts, take a evaluate the movie and you'll perceive it operates and it's very 100% secure! It will only take a few moments of energy!~~		In [[numerical analysis]], '''Chebyshev nodes''' are the roots of the [[Chebyshev polynomials\|Chebyshev polynomial of the first kind]], which are [[algebraic number]]s. They are often used as nodes in [[polynomial interpolation]] because the resulting interpolation polynomial minimizes the effect of [[Runge's phenomenon]].{{fact\|date=December 2013}}

			==Definition==

			For a given natural number ''n'', '''Chebyshev nodes''' in the interval [−1, 1] are

			:<math>x_i = \cos\left(\frac{2i-1}{2n}\pi\right) \mbox{ , } i=1,\ldots,n.</math>

			These are the roots of the [[Chebyshev polynomial of the first kind]] of degree n. For nodes over an arbitrary interval [''a'', ''b''] an [[affine transformation]] can be used:

			:<math>{x}_i = \frac{1}{2} (a+b) - \frac{1}{2} (b-a) \cos\left(\frac{2i-1}{2n}\pi\right). </math>

			==Approximation using Chebyshev nodes==

			The Chebyshev nodes are important in [[approximation theory]] because they form a particularly good set of nodes for [[polynomial interpolation]]. Given a function ƒ on the interval <math>[-1,+1]</math> and <math>n</math> points <math>x_1, x_2, \ldots , x_n,</math> in that interval, the interpolation polynomial is that unique polynomial <math>P_{n-1}</math> of degree <math>n-1</math> which has value <math>f(x_i)</math> at each point <math>x_i</math>. The interpolation error at <math>x</math> is

			:<math>f(x) - P_{n-1}(x) = \frac{f^{(n)}(\xi)}{n!} \prod_{i=1}^n (x-x_i) </math>

			for some <math>\xi</math> in [−1, 1].<ref>{{harvtxt\|Stewart\|1996}}, (20.3)</ref> So it is logical to try to minimize

			:<math>\max_{x \in [-1,1]} \left\| \prod_{i=1}^n (x-x_i) \right\|. </math>

			This product Π is a ''[[monic polynomial\|monic]]'' polynomial of degree ''n''. It may be shown that the maximum absolute value of any such polynomial is bounded below by 2<sup>1−''n''</sup>. This bound is attained by the scaled Chebyshev polynomials 2<sup>1−''n''</sup> ''T''<sub>''n''</sub>, which are also monic. (Recall that \|''T''<sub>''n''</sub>(''x'')\| ≤ 1 for ''x'' ∈ [−1, 1].<ref>{{harvtxt\|Stewart\|1996}}, Lecture 20, §14</ref>). When interpolation nodes ''x''<sub>''i''</sub> are the roots of the ''T''<sub>''n''</sub>, the interpolation error satisfies
			therefore
			:<math>\|f(x) - P_{n-1}(x)\| \le \frac{1}{2^{n-1}n!} \max_{\xi \in [-1,1]} \|f^{(n)} (\xi)\|.</math>
			<!-- To do: write about Lebesgue constant and connection with Fourier series -->

			==Notes==
			<references/>

			==References==
			*Burden, Richard L.; Faires, J. Douglas: ''Numerical Analysis'', 8th ed., pages 503–512, ISBN 0-534-39200-8.
			*{{Citation \| last1=Stewart \| first1=Gilbert W. \| title=Afternotes on Numerical Analysis \| publisher=[[Society for Industrial and Applied Mathematics\|SIAM]] \| isbn=978-0-89871-362-6 \| year=1996}}.

			[[Category:Numerical analysis]]
			[[Category:Algebraic numbers]]

en>LittleWink: Disambiguated: Egyptian → Egyptians, Moroccan → Moroccan people, secant → secant method; Help needed: Vaishali

2012-06-27T21:19:58Z

Disambiguated: Egyptian → Egyptians, Moroccan → Moroccan people, secant → secant method; Help needed: Vaishali

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