Cubic graph - Revision history

en>Tomruen at 06:31, 11 January 2015

2015-01-11T06:31:44Z

← Older revision		Revision as of 08:31, 11 January 2015
Line 1:		Line 1:
	~~In [[number theory]]~~, the ~~'''integer square root''' (isqrt) of a [[positive integer]] ''n''~~ is ~~the positive integer ''m'' which is the [[floor~~ and ~~ceiling functions\|greatest integer less than or equal]] to the~~ [~~[square root]] of ''n'',~~		I woke up last week and noticed - Today I have been single for some time and following much bullying from pals I today locate myself opted for online dating. They guaranteed me that there are plenty of pleasant, regular and enjoyable folks to fulfill, therefore the pitch is gone by here!<br>My buddies and fam are magnificent and hanging out together at [http://Www.Bargigabytes.com/ bar gigabytes] or dishes is constantly a must. I haven't ever [http://lukebryantickets.iczmpbangladesh.org tickets to luke bryan] been into night clubs as I find that one can not have a [http://www.sharkbayte.com/keyword/decent+dialog decent dialog] using the sound. I also have 2 quite cute and undoubtedly cheeky canines who are consistently enthusiastic to meet new individuals.<br>I try and maintain as toned as possible being at the fitness center several times weekly. I appreciate [http://lukebryantickets.flicense.com luke bryan concert tickets cheap] my sports and attempt to perform or see as many a potential. Being winter I shall often at Hawthorn fits. Note: I have observed the carnage of wrestling suits at stocktake revenue, In case that you really [http://lukebryantickets.lazintechnologies.com who has luke bryan toured with] contemplated buying a sport I really don't brain.<br><br>my site [http://lukebryantickets.pyhgy.com luke bryan new tour dates]

	: ~~<math>\mbox{isqrt}( n ) = \lfloor \sqrt n \rfloor.<~~/~~math>~~

	~~For example, <math>\mbox{isqrt}(27) = 5<~~/~~math> because <math>5\cdot 5=25 \le 27</math> and <math>6\cdot 6=36 > 27</math>~~.

	~~==Algorithm==~~
	~~One way of calculating <math>\sqrt{n}<~~/~~math> and <math>\mbox{isqrt}( n )</math>~~ is ~~to use [[Newton~~'~~s method]] to find a solution for the equation <math>x^{2} - n = 0</math>, giving the~~ [~~[Recursion\|recursive]] formula~~

	: ~~<math>{x}_{k+1} = \frac{1}{2}\left(x_k + \frac{ n }{x_k}\right), \quad k \ge 0, \quad x_0 > 0.<~~/~~math>~~

	~~The [[sequence]] <math>\{ x_k \}<~~/~~math> [[Limit (mathematics)\|converges]] [[Rate of convergence\|quadratically~~]~~] to <math>\sqrt{n}</math>~~ as ~~<math>k\to \infty</math>. It can be proven~~ that ~~if <math>x_{0} = n</math> is chosen as the initial guess,~~ one can ~~stop as soon as~~
	:~~<math>\| x_{k+1}-x_{k}\| < 1<~~/~~math>~~
	~~to ensure that <math>\lfloor x_{k+1} \rfloor=\lfloor \sqrt n \rfloor~~.</~~math>~~

	~~==Domain of computation==~~
	~~Although <math>\sqrt{n}<~~/~~math> is [[irrational number\|irrational]] for [[almost all~~]~~] <math>n</math>,~~ the ~~sequence <math>\{ x_k \}</math> contains only [[rational number\|rational]] terms when <math> x_0 </math> is rational~~. ~~Thus, with this method it is unnecessary~~ to ~~exit the [[field (mathematics)\|field]] of rational numbers in order to calculate <math>\mbox{isqrt}( n )</math>, a fact which has some theoretical advantages~~.

	~~==Stopping criterion==~~
	~~One can prove that~~ <~~math~~>~~c=1</math> is the largest~~ possible ~~number for which~~ the ~~stopping criterion~~
	:~~<math>\|x_{k+1} - x_{k}\| < c\ <~~/~~math>~~
	~~ensures <math>\lfloor x_{k+1} \rfloor=\lfloor \sqrt n \rfloor<~~/~~math>~~
	~~in the algorithm above~~.

	~~In implementations which use number formats that cannot represent all rational numbers exactly (for example, floating point),~~ a ~~stopping constant less than one should be used to protect against roundoff errors~~.

	~~== See also ==~~
	* [[Methods of ~~computing square roots]]~~

	~~==External links==~~
	*[http://~~mathcentral~~.~~uregina~~.~~ca/RR~~/~~database~~/RR.~~09.95/grzesina1~~.~~html A geometric view of the square root algorithm]~~

	~~{{number theoretic algorithms}}~~

	~~[[Category:Number theoretic algorithms]]~~
	~~[[Category:Number theory]]~~
	~~[[Category:Root-finding algorithms]~~]

en>David Eppstein: Undid revision 558848965 by 24.185.3.4 (talk) there IS a number, there ARE infinitely many, but not there ARE a number nor there IS infinitely many

2013-06-08T03:17:04Z

Undid revision 558848965 by 24.185.3.4 (talk) there IS a number, there ARE infinitely many, but not there ARE a number nor there IS infinitely many

← Older revision		Revision as of 05:17, 8 June 2013
Line 1:		Line 1:
	It is ~~time to address~~ the ~~slow computer issues even if we never know how. Just considering a computer~~ is ~~functioning thus slow or keeps freezing up; refuses to indicate that you can not address~~ the ~~issue plus fix it. You may~~ or ~~could not be aware which any computer owner must learn that there are certain factors which a computer needs~~ to ~~keep~~ the ~~greatest performance~~. ~~The sad truth is that numerous persons whom own a system have no idea that it requires routine repair simply like their cars~~.<br><br>~~Registry~~ is ~~not furthermore significant~~ to ~~fast computer boot up, however, also important~~ to the ~~overall performance of the computer. If you have a registry error~~, ~~we might face blue screen, freezing or even crash. It's important to regularly clean up~~ the ~~invalid~~, ~~missing~~, ~~junk registry keys to keep the computer healthy and running swiftly~~.<br><br>~~The error is basically a outcome~~ of ~~problem with Windows Installer package. The Windows Installer is a tool chosen~~ to ~~install, uninstall plus repair the many programs on a computer. Let you discuss a limited aspects which helped a lot of folks who facing the similar problem.~~<br><br>~~Always see~~ to it that ~~you have installed antivirus~~, ~~anti-spyware and anti-adware programs plus have them up-to-date regularly. This~~ can ~~help~~ stop ~~windows XP running slow.~~<br><br>~~Another prevalent cause of PC slow down is a corrupt registry~~. ~~The registry is a important component~~ of ~~computers running on Windows platform. When this gets corrupted a PC will slowdown, or worse, not commence at all. Fixing the registry~~ is ~~easy with~~ all ~~the employ of a program plus [http://bestregistrycleanerfix.com/tune-up-utilities tuneup utilities~~].<br><br>~~Software mistakes or hardware mistakes that happen whenever running Windows and intermittent mistakes are~~ the ~~general factors for a blue screen physical memory dump. New software or drivers that have been installed or changes in the registry settings are the typical s~~/~~w causes. Intermittent errors refer to failed system memory/ hard disk or over heated processor plus these too may result the blue screen bodily memory dump error.~~<br><br>~~The 'registry'~~ is ~~merely the central database that shops all your settings plus choices~~. ~~It's a truly important part of the XP system~~, ~~meaning which Windows~~ is ~~regularly adding and updating~~ the ~~files inside it. The problems occur whenever Windows really corrupts & loses some~~ of ~~these files. This makes your computer run slow, because it tries hard~~ to ~~find them again.~~<br><br>~~Next~~, ~~there is an easy means to deal with this problem~~. ~~You~~ can ~~install a registry cleaner~~ that ~~there are it on~~ the ~~internet. This software will help you see out these errors~~ in the ~~computer plus clean them~~. ~~It furthermore can figure out these malware plus different threats~~ which ~~influence the speed~~ of ~~your computer~~. ~~So this software could speed up PC simpler~~. ~~We can choose one~~ of ~~these techniques to accelerate you computer.~~		In [[number theory]], the '''integer square root''' (isqrt) of a [[positive integer]] ''n'' is the positive integer ''m'' which is the [[floor and ceiling functions\|greatest integer less than or equal]] to the [[square root]] of ''n'',

			: <math>\mbox{isqrt}( n ) = \lfloor \sqrt n \rfloor.</math>

			For example, <math>\mbox{isqrt}(27) = 5</math> because <math>5\cdot 5=25 \le 27</math> and <math>6\cdot 6=36 > 27</math>.

			==Algorithm==
			One way of calculating <math>\sqrt{n}</math> and <math>\mbox{isqrt}( n )</math> is to use [[Newton's method]] to find a solution for the equation <math>x^{2} - n = 0</math>, giving the [[Recursion\|recursive]] formula

			: <math>{x}_{k+1} = \frac{1}{2}\left(x_k + \frac{ n }{x_k}\right), \quad k \ge 0, \quad x_0 > 0.</math>

			The [[sequence]] <math>\{ x_k \}</math> [[Limit (mathematics)\|converges]] [[Rate of convergence\|quadratically]] to <math>\sqrt{n}</math> as <math>k\to \infty</math>. It can be proven that if <math>x_{0} = n</math> is chosen as the initial guess, one can stop as soon as
			:<math>\| x_{k+1}-x_{k}\| < 1</math>
			to ensure that <math>\lfloor x_{k+1} \rfloor=\lfloor \sqrt n \rfloor.</math>

			==Domain of computation==
			Although <math>\sqrt{n}</math> is [[irrational number\|irrational]] for [[almost all]] <math>n</math>, the sequence <math>\{ x_k \}</math> contains only [[rational number\|rational]] terms when <math> x_0 </math> is rational. Thus, with this method it is unnecessary to exit the [[field (mathematics)\|field]] of rational numbers in order to calculate <math>\mbox{isqrt}( n )</math>, a fact which has some theoretical advantages.

			==Stopping criterion==
			One can prove that <math>c=1</math> is the largest possible number for which the stopping criterion
			:<math>\|x_{k+1} - x_{k}\| < c\ </math>
			ensures <math>\lfloor x_{k+1} \rfloor=\lfloor \sqrt n \rfloor</math>
			in the algorithm above.

			In implementations which use number formats that cannot represent all rational numbers exactly (for example, floating point), a stopping constant less than one should be used to protect against roundoff errors.

			== See also ==
			* [[Methods of computing square roots]]

			==External links==
			*[http://mathcentral.uregina.ca/RR/database/RR.09.95/grzesina1.html A geometric view of the square root algorithm]

			{{number theoretic algorithms}}

			[[Category:Number theoretic algorithms]]
			[[Category:Number theory]]
			[[Category:Root-finding algorithms]]

en>David Eppstein: clean up citation postscripts, link journal names

2011-12-25T16:34:36Z

clean up citation postscripts, link journal names

New page

It is time to address the slow computer issues even if we never know how. Just considering a computer is functioning thus slow or keeps freezing up; refuses to indicate that you can not address the issue plus fix it. You may or could not be aware which any computer owner must learn that there are certain factors which a computer needs to keep the greatest performance. The sad truth is that numerous persons whom own a system have no idea that it requires routine repair simply like their cars. Registry is not furthermore significant to fast computer boot up, however, also important to the overall performance of the computer. If you have a registry error, we might face blue screen, freezing or even crash. It's important to regularly clean up the invalid, missing, junk registry keys to keep the computer healthy and running swiftly. The error is basically a outcome of problem with Windows Installer package. The Windows Installer is a tool chosen to install, uninstall plus repair the many programs on a computer. Let you discuss a limited aspects which helped a lot of folks who facing the similar problem. Always see to it that you have installed antivirus, anti-spyware and anti-adware programs plus have them up-to-date regularly. This can help stop windows XP running slow. Another prevalent cause of PC slow down is a corrupt registry. The registry is a important component of computers running on Windows platform. When this gets corrupted a PC will slowdown, or worse, not commence at all. Fixing the registry is easy with all the employ of a program plus [http://bestregistrycleanerfix.com/tune-up-utilities tuneup utilities]. Software mistakes or hardware mistakes that happen whenever running Windows and intermittent mistakes are the general factors for a blue screen physical memory dump. New software or drivers that have been installed or changes in the registry settings are the typical s/w causes. Intermittent errors refer to failed system memory/ hard disk or over heated processor plus these too may result the blue screen bodily memory dump error. The 'registry' is merely the central database that shops all your settings plus choices. It's a truly important part of the XP system, meaning which Windows is regularly adding and updating the files inside it. The problems occur whenever Windows really corrupts & loses some of these files. This makes your computer run slow, because it tries hard to find them again. Next, there is an easy means to deal with this problem. You can install a registry cleaner that there are it on the internet. This software will help you see out these errors in the computer plus clean them. It furthermore can figure out these malware plus different threats which influence the speed of your computer. So this software could speed up PC simpler. We can choose one of these techniques to accelerate you computer.