Turbulent diffusion - Revision history

en>Phleg1: /* Eulerian approach */ dab, capitalization

2014-10-23T06:44:41Z

Eulerian approach: dab, capitalization

← Older revision		Revision as of 08:44, 23 October 2014
Line 1:		Line 1:
	~~'''Iterative refinement'''~~ is ~~an [[iterative method]] proposed by James H~~. ~~Wilkinson~~ to ~~improve the accuracy of numerical solutions~~ to ~~[[systems~~ of ~~linear equations]]~~.		by Nas, is very fitting and the film agrees with it. The next step is to visit your Word - Press blog dashboard. These templates are professionally designed and are also Adsense ready. Transforming your designs to Word - Press blogs is not that easy because of the simplified way in creating your very own themes. It's as simple as hiring a Wordpress plugin developer or learning how to create what is needed. <br><br>These folders as well as files have to copied and the saved. If you wish to sell your services or products via internet using your website, you have to put together on the website the facility for trouble-free payment transfer between customers and the company. We also help to integrate various plug-ins to expand the functionalities of the web application. Furthermore, with the launch of Windows 7 Phone is the smart phone market nascent App. This can be done by using a popular layout format and your unique Word - Press design can be achieved in other elements of the blog. <br><br>Usually, Wordpress owners selling the ad space on monthly basis and this means a residual income source. After sending these details, your Word - Press blog will be setup within a few days. Use this section to change many formatting elements. Newer programs allow website owners and internet marketers to automatically and dynamically change words in their content to match the keywords entered by their web visitors in their search queries'a feat that they cannot easily achieve with older software. If you've hosted your Word - Press website on a shared hosting server then it'll be easier for you to confirm the restricted access to your site files. <br><br>In the event you adored this article and you wish to get more info with regards to [http://iz.sa/wordpressdropboxbackup965560 wordpress backup plugin] kindly go to the web-page. The primary differences are in the plugins that I install, as all sites don't need all the normal plugins. I have compiled a few tips on how you can start a food blog and hopefully the following information and tips can help you to get started on your food blogging creative journey. One of the great features of Wordpress is its ability to integrate SEO into your site. It supports backup scheduling and allows you to either download the backup file or email it to you. Make sure you have the latest versions of all your plugins are updated. <br><br>Someone with a basic knowledge of setting up a website should be able to complete the process in a couple of minutes however even basic users should find they are able to complete the installation in around 20 minutes by following the step by step guide online. Here's a list of some exciting Word - Press features that have created waves in the web development industry:. Must being, it's beneficial because I don't know about you, but loading an old website on a mobile, having to scroll down, up, and sideways' I find links being clicked and bounced around like I'm on a freaking trampoline. with posts or testimonials updated as they are uploaded to a particular section of the website. Press CTRL and the numbers one to six to choose your option.

	~~When solving~~ a ~~linear system {{math\|'''~~<~~var~~>Ax<~~/var~~>~~''' {{=}} '''<var>b</var>'''}}~~, ~~due~~ to the ~~presence~~ of ~~[[Round-off error\|rounding error]]s~~, the ~~computed solution {{math\|'''<var>x</var>̂'''}} may sometimes deviate from~~ the ~~exact solution {{math\|'''<var>x</var>'''<sup></sup>}}~~. ~~Starting with {{math\|'''~~<~~var~~>x<~~/var>'''<sub>1</sub> {{=}} '''<var~~>~~x</var>̂'''}}~~, ~~iterative refinement computes~~ a ~~sequence {{math\|{'''<var>x</var>'''<sub>1</sub>~~,~~'''<var>x</var>'''<sub>2</sub>,'''<var>x</var>'''<sub>3</sub>,…}}} which converges~~ to ~~{{math\|'''<var>x</var>'''<sup></sup>}} when certain assumptions are met~~.

	~~==Description==~~
	~~For {{math\|<var>m</var> {{=}} 1,2,…}},~~ the ~~<math>m</math>th iteration of iterative refinement consists of three steps:~~
	~~# Compute the residual<br>{{math\|~~'''~~<var>r</var>'''<sub><var>m</var></sub> {{=}} '''<var>b</var>''' − '''<var>Ax</var>'''<sub><var>m</var></sub>}}~~
	~~# Solve~~ the ~~system~~<br>~~{{math\|'''~~<~~var>Ad</var>'''<sub><var>m</var~~>~~</sub> {{=}} '''<var>r</var>'''<sub><var>m</var></sub>}}~~
	~~# Add~~ the ~~correction<br>{{math\|'''<var>x<~~/~~var>'''<sub><var>m<~~/~~var>+1</sub> {{=}} '''<var>x</var>'''<sub><var>m<~~/~~var></sub> + '''<var>d</var>'''<sub><var>m</var></sub>}}~~

	~~==Error analysis==~~
	~~As a rule of thumb, iterative refinement for [[Gaussian elimination]~~] ~~produces a solution correct~~ to ~~working precision if double~~ the ~~working precision is used~~ in the ~~computation of {{math\|<var>r</var>}}~~, e.g. by using [[Quadruple-precision floating-point format\|quad]] or [[extended precision\|double extended]] precision [[IEEE 754]] [[floating point]], and if {{math\|<var>A</var>}} is not too ill-conditioned (and the ~~iteration~~ and ~~the rate~~ of ~~convergence are determined by~~ the ~~condition number~~ of ~~{{math\|<var>A</var>}})~~.~~<ref>{{cite book\|first=Nicholas \| last=Higham \|title=Accuracy~~ and ~~Stability of Numerical Algorithms (2 ed)\| publisher=SIAM\|year=2002 \| pages=232 }}</ref>~~

	~~More formally, assuming that each solve step is reasonably accurate, i.e~~.~~, in mathematical terms, for every {{math\|<var>m</var>}}, we~~ have

	~~:{{math\|'''<var>A</var>'''('''<var>I</var>''' + <var>'''F'''<sub>m</sub></var>)<var>'''d'''<sub>m</sub></var> {{=}} <var>'''r'''<sub>m</sub></var>}}~~

	~~where {{math\|‖<var>'''F'''<sub>m</sub></var>‖<sub>∞</sub> < 1}},~~ the ~~[[Approximation error\|relative error]] in the <math>m</math>th iterate~~ of ~~iterative refinement satisfies~~

	:<~~math~~>\frac{\lVert\boldsymbol{x}_m-\boldsymbol{x}^\ast\rVert_\infty}{\lVert\boldsymbol{x}^\ast\rVert_\infty}\leq\bigl(\sigma\kappa_\infty(\boldsymbol{A})\epsilon_1\bigr)^m+\mu_1\epsilon_1+\mu_2n\kappa_\infty(\boldsymbol{A})\epsilon_2<~~/math~~>

	~~where~~
	* {{math\|‖·‖<sub>∞</sub>}} denotes the [[Uniform norm\|{{math\|∞}}-norm]] of a ~~vector,~~
	* {{math\|<var>κ</var><sub>∞</sub>('''<var>A</var>''')}} is the ~~{{math\|∞}}-[[condition number]]~~ of ~~{{math\|~~'~~''<var>A</var>'''}}~~,
	* <math>n</math> is the order of {{math\|''~~'<var>A</var>'''}}~~,
	* {{math\|<var>ε</var><sub>1</sub>}} and {{math\|<var>ε</var><sub>2</sub>}} are [[Machine epsilon\|unit round-offs]] of [[Floating point\|floating-point]] arithmetic operations,
	* {{math\|<var>σ</var>}}, ~~{{math\|<var>μ</var><sub>1</sub>}} and {{math\|<var>μ</var><sub>2</sub>}} are constants depending on {{math\|'''<var>A</var>'''}}~~, ~~{{math\|<var>ε</var><sub>1</sub>}}~~ and ~~{{math\|<var>ε</var><sub>2</sub>}}~~
	~~if {{math\|~~'~~''<var>A</var>'''}} is “not too badly conditioned”, which in this context means~~

	~~:{{math\|0 < <var>σκ</var><sub>∞</sub>('''<var>A</var>''')<var>ε</var><sub>1</sub> ≪ 1}}~~

	and ~~implies that {{math\|<var>μ</var><sub>1</sub>}} and {{math\|<var>μ</var><sub>2</sub>}} are of order unity.~~

	~~The distinction of {{math\|<var>ε</var><sub>1</sub>}} and {{math\|<var>ε</var><sub>2</sub>}} is intended to allow mixed-precision evaluation of {{math\|''~~'~~<var>r</var>'''<sub><var>~~m~~</var></sub>}} where intermediate results are computed~~ with ~~unit round-off {{math\|<var>ε</var><sub>2</sub>}} before the final result is rounded (~~or ~~truncated) with unit round-off {{math\|<var>ε</var><sub>1</sub>}}. All other computations~~ are ~~assumed~~ to ~~be carried out with unit round-off {{math\|<var>ε</var><sub>1</sub>}}.~~

	~~==Notes==~~
	~~{{Reflist}}~~

	~~==References==~~
	*{{cite book\|title=Rounding Errors in Algebraic Processes\|first=James H.\|last=Wilkinson\|publisher=[[Prentice Hall]]\|location= Englewood Cliffs, NJ\|year=1963}}
	*{{cite journal\|title=Iterative Refinement in Floating Point\|first=Cleve B.\|last=Moler\|publisher=[[Association for Computing Machinery]]\|location=New York, NY\|date=April 1967\|journal=[[Journal of the ~~ACM]]\|volume=14\|issue=2\|pages=316–321\|url=http://portal~~.~~acm.org/citation.cfm?id=321394\|doi=10.1145/321386~~.~~321394}}~~

	~~{{Numerical linear algebra}}~~

	~~[[Category:Numerical linear algebra]]~~


	~~{{comp-sci-stub}}~~

en>Femkemilene: /* Eulerian approach */ interpunction

2014-02-03T08:30:43Z

Eulerian approach: interpunction

← Older revision		Revision as of 10:30, 3 February 2014
Line 1:		Line 1:
	~~If you are looking for a specific plugin, then you can just search for the name of the plugin~~. ~~Good luck on continue learning how to make a wordpress website. PSD files are incompatible to browsers and are suppose~~ to ~~be converted into wordpress compatible files so that it opens up in browser. Out of~~ the ~~various designs~~ of photography identified these days, sports photography is preferred most, probably for the enjoyment and enjoyment associated with it. In the most current edition you can customize your retailer layout and display hues and fonts similar to ~~your site or blog~~. ~~<br><br>~~		'''Iterative refinement''' is an [[iterative method]] proposed by James H. Wilkinson to improve the accuracy of numerical solutions to [[systems of linear equations]].

	~~Right starting from social media support~~ to ~~search engine optimization~~, ~~such plugins are easily available within~~ the ~~Word - Press open source platform~~. ~~After all~~, ~~Word - Press is free~~, ~~many of the enhancements for Word - Press like themes and plugins~~ are ~~also free~~, ~~and there is plenty~~ of ~~free information online about how to use Word - Press. It allows Word - Press users to easily use HTML5~~ the ~~element enable native video playback within~~ the ~~browser. These four plugins will make this effort easier and~~ the ~~sites run effectively as well as make other widgets added to~~ a ~~site easier~~ to ~~configure~~. ~~Once you've installed the program you can quickly begin~~ by ~~adding content and editing it with features such as bullet pointing~~, ~~text alignment~~ and ~~effects without having to do all the coding yourself.~~ <br><br>~~ve labored so hard to publish~~ and ~~put up on their website. The nominee in each category with~~ the ~~most votes was crowned~~ the ~~2010 Parents Picks Awards WINNER~~ and ~~has been established as the best product~~, ~~tip or place in~~ that ~~category. After age 35~~, ~~18% of pregnancies will end in miscarriage~~. ~~You can allow visitors to post comments, or you can even allow your visitors to register and create their own personal blogs~~. ~~Purchase these from our site~~, ~~or bring your own~~, ~~it doesn't matter~~, we ~~will still give you free installation and configuration.~~ <br><br>~~If you liked this article and you would like to receive a lot more facts with regards to [http:~~//~~www.shortlinki.com~~/~~wordpress_dropbox_backup_462349 backup plugin~~] ~~kindly check out our web-page. The next thing I did after installing Wordpress was to find myself a free good-looking Wordpress-theme offering the functionality I was after. php file~~ in the ~~Word~~ - ~~Press root folder and look for this line~~ (~~line 73 in our example~~)~~:. Websites that do rank highly~~, ~~do so becaue they use keyword~~-~~heavy post titles. So~~, ~~we have to add our social media sharing buttons in website. Word~~ - ~~Press offers constant updated services~~ and ~~products~~, ~~that~~ too, ~~absolutely free~~ of ~~cost~~. <br><br>~~Instead, you can easily just include it with our bodies integration field in e - Panel. s ability~~ to ~~use different themes and skins known as Word - Press Templates or Themes. Useful Plugins Uber~~ - ~~Menu Top Megamenu Now it is the time~~ of ~~sticky Top navbar. Working~~ with ~~a Word~~ - ~~Press blog and~~ the ~~appropriate cost~~-~~free Word~~ - ~~Press theme~~, ~~you can get a professional internet site up and published~~ in ~~no time at all~~. ~~Article Source~~: ~~Hostgator discount coupons for your Wordpress site here~~.		When solving a linear system {{math\|'''<var>Ax</var>''' {{=}} '''<var>b</var>'''}}, due to the presence of [[Round-off error\|rounding error]]s, the computed solution {{math\|'''<var>x</var>̂'''}} may sometimes deviate from the exact solution {{math\|'''<var>x</var>'''<sup></sup>}}. Starting with {{math\|'''<var>x</var>'''<sub>1</sub> {{=}} '''<var>x</var>̂'''}}, iterative refinement computes a sequence {{math\|{'''<var>x</var>'''<sub>1</sub>,'''<var>x</var>'''<sub>2</sub>,'''<var>x</var>'''<sub>3</sub>,…}}} which converges to {{math\|'''<var>x</var>'''<sup></sup>}} when certain assumptions are met.

			==Description==
			For {{math\|<var>m</var> {{=}} 1,2,…}}, the <math>m</math>th iteration of iterative refinement consists of three steps:
			# Compute the residual<br>{{math\|'''<var>r</var>'''<sub><var>m</var></sub> {{=}} '''<var>b</var>''' − '''<var>Ax</var>'''<sub><var>m</var></sub>}}
			# Solve the system<br>{{math\|'''<var>Ad</var>'''<sub><var>m</var></sub> {{=}} '''<var>r</var>'''<sub><var>m</var></sub>}}
			# Add the correction<br>{{math\|'''<var>x</var>'''<sub><var>m</var>+1</sub> {{=}} '''<var>x</var>'''<sub><var>m</var></sub> + '''<var>d</var>'''<sub><var>m</var></sub>}}

			==Error analysis==
			As a rule of thumb, iterative refinement for [[Gaussian elimination]] produces a solution correct to working precision if double the working precision is used in the computation of {{math\|<var>r</var>}}, e.g. by using [[Quadruple-precision floating-point format\|quad]] or [[extended precision\|double extended]] precision [[IEEE 754]] [[floating point]], and if {{math\|<var>A</var>}} is not too ill-conditioned (and the iteration and the rate of convergence are determined by the condition number of {{math\|<var>A</var>}}).<ref>{{cite book\|first=Nicholas \| last=Higham \|title=Accuracy and Stability of Numerical Algorithms (2 ed)\| publisher=SIAM\|year=2002 \| pages=232 }}</ref>

			More formally, assuming that each solve step is reasonably accurate, i.e., in mathematical terms, for every {{math\|<var>m</var>}}, we have

			:{{math\|'''<var>A</var>'''('''<var>I</var>''' + <var>'''F'''<sub>m</sub></var>)<var>'''d'''<sub>m</sub></var> {{=}} <var>'''r'''<sub>m</sub></var>}}

			where {{math\|‖<var>'''F'''<sub>m</sub></var>‖<sub>∞</sub> < 1}}, the [[Approximation error\|relative error]] in the <math>m</math>th iterate of iterative refinement satisfies

			:<math>\frac{\lVert\boldsymbol{x}_m-\boldsymbol{x}^\ast\rVert_\infty}{\lVert\boldsymbol{x}^\ast\rVert_\infty}\leq\bigl(\sigma\kappa_\infty(\boldsymbol{A})\epsilon_1\bigr)^m+\mu_1\epsilon_1+\mu_2n\kappa_\infty(\boldsymbol{A})\epsilon_2</math>

			where
			* {{math\|‖·‖<sub>∞</sub>}} denotes the [[Uniform norm\|{{math\|∞}}-norm]] of a vector,
			* {{math\|<var>κ</var><sub>∞</sub>('''<var>A</var>''')}} is the {{math\|∞}}-[[condition number]] of {{math\|'''<var>A</var>'''}},
			* <math>n</math> is the order of {{math\|'''<var>A</var>'''}},
			* {{math\|<var>ε</var><sub>1</sub>}} and {{math\|<var>ε</var><sub>2</sub>}} are [[Machine epsilon\|unit round-offs]] of [[Floating point\|floating-point]] arithmetic operations,
			* {{math\|<var>σ</var>}}, {{math\|<var>μ</var><sub>1</sub>}} and {{math\|<var>μ</var><sub>2</sub>}} are constants depending on {{math\|'''<var>A</var>'''}}, {{math\|<var>ε</var><sub>1</sub>}} and {{math\|<var>ε</var><sub>2</sub>}}
			if {{math\|'''<var>A</var>'''}} is “not too badly conditioned”, which in this context means

			:{{math\|0 < <var>σκ</var><sub>∞</sub>('''<var>A</var>''')<var>ε</var><sub>1</sub> ≪ 1}}

			and implies that {{math\|<var>μ</var><sub>1</sub>}} and {{math\|<var>μ</var><sub>2</sub>}} are of order unity.

			The distinction of {{math\|<var>ε</var><sub>1</sub>}} and {{math\|<var>ε</var><sub>2</sub>}} is intended to allow mixed-precision evaluation of {{math\|'''<var>r</var>'''<sub><var>m</var></sub>}} where intermediate results are computed with unit round-off {{math\|<var>ε</var><sub>2</sub>}} before the final result is rounded (or truncated) with unit round-off {{math\|<var>ε</var><sub>1</sub>}}. All other computations are assumed to be carried out with unit round-off {{math\|<var>ε</var><sub>1</sub>}}.

			==Notes==
			{{Reflist}}

			==References==
			*{{cite book\|title=Rounding Errors in Algebraic Processes\|first=James H.\|last=Wilkinson\|publisher=[[Prentice Hall]]\|location= Englewood Cliffs, NJ\|year=1963}}
			*{{cite journal\|title=Iterative Refinement in Floating Point\|first=Cleve B.\|last=Moler\|publisher=[[Association for Computing Machinery]]\|location=New York, NY\|date=April 1967\|journal=[[Journal of the ACM]]\|volume=14\|issue=2\|pages=316–321\|url=http://portal.acm.org/citation.cfm?id=321394\|doi=10.1145/321386.321394}}

			{{Numerical linear algebra}}

			[[Category:Numerical linear algebra]]


			{{comp-sci-stub}}

en>Helpful Pixie Bot: ISBNs (Build KE)

2012-05-11T07:15:24Z

ISBNs (Build KE)

New page

If you are looking for a specific plugin, then you can just search for the name of the plugin. Good luck on continue learning how to make a wordpress website. PSD files are incompatible to browsers and are suppose to be converted into wordpress compatible files so that it opens up in browser. Out of the various designs of photography identified these days, sports photography is preferred most, probably for the enjoyment and enjoyment associated with it. In the most current edition you can customize your retailer layout and display hues and fonts similar to your site or blog. 

Right starting from social media support to search engine optimization, such plugins are easily available within the Word - Press open source platform. After all, Word - Press is free, many of the enhancements for Word - Press like themes and plugins are also free, and there is plenty of free information online about how to use Word - Press. It allows Word - Press users to easily use HTML5 the element enable native video playback within the browser. These four plugins will make this effort easier and the sites run effectively as well as make other widgets added to a site easier to configure. Once you've installed the program you can quickly begin by adding content and editing it with features such as bullet pointing, text alignment and effects without having to do all the coding yourself. ve labored so hard to publish and put up on their website. The nominee in each category with the most votes was crowned the 2010 Parents Picks Awards WINNER and has been established as the best product, tip or place in that category. After age 35, 18% of pregnancies will end in miscarriage. You can allow visitors to post comments, or you can even allow your visitors to register and create their own personal blogs. Purchase these from our site, or bring your own, it doesn't matter, we will still give you free installation and configuration. If you liked this article and you would like to receive a lot more facts with regards to [http://www.shortlinki.com/wordpress_dropbox_backup_462349 backup plugin] kindly check out our web-page. The next thing I did after installing Wordpress was to find myself a free good-looking Wordpress-theme offering the functionality I was after. php file in the Word - Press root folder and look for this line (line 73 in our example):. Websites that do rank highly, do so becaue they use keyword-heavy post titles. So, we have to add our social media sharing buttons in website. Word - Press offers constant updated services and products, that too, absolutely free of cost. Instead, you can easily just include it with our bodies integration field in e - Panel. s ability to use different themes and skins known as Word - Press Templates or Themes. Useful Plugins Uber - Menu Top Megamenu Now it is the time of sticky Top navbar. Working with a Word - Press blog and the appropriate cost-free Word - Press theme, you can get a professional internet site up and published in no time at all. Article Source: Hostgator discount coupons for your Wordpress site here.

← Older revision		Revision as of 08:44, 23 October 2014
Line 1:		Line 1:
	~~'''Iterative refinement'''~~ is ~~an [[iterative method]] proposed by James H~~. ~~Wilkinson~~ to ~~improve the accuracy of numerical solutions~~ to ~~[[systems~~ of ~~linear equations]]~~.		by Nas, is very fitting and the film agrees with it. The next step is to visit your Word - Press blog dashboard. These templates are professionally designed and are also Adsense ready. Transforming your designs to Word - Press blogs is not that easy because of the simplified way in creating your very own themes. It's as simple as hiring a Wordpress plugin developer or learning how to create what is needed. <br><br>These folders as well as files have to copied and the saved. If you wish to sell your services or products via internet using your website, you have to put together on the website the facility for trouble-free payment transfer between customers and the company. We also help to integrate various plug-ins to expand the functionalities of the web application. Furthermore, with the launch of Windows 7 Phone is the smart phone market nascent App. This can be done by using a popular layout format and your unique Word - Press design can be achieved in other elements of the blog. <br><br>Usually, Wordpress owners selling the ad space on monthly basis and this means a residual income source. After sending these details, your Word - Press blog will be setup within a few days. Use this section to change many formatting elements. Newer programs allow website owners and internet marketers to automatically and dynamically change words in their content to match the keywords entered by their web visitors in their search queries'a feat that they cannot easily achieve with older software. If you've hosted your Word - Press website on a shared hosting server then it'll be easier for you to confirm the restricted access to your site files. <br><br>In the event you adored this article and you wish to get more info with regards to [http://iz.sa/wordpressdropboxbackup965560 wordpress backup plugin] kindly go to the web-page. The primary differences are in the plugins that I install, as all sites don't need all the normal plugins. I have compiled a few tips on how you can start a food blog and hopefully the following information and tips can help you to get started on your food blogging creative journey. One of the great features of Wordpress is its ability to integrate SEO into your site. It supports backup scheduling and allows you to either download the backup file or email it to you. Make sure you have the latest versions of all your plugins are updated. <br><br>Someone with a basic knowledge of setting up a website should be able to complete the process in a couple of minutes however even basic users should find they are able to complete the installation in around 20 minutes by following the step by step guide online. Here's a list of some exciting Word - Press features that have created waves in the web development industry:. Must being, it's beneficial because I don't know about you, but loading an old website on a mobile, having to scroll down, up, and sideways' I find links being clicked and bounced around like I'm on a freaking trampoline. with posts or testimonials updated as they are uploaded to a particular section of the website. Press CTRL and the numbers one to six to choose your option.

	~~When solving~~ a ~~linear system {{math\|'''~~<~~var~~>Ax<~~/var~~>~~''' {{=}} '''<var>b</var>'''}}~~, ~~due~~ to the ~~presence~~ of ~~[[Round-off error\|rounding error]]s~~, the ~~computed solution {{math\|'''<var>x</var>̂'''}} may sometimes deviate from~~ the ~~exact solution {{math\|'''<var>x</var>'''<sup></sup>}}~~. ~~Starting with {{math\|'''~~<~~var~~>x<~~/var>'''<sub>1</sub> {{=}} '''<var~~>~~x</var>̂'''}}~~, ~~iterative refinement computes~~ a ~~sequence {{math\|{'''<var>x</var>'''<sub>1</sub>~~,~~'''<var>x</var>'''<sub>2</sub>,'''<var>x</var>'''<sub>3</sub>,…}}} which converges~~ to ~~{{math\|'''<var>x</var>'''<sup></sup>}} when certain assumptions are met~~.

	~~==Description==~~
	~~For {{math\|<var>m</var> {{=}} 1,2,…}},~~ the ~~<math>m</math>th iteration of iterative refinement consists of three steps:~~
	~~# Compute the residual<br>{{math\|~~'''~~<var>r</var>'''<sub><var>m</var></sub> {{=}} '''<var>b</var>''' − '''<var>Ax</var>'''<sub><var>m</var></sub>}}~~
	~~# Solve~~ the ~~system~~<br>~~{{math\|'''~~<~~var>Ad</var>'''<sub><var>m</var~~>~~</sub> {{=}} '''<var>r</var>'''<sub><var>m</var></sub>}}~~
	~~# Add~~ the ~~correction<br>{{math\|'''<var>x<~~/~~var>'''<sub><var>m<~~/~~var>+1</sub> {{=}} '''<var>x</var>'''<sub><var>m<~~/~~var></sub> + '''<var>d</var>'''<sub><var>m</var></sub>}}~~

	~~==Error analysis==~~
	~~As a rule of thumb, iterative refinement for [[Gaussian elimination]~~] ~~produces a solution correct~~ to ~~working precision if double~~ the ~~working precision is used~~ in the ~~computation of {{math\|<var>r</var>}}~~, e.g. by using [[Quadruple-precision floating-point format\|quad]] or [[extended precision\|double extended]] precision [[IEEE 754]] [[floating point]], and if {{math\|<var>A</var>}} is not too ill-conditioned (and the ~~iteration~~ and ~~the rate~~ of ~~convergence are determined by~~ the ~~condition number~~ of ~~{{math\|<var>A</var>}})~~.~~<ref>{{cite book\|first=Nicholas \| last=Higham \|title=Accuracy~~ and ~~Stability of Numerical Algorithms (2 ed)\| publisher=SIAM\|year=2002 \| pages=232 }}</ref>~~

	~~More formally, assuming that each solve step is reasonably accurate, i.e~~.~~, in mathematical terms, for every {{math\|<var>m</var>}}, we~~ have

	~~:{{math\|'''<var>A</var>'''('''<var>I</var>''' + <var>'''F'''<sub>m</sub></var>)<var>'''d'''<sub>m</sub></var> {{=}} <var>'''r'''<sub>m</sub></var>}}~~

	~~where {{math\|‖<var>'''F'''<sub>m</sub></var>‖<sub>∞</sub> < 1}},~~ the ~~[[Approximation error\|relative error]] in the <math>m</math>th iterate~~ of ~~iterative refinement satisfies~~

	:<~~math~~>\frac{\lVert\boldsymbol{x}_m-\boldsymbol{x}^\ast\rVert_\infty}{\lVert\boldsymbol{x}^\ast\rVert_\infty}\leq\bigl(\sigma\kappa_\infty(\boldsymbol{A})\epsilon_1\bigr)^m+\mu_1\epsilon_1+\mu_2n\kappa_\infty(\boldsymbol{A})\epsilon_2<~~/math~~>

	~~where~~
	* {{math\|‖·‖<sub>∞</sub>}} denotes the [[Uniform norm\|{{math\|∞}}-norm]] of a ~~vector,~~
	* {{math\|<var>κ</var><sub>∞</sub>('''<var>A</var>''')}} is the ~~{{math\|∞}}-[[condition number]]~~ of ~~{{math\|~~'~~''<var>A</var>'''}}~~,
	* <math>n</math> is the order of {{math\|''~~'<var>A</var>'''}}~~,
	* {{math\|<var>ε</var><sub>1</sub>}} and {{math\|<var>ε</var><sub>2</sub>}} are [[Machine epsilon\|unit round-offs]] of [[Floating point\|floating-point]] arithmetic operations,
	* {{math\|<var>σ</var>}}, ~~{{math\|<var>μ</var><sub>1</sub>}} and {{math\|<var>μ</var><sub>2</sub>}} are constants depending on {{math\|'''<var>A</var>'''}}~~, ~~{{math\|<var>ε</var><sub>1</sub>}}~~ and ~~{{math\|<var>ε</var><sub>2</sub>}}~~
	~~if {{math\|~~'~~''<var>A</var>'''}} is “not too badly conditioned”, which in this context means~~

	~~:{{math\|0 < <var>σκ</var><sub>∞</sub>('''<var>A</var>''')<var>ε</var><sub>1</sub> ≪ 1}}~~

	and ~~implies that {{math\|<var>μ</var><sub>1</sub>}} and {{math\|<var>μ</var><sub>2</sub>}} are of order unity.~~

	~~The distinction of {{math\|<var>ε</var><sub>1</sub>}} and {{math\|<var>ε</var><sub>2</sub>}} is intended to allow mixed-precision evaluation of {{math\|''~~'~~<var>r</var>'''<sub><var>~~m~~</var></sub>}} where intermediate results are computed~~ with ~~unit round-off {{math\|<var>ε</var><sub>2</sub>}} before the final result is rounded (~~or ~~truncated) with unit round-off {{math\|<var>ε</var><sub>1</sub>}}. All other computations~~ are ~~assumed~~ to ~~be carried out with unit round-off {{math\|<var>ε</var><sub>1</sub>}}.~~

	~~==Notes==~~
	~~{{Reflist}}~~

	~~==References==~~
	*{{cite book\|title=Rounding Errors in Algebraic Processes\|first=James H.\|last=Wilkinson\|publisher=[[Prentice Hall]]\|location= Englewood Cliffs, NJ\|year=1963}}
	*{{cite journal\|title=Iterative Refinement in Floating Point\|first=Cleve B.\|last=Moler\|publisher=[[Association for Computing Machinery]]\|location=New York, NY\|date=April 1967\|journal=[[Journal of the ~~ACM]]\|volume=14\|issue=2\|pages=316–321\|url=http://portal~~.~~acm.org/citation.cfm?id=321394\|doi=10.1145/321386~~.~~321394}}~~

	~~{{Numerical linear algebra}}~~

	~~[[Category:Numerical linear algebra]]~~


	~~{{comp-sci-stub}}~~

← Older revision		Revision as of 10:30, 3 February 2014
Line 1:		Line 1:
	~~If you are looking for a specific plugin, then you can just search for the name of the plugin~~. ~~Good luck on continue learning how to make a wordpress website. PSD files are incompatible to browsers and are suppose~~ to ~~be converted into wordpress compatible files so that it opens up in browser. Out of~~ the ~~various designs~~ of photography identified these days, sports photography is preferred most, probably for the enjoyment and enjoyment associated with it. In the most current edition you can customize your retailer layout and display hues and fonts similar to ~~your site or blog~~. ~~<br><br>~~		'''Iterative refinement''' is an [[iterative method]] proposed by James H. Wilkinson to improve the accuracy of numerical solutions to [[systems of linear equations]].

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			==Description==
			For {{math\|<var>m</var> {{=}} 1,2,…}}, the <math>m</math>th iteration of iterative refinement consists of three steps:
			# Compute the residual<br>{{math\|'''<var>r</var>'''<sub><var>m</var></sub> {{=}} '''<var>b</var>''' − '''<var>Ax</var>'''<sub><var>m</var></sub>}}
			# Solve the system<br>{{math\|'''<var>Ad</var>'''<sub><var>m</var></sub> {{=}} '''<var>r</var>'''<sub><var>m</var></sub>}}
			# Add the correction<br>{{math\|'''<var>x</var>'''<sub><var>m</var>+1</sub> {{=}} '''<var>x</var>'''<sub><var>m</var></sub> + '''<var>d</var>'''<sub><var>m</var></sub>}}

			==Error analysis==
			As a rule of thumb, iterative refinement for [[Gaussian elimination]] produces a solution correct to working precision if double the working precision is used in the computation of {{math\|<var>r</var>}}, e.g. by using [[Quadruple-precision floating-point format\|quad]] or [[extended precision\|double extended]] precision [[IEEE 754]] [[floating point]], and if {{math\|<var>A</var>}} is not too ill-conditioned (and the iteration and the rate of convergence are determined by the condition number of {{math\|<var>A</var>}}).<ref>{{cite book\|first=Nicholas \| last=Higham \|title=Accuracy and Stability of Numerical Algorithms (2 ed)\| publisher=SIAM\|year=2002 \| pages=232 }}</ref>

			More formally, assuming that each solve step is reasonably accurate, i.e., in mathematical terms, for every {{math\|<var>m</var>}}, we have

			:{{math\|'''<var>A</var>'''('''<var>I</var>''' + <var>'''F'''<sub>m</sub></var>)<var>'''d'''<sub>m</sub></var> {{=}} <var>'''r'''<sub>m</sub></var>}}

			where {{math\|‖<var>'''F'''<sub>m</sub></var>‖<sub>∞</sub> < 1}}, the [[Approximation error\|relative error]] in the <math>m</math>th iterate of iterative refinement satisfies

			:<math>\frac{\lVert\boldsymbol{x}_m-\boldsymbol{x}^\ast\rVert_\infty}{\lVert\boldsymbol{x}^\ast\rVert_\infty}\leq\bigl(\sigma\kappa_\infty(\boldsymbol{A})\epsilon_1\bigr)^m+\mu_1\epsilon_1+\mu_2n\kappa_\infty(\boldsymbol{A})\epsilon_2</math>

			where
			* {{math\|‖·‖<sub>∞</sub>}} denotes the [[Uniform norm\|{{math\|∞}}-norm]] of a vector,
			* {{math\|<var>κ</var><sub>∞</sub>('''<var>A</var>''')}} is the {{math\|∞}}-[[condition number]] of {{math\|'''<var>A</var>'''}},
			* <math>n</math> is the order of {{math\|'''<var>A</var>'''}},
			* {{math\|<var>ε</var><sub>1</sub>}} and {{math\|<var>ε</var><sub>2</sub>}} are [[Machine epsilon\|unit round-offs]] of [[Floating point\|floating-point]] arithmetic operations,
			* {{math\|<var>σ</var>}}, {{math\|<var>μ</var><sub>1</sub>}} and {{math\|<var>μ</var><sub>2</sub>}} are constants depending on {{math\|'''<var>A</var>'''}}, {{math\|<var>ε</var><sub>1</sub>}} and {{math\|<var>ε</var><sub>2</sub>}}
			if {{math\|'''<var>A</var>'''}} is “not too badly conditioned”, which in this context means

			:{{math\|0 < <var>σκ</var><sub>∞</sub>('''<var>A</var>''')<var>ε</var><sub>1</sub> ≪ 1}}

			and implies that {{math\|<var>μ</var><sub>1</sub>}} and {{math\|<var>μ</var><sub>2</sub>}} are of order unity.

			The distinction of {{math\|<var>ε</var><sub>1</sub>}} and {{math\|<var>ε</var><sub>2</sub>}} is intended to allow mixed-precision evaluation of {{math\|'''<var>r</var>'''<sub><var>m</var></sub>}} where intermediate results are computed with unit round-off {{math\|<var>ε</var><sub>2</sub>}} before the final result is rounded (or truncated) with unit round-off {{math\|<var>ε</var><sub>1</sub>}}. All other computations are assumed to be carried out with unit round-off {{math\|<var>ε</var><sub>1</sub>}}.

			==Notes==
			{{Reflist}}

			==References==
			*{{cite book\|title=Rounding Errors in Algebraic Processes\|first=James H.\|last=Wilkinson\|publisher=[[Prentice Hall]]\|location= Englewood Cliffs, NJ\|year=1963}}
			*{{cite journal\|title=Iterative Refinement in Floating Point\|first=Cleve B.\|last=Moler\|publisher=[[Association for Computing Machinery]]\|location=New York, NY\|date=April 1967\|journal=[[Journal of the ACM]]\|volume=14\|issue=2\|pages=316–321\|url=http://portal.acm.org/citation.cfm?id=321394\|doi=10.1145/321386.321394}}

			{{Numerical linear algebra}}

			[[Category:Numerical linear algebra]]


			{{comp-sci-stub}}