Rodrigues equation - Revision history

en>Fadesga: /* References */

2014-03-25T00:19:51Z

References

← Older revision		Revision as of 02:19, 25 March 2014
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	~~{{for\|physical models of polyhedra\|Polyhedron model}}~~		The author is known by the name of Numbers Lint. She is [http://xrambo.com/user/RKDBG home std test] kit [http://www.dynast.co.kr/xe/?document_srl=150618 at home std testing] home std test a librarian but she's usually needed her personal company. One of the very best things in the globe for me is to do aerobics and I've been [http://www.Itsyoursexlife.com/gyt/std-and-testing-faqs/ performing] it for fairly a while. Minnesota has always been his house but his spouse wants them to move.<br><br>Also visit my site :: [https://www.machlitim.org.il/subdomain/megila/end/node/9243 https://www.machlitim.org.il/subdomain/megila/end/node/9243]
	The ~~'''polyhedral model''' (also called the '''polytope method''')~~ is a mathematical framework for [[loop nest optimization]] in [[program optimization]]. The polytope method treats each loop iteration within nested loops as lattice points inside mathematical objects called [[polytope]]s, performs [[affine transformation]]s or more general non-affine transformations such as tiling on the ~~polytopes, and then converts the transformed polytopes into equivalent, but optimized (depending on targeted optimization goal), loop nests through polyhedra scanning~~.

	~~== Detailed example ==~~


	[~~[Image~~:~~Polytope model unskewed~~.~~svg\|thumb\|right\|The dependencies of <code>src<~~/~~code>, before [[Loop_optimization#Common_loop_transformations\|loop skewing~~]~~]. The red dot corresponds to <code>src~~[~~1][0]<~~/~~code>; the purple dot corresponds to <code>src[2][2]<~~/~~code>~~.]]
	~~The following [[C (programming language)\|C]] code implements a form of error-distribution [[dither]]ing similar to [[Floyd–Steinberg dithering]], but modified for pedagogical reasons~~. ~~The two-dimensional array <code>src<~~/~~code> contains <code>h<~~/~~code> rows of <code>w</code> pixels, each pixel having~~ a [[grayscale]] value between 0 and 255 inclusive. After the routine has finished, the output array <code>dst</code> will contain only pixels with value 0 or value 255. During the computation, each pixel's ~~dithering error is collected by adding it back into the <code>src</code> array~~. ~~(Notice that <code>src</code> and <code>dst</code> are both read and written during the computation; <code>src</code> is not read-only, and <code>dst</code> is not write-only.)~~

	~~Each iteration~~ of the ~~[[inner loop]] modifies the values~~ in ~~<code>src[i][j]</code> based on~~ the ~~values of <code>src[i-1][j]</code>, <code>src[i][j-1]</code>, and <code>src[i+1][j-1]</code>. (The same dependencies apply~~ to ~~<code>dst[i][j]</code>. For the purposes of [[Loop_optimization#Common_loop_transformations\|loop skewing]], we can think of <code>src[i][j]</code>~~ and ~~<code>dst~~[~~i][j]<~~/~~code> as the same element.) We can illustrate the dependencies of <code>src[i][j]<~~/~~code> graphically, as in the diagram on the right~~.

	~~<!-- This table nonsense makes the page look slightly better in Firefox on Windows XP~~. ~~Is there a more portable and self-explanatory way to get the code and images not to overlap with each other? --~~~~ -->~~
	~~<table><tr><td width="50%">~~
	~~<source lang=C>~~
	~~#define ERR(x,y) (dst[x][y] - src[x][y])~~

	~~void dither(unsigned char src, unsigned char dst, int w, int h)~~
	{
	~~int i, j;~~
	~~for (j = 0; j < h; ++j) {~~
	~~for (i = 0; i < w; ++i) {~~
	~~int v = src[i][j];~~
	~~if (i > 0)~~
	~~v -= ERR(i - 1, j)~~ /2;
	~~if (j > 0)~~
	~~v -= ERR(i, j - 1)~~ / 4;
	~~if (j > 0 && i < w~~ - 1)
	v -~~= ERR(i + 1, j~~ - 1) / 4;
	~~dst[i~~]~~[j] = (v < 128) ? 0 : 255;~~
	~~src[i][j] = (v < 0) ? 0 : (v < 255) ? v : 255;~~
	}
	}
	}
	~~</source>~~
	~~</td><td></td></tr></table>~~

	~~[[Image:Polytope model skewed.svg\|thumb\|right\|The dependencies of <code>src</code>, after loop skewing. The array elements will be processed in the order ''gray, red, green, blue, yellow...'']]~~

	~~Performing the affine transformation <math>(p,\, t) = (i,\, 2j+i)</math> on the original dependency diagram gives us~~ a ~~new diagram, which is shown in the next image~~. ~~We can then rewrite the code~~ to ~~loop on <code>p</code> and <code>t</code> instead of <code>i</code> and <code>j</code>, obtaining the following "skewed" routine.~~

	~~<!-- Please don't break this code~~. ~~Test before committing! -->~~
	<~~table~~><~~tr><td width="50%">~~
	~~<source lang=C>~~
	~~void dither_skewed(unsigned char src, unsigned char dst, int w, int h)~~
	{
	~~int t, p;~~
	~~for (t = 0; t < (w + (2 * h)); ++t) {~~
	~~int pmin = max(t % 2, t - (2 * h) + 2);~~
	~~int pmax = min(t, w - 1);~~
	~~for (p = pmin; p <= pmax; p += 2) {~~
	~~int i = p;~~
	~~int j = (t - p) / 2;~~
	~~int v = src[i][j];~~
	~~if (i > 0)~~
	~~v -= ERR(i - 1, j) / 2;~~
	~~if (j > 0)~~
	~~v -= ERR(i, j - 1) / 4;~~
	~~if (j~~ > ~~0 && i < w - 1)~~
	~~v -= ERR(i + 1, j - 1) / 4;~~
	~~dst[i][j] = (v < 128) ? 0 : 255;~~
	~~src[i][j] = (v < 0) ? 0~~ : ~~(v < 255) ? v~~ : ~~255;~~
	}
	}
	}
	~~</source>~~
	~~</td><td></td></tr></table>~~

	~~==See also==~~
	*[[Frameworks supporting the polyhedral model]]
	*[[Loop nest optimization]]
	*[~~[Loop unrolling]]~~
	*[[Loop reversal]]
	*[[Loop tiling]]

	~~==External links and references==~~
	*[http://www.~~infosun~~.~~fmi.uni-passau~~.de/cl/~~loopo~~/~~doc~~/~~loopo_doc~~/~~node3.html "The basic polytope method"], tutorial by Martin Griebl containing diagrams of the pseudocode example above~~
	*[[Frameworks supporting the polyhedral model#Transitive closure operation\|"Framework for polyhedral model"]]
	*[http://~~citeseer~~.~~ist~~.~~psu~~.~~edu~~/~~viewdoc~~/~~summary?doi=10.1.1.30.5675 "Code Generation in the Polytope Model"] (1998). Martin Griebl, Christian Lengauer, and Sabine Wetzel~~
	*[http://www.cloog.org/ ~~"The CLooG Polyhedral Code Generator"]~~
	*[http://~~www.chunchen.info/omega "CodeGen+: Z-polyhedra scanning"]~~

	~~[[Category:Compiler optimizations]]~~
	~~[[Category:Articles with example pseudocode]]~~
	~~[[Category:Articles with example C code]~~]

en>Addbot: Bot: Removing Orphan Tag (Nolonger an Orphan) (Report Errors)

2013-01-09T02:07:50Z

Bot: Removing Orphan Tag (Nolonger an Orphan) (Report Errors)

← Older revision		Revision as of 04:07, 9 January 2013
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	~~Alyson~~ is the ~~name people use to contact me~~ and ~~I believe online reader~~ ([~~http~~://~~www~~.~~010~~-~~5260~~-~~5333~~.~~com~~/~~index~~.~~php~~?~~document_srl~~=~~1880&mid~~=~~board_ALMP66 http:~~//~~www.010~~-~~5260~~-~~5333.com~~/]) ~~it sounds fairly great when you say it~~. ~~I am an invoicing officer and I~~'~~ll be promoted quickly~~. ~~I am really fond of to go to karaoke but I~~'~~ve been taking~~ on new ~~things recently~~. ~~Some time in~~ the ~~past he selected~~ to ~~reside in North Carolina~~ and ~~he doesn~~'t ~~plan on altering it~~.<br><br>~~my web~~-~~site~~ :: ~~online~~ [http://www.~~prograd~~.~~uff~~.br/~~novo~~/~~facts-about-growing-greater-organic-garden telephone psychic~~] ~~reading (~~[http://~~jplusfn~~.~~gaplus~~.kr/xe/~~qna/78647~~ http://~~jplusfn~~.~~gaplus~~.kr/xe/~~qna~~/~~78647~~])		{{for\|physical models of polyhedra\|Polyhedron model}}
			The '''polyhedral model''' (also called the '''polytope method''') is a mathematical framework for [[loop nest optimization]] in [[program optimization]]. The polytope method treats each loop iteration within nested loops as lattice points inside mathematical objects called [[polytope]]s, performs [[affine transformation]]s or more general non-affine transformations such as tiling on the polytopes, and then converts the transformed polytopes into equivalent, but optimized (depending on targeted optimization goal), loop nests through polyhedra scanning.

			== Detailed example ==


			[[Image:Polytope model unskewed.svg\|thumb\|right\|The dependencies of <code>src</code>, before [[Loop_optimization#Common_loop_transformations\|loop skewing]]. The red dot corresponds to <code>src[1][0]</code>; the purple dot corresponds to <code>src[2][2]</code>.]]
			The following [[C (programming language)\|C]] code implements a form of error-distribution [[dither]]ing similar to [[Floyd–Steinberg dithering]], but modified for pedagogical reasons. The two-dimensional array <code>src</code> contains <code>h</code> rows of <code>w</code> pixels, each pixel having a [[grayscale]] value between 0 and 255 inclusive. After the routine has finished, the output array <code>dst</code> will contain only pixels with value 0 or value 255. During the computation, each pixel's dithering error is collected by adding it back into the <code>src</code> array. (Notice that <code>src</code> and <code>dst</code> are both read and written during the computation; <code>src</code> is not read-only, and <code>dst</code> is not write-only.)

			Each iteration of the [[inner loop]] modifies the values in <code>src[i][j]</code> based on the values of <code>src[i-1][j]</code>, <code>src[i][j-1]</code>, and <code>src[i+1][j-1]</code>. (The same dependencies apply to <code>dst[i][j]</code>. For the purposes of [[Loop_optimization#Common_loop_transformations\|loop skewing]], we can think of <code>src[i][j]</code> and <code>dst[i][j]</code> as the same element.) We can illustrate the dependencies of <code>src[i][j]</code> graphically, as in the diagram on the right.

			<!-- This table nonsense makes the page look slightly better in Firefox on Windows XP. Is there a more portable and self-explanatory way to get the code and images not to overlap with each other? --~~~~ -->
			<table><tr><td width="50%">
			<source lang=C>
			#define ERR(x,y) (dst[x][y] - src[x][y])

			void dither(unsigned char src, unsigned char dst, int w, int h)
			{
			int i, j;
			for (j = 0; j < h; ++j) {
			for (i = 0; i < w; ++i) {
			int v = src[i][j];
			if (i > 0)
			v -= ERR(i - 1, j) /2;
			if (j > 0)
			v -= ERR(i, j - 1) / 4;
			if (j > 0 && i < w - 1)
			v -= ERR(i + 1, j - 1) / 4;
			dst[i][j] = (v < 128) ? 0 : 255;
			src[i][j] = (v < 0) ? 0 : (v < 255) ? v : 255;
			}
			}
			}
			</source>
			</td><td></td></tr></table>

			[[Image:Polytope model skewed.svg\|thumb\|right\|The dependencies of <code>src</code>, after loop skewing. The array elements will be processed in the order ''gray, red, green, blue, yellow...'']]

			Performing the affine transformation <math>(p,\, t) = (i,\, 2j+i)</math> on the original dependency diagram gives us a new diagram, which is shown in the next image. We can then rewrite the code to loop on <code>p</code> and <code>t</code> instead of <code>i</code> and <code>j</code>, obtaining the following "skewed" routine.

			<!-- Please don't break this code. Test before committing! -->
			<table><tr><td width="50%">
			<source lang=C>
			void dither_skewed(unsigned char src, unsigned char dst, int w, int h)
			{
			int t, p;
			for (t = 0; t < (w + (2 * h)); ++t) {
			int pmin = max(t % 2, t - (2 * h) + 2);
			int pmax = min(t, w - 1);
			for (p = pmin; p <= pmax; p += 2) {
			int i = p;
			int j = (t - p) / 2;
			int v = src[i][j];
			if (i > 0)
			v -= ERR(i - 1, j) / 2;
			if (j > 0)
			v -= ERR(i, j - 1) / 4;
			if (j > 0 && i < w - 1)
			v -= ERR(i + 1, j - 1) / 4;
			dst[i][j] = (v < 128) ? 0 : 255;
			src[i][j] = (v < 0) ? 0 : (v < 255) ? v : 255;
			}
			}
			}
			</source>
			</td><td></td></tr></table>

			==See also==
			*[[Frameworks supporting the polyhedral model]]
			*[[Loop nest optimization]]
			*[[Loop unrolling]]
			*[[Loop reversal]]
			*[[Loop tiling]]

			==External links and references==
			*[http://www.infosun.fmi.uni-passau.de/cl/loopo/doc/loopo_doc/node3.html "The basic polytope method"], tutorial by Martin Griebl containing diagrams of the pseudocode example above
			*[[Frameworks supporting the polyhedral model#Transitive closure operation\|"Framework for polyhedral model"]]
			*[http://citeseer.ist.psu.edu/viewdoc/summary?doi=10.1.1.30.5675 "Code Generation in the Polytope Model"] (1998). Martin Griebl, Christian Lengauer, and Sabine Wetzel
			*[http://www.cloog.org/ "The CLooG Polyhedral Code Generator"]
			*[http://www.chunchen.info/omega "CodeGen+: Z-polyhedra scanning"]

			[[Category:Compiler optimizations]]
			[[Category:Articles with example pseudocode]]
			[[Category:Articles with example C code]]

en>Filipedireito: /* Equaition */

2011-10-11T22:08:54Z

Equaition

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Alyson is the name people use to contact me and I believe online reader ([http://www.010-5260-5333.com/index.php?document_srl=1880&mid=board_ALMP66 http://www.010-5260-5333.com/]) it sounds fairly great when you say it. I am an invoicing officer and I'll be promoted quickly. I am really fond of to go to karaoke but I've been taking on new things recently. Some time in the past he selected to reside in North Carolina and he doesn't plan on altering it.<br><br>my web-site :: online [http://www.prograd.uff.br/novo/facts-about-growing-greater-organic-garden telephone psychic] reading ([http://jplusfn.gaplus.kr/xe/qna/78647 http://jplusfn.gaplus.kr/xe/qna/78647])