07-401/Andrei Litvin Poly Solution - Revision history

Drorbn at 14:40, 28 March 2007

2007-03-28T14:40:21Z

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			{{Begin Side Note\|30%}}
⚫	~~<div align=right>(~~See the "Just for Fun" part of [[07-401/Homework Assignment 7]]~~)</div>~~
			<center><math>\sqrt[3]{7} - \sqrt[4]{\sqrt{5}\left/\sqrt[3]{\sqrt{2}+\sqrt[5]{7}}\right.}</math></center>
		⚫	See the "Just for Fun" part of [[07-401/Homework Assignment 7]]
			{{End Side Note}}

	Here is what I have so far in terms of polynomials:		Here is what I have so far in terms of polynomials:

Drorbn at 14:35, 28 March 2007

2007-03-28T14:35:15Z

← Older revision		Revision as of 10:35, 28 March 2007
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			<div align=right>(See the "Just for Fun" part of [[07-401/Homework Assignment 7]])</div>

	Here is what I have so far in terms of polynomials:		Here is what I have so far in terms of polynomials:

Drorbn: /* Program */

2007-03-28T14:30:29Z

Program

← Older revision		Revision as of 10:30, 28 March 2007
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	** if <math>p(a) = 0</math>, then return q, such that <math>q(\sqrt[n]{a}) = 0</math>		** if <math>p(a) = 0</math>, then return q, such that <math>q(\sqrt[n]{a}) = 0</math>
	* "solve_sum(p1, p2)"		* "solve_sum(p1, p2)"
	** if <math>p1(a) = 0</math> and <math>p2(a) == 0</math>, then return q, such that <math>q(a+b) = 0</math>		** if <math>p1(a) = 0</math> and <math>p2(a) = 0</math>, then return q, such that <math>q(a+b) = 0</math>
	* "solve_diff(p1, p2)"		* "solve_diff(p1, p2)"
	** if <math>p1(a) = 0</math> and <math>p2(a) == 0</math>, then return q, such that <math>q(a-b) = 0</math>		** if <math>p1(a) = 0</math> and <math>p2(a) = 0</math>, then return q, such that <math>q(a-b) = 0</math>
	* "solve_prod(p1, p2)"		* "solve_prod(p1, p2)"
	** if <math>p1(a) = 0</math> and <math>p2(a) == 0</math>, then return q, such that <math>q(a*b) = 0</math>		** if <math>p1(a) = 0</math> and <math>p2(a) = 0</math>, then return q, such that <math>q(a*b) = 0</math>

	Now you are ready to do things like:		Now you are ready to do things like:

76.64.140.109: /* Program */

2007-03-28T00:55:33Z

Program

← Older revision		Revision as of 20:55, 27 March 2007
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	After you have built one or more polynomials (the sample starts with simple ones, like "x - 7" and " x- 5), you have an array of functions that you can build new polynomials. These are:		After you have built one or more polynomials (the sample starts with simple ones, like "x - 7" and " x- 5), you have an array of functions that you can build new polynomials. These are:

	* "solve_inverse(p)" ~~- if "p(a) == 0", then return q, such that "q(1/a) == 0"~~		* "solve_inverse(p)":
	* ~~"solve_sqrt(p, n)" -~~ if "p(a) == 0", then return q, such that ~~"q(~~<math>\~~sqrt[n]{~~a}~~</math>~~) == 0"		** if <math>p(a) = 0</math> then return q, such that <math>q({1 \over a}) = 0</math>
			* "solve_sqrt(p, n)":
⚫	* ~~"solve_sum(p1, p2)" -~~ if "p1(a) == 0" and "p2(a) == 0", then return q, such that "q(a+b) == 0"
	* ~~"solve_diff(p1, p2)" -~~ if ~~"p1~~(a) ~~== 0" and "p2(a) =~~= 0", then return q, such that "q(a-b) == 0"		** if <math>p(a) = 0</math>, then return q, such that <math>q(\sqrt[n]{a}) = 0</math>
			* "solve_sum(p1, p2)"
⚫	* ~~"solve_prod(p1, p2)" -~~ if "p1(a) == 0" and "p2(a) == 0", then return q, such that "q(a*b) == 0"
		⚫	** if <math>p1(a) = 0</math> and <math>p2(a) == 0</math>, then return q, such that <math>q(a+b) = 0</math>
			* "solve_diff(p1, p2)"
			** if <math>p1(a) = 0</math> and <math>p2(a) == 0</math>, then return q, such that <math>q(a-b) = 0</math>
			* "solve_prod(p1, p2)"
		⚫	** if <math>p1(a) = 0</math> and <math>p2(a) == 0</math>, then return q, such that <math>q(a*b) = 0</math>

	Now you are ready to do things like:		Now you are ready to do things like:

76.64.140.109 at 00:51, 28 March 2007

2007-03-28T00:51:27Z

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	:::<math>8665959513826347108430640406618155029520076750443339789185589187423680</math>		:::<math>8665959513826347108430640406618155029520076750443339789185589187423680</math>
	(223 decimal digits)		(223 decimal digits)

			==Program==

			A zip file containing the program can be downloaded [http://www.bitrover.com/polysolve/polysolve_andrei_litvin2.zip here] (I hope this works .. if not send me an e-mail).

			It does require [http://www.python.org/download/ Python] (I developed and tested it with version 2.5) and the file you probably want to edit is "test.py" (or you can also create your own program files if you wish).

			It starts with a statement of the form "x = Poly()" which by default creates the polynomial "x". The object overloads +,-,* and (power),so that you can build your own fancy polynomials (e.g. "x2 + 4x + 1" or "(x + 256)((x34 + 1)3) + 1).

			After you have built one or more polynomials (the sample starts with simple ones, like "x - 7" and " x- 5), you have an array of functions that you can build new polynomials. These are:

			* "solve_inverse(p)" - if "p(a) == 0", then return q, such that "q(1/a) == 0"
			* "solve_sqrt(p, n)" - if "p(a) == 0", then return q, such that "q(<math>\sqrt[n]{a}</math>) == 0"
			* "solve_sum(p1, p2)" - if "p1(a) == 0" and "p2(a) == 0", then return q, such that "q(a+b) == 0"
			* "solve_diff(p1, p2)" - if "p1(a) == 0" and "p2(a) == 0", then return q, such that "q(a-b) == 0"
			* "solve_prod(p1, p2)" - if "p1(a) == 0" and "p2(a) == 0", then return q, such that "q(a*b) == 0"

			Now you are ready to do things like:

			from poly import * # these imports are needed
			from polysolve import *

			x = Poly() # the most basic poly you can get

			p1 = solve_sum(x3 - 3, x5 - 6)
			p2 = solve_sum(x**2 - 2, x - 1)
			p3 = solve_sum(p1, p2)

			print p3

Drorbn at 18:15, 27 March 2007

2007-03-27T18:15:25Z

← Older revision		Revision as of 14:15, 27 March 2007
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	::<math>100118652346710314482731150749632703657021113788752749525948078496866540654\backslash</math>		::<math>100118652346710314482731150749632703657021113788752749525948078496866540654\backslash</math>
	:::<math>8665959513826347108430640406618155029520076750443339789185589187423680</math>		:::<math>8665959513826347108430640406618155029520076750443339789185589187423680</math>
			(223 decimal digits)

Drorbn at 18:13, 27 March 2007

2007-03-27T18:13:23Z

← Older revision		Revision as of 14:13, 27 March 2007
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	Polynomial with root <math>\sqrt{2}</math>:		Polynomial with root <math>\sqrt{2}</math>:

	<math>{x}^{2} - 2</math> (duh)		:<math>{x}^{2} - 2</math> (duh)

	Polynomial with root <math>\sqrt[5]{7}</math>:		Polynomial with root <math>\sqrt[5]{7}</math>:

	<math>x^5 - 7</math> (duh as well)		:<math>x^5 - 7</math> (duh as well)

	Polynomial with root <math>\sqrt{2} + \sqrt[5]{7}</math>:		Polynomial with root <math>\sqrt{2} + \sqrt[5]{7}</math>:

	<math>x^{10} - ~~10x~~^8 + ~~40x~~^6 - ~~14x~~^5 - ~~80x~~^4 - ~~280x~~^3 + ~~80x~~^2 - ~~280*x~~ + 17		:<math>x^{10} - 10x^8 + 40x^6 - 14x^5 - 80x^4 - 280x^3 + 80x^2 - 280x + 17</math> (more interesting!)
	</math> (more interesting!)

	Polynomial with root <math>\sqrt[3]{\sqrt{2} + \sqrt[5]{7}}</math>:		Polynomial with root <math>\sqrt[3]{\sqrt{2} + \sqrt[5]{7}}</math>:

	<math>x^{30} - ~~10x~~^{24} + ~~40x~~^{18} - ~~14x~~^{15} - ~~80x~~^{12} - ~~280x~~^9 + ~~80x~~^6 - ~~280*x~~^3 + 17		:<math>x^{30} - 10x^{24} + 40x^{18} - 14x^{15} - 80x^{12} - 280x^9 + 80x^6 - 280x^3 + 17</math> (similar to the above)
	</math> (similar to the above)

	Polynomial with root <math> {1} \over {\sqrt[3]{\sqrt{2} + \sqrt[5]{7}}}</math>:		Polynomial with root <math> {1} \over {\sqrt[3]{\sqrt{2} + \sqrt[5]{7}}}</math>:

		⚫	:<math>17x^{30} - 280x^{27} + 80x^{24} - 280x^{21} - 80x^{18} - 14x^{15} + 40x^{12} - 10x^6 + 1</math>
⚫	<math>
		⚫	::(just invert the above .. quite the same as the above 2)
⚫	~~17x~~^{30} - ~~280x~~^{27} + ~~80x~~^{24} - ~~280x~~^{21} - ~~80x~~^{18} - ~~14x~~^{15} + ~~40x~~^{12} - ~~10x~~^6 + 1
⚫	~~</math>~~ (just invert the above .. quite the same as the above 2)

	Polynomial with root <math> {\sqrt{5}} \over {\sqrt[3]{\sqrt{2} + \sqrt[5]{7}}}</math>:		Polynomial with root <math> {\sqrt{5}} \over {\sqrt[3]{\sqrt{2} + \sqrt[5]{7}}}</math>:

			:<math>289x^{60} - 9460000x^{54} - 2392500000x^{48} - 190781250000x^{42} + 1127929687500x^{36}</math>
⚫	<math>
			::<math> - 249084472656250x^{30} + 12817382812500000x^{24} - 457763671875000000x^{18}</math>
	289x^{60} - 9460000x^{54} - 2392500000x^{48} - 190781250000x^{42} + 1127929687500x^{36} - 249084472656250x^{30} + 12817382812500000x^{24} - 457763671875000000x^{18} + 10728836059570312500x^{12} - 149011611938476562500x^6 + 931322574615478515625
			::<math>+ 10728836059570312500x^{12} - 149011611938476562500x^6 + 931322574615478515625
⚫	~~</math>~~ (ok .. this is ugly)
		⚫	</math>
		⚫	::(ok .. this is ugly)

	Polynomial with root <math> \sqrt[4]{{\sqrt{5}} \over {\sqrt[3]{\sqrt{2} + \sqrt[5]{7}}}}</math>:		Polynomial with root <math> \sqrt[4]{{\sqrt{5}} \over {\sqrt[3]{\sqrt{2} + \sqrt[5]{7}}}}</math>:

		⚫	<math>289x^{240} - 9460000x^{216} - 2392500000x^{192} - 190781250000x^{168} +
	<math>
			1127929687500x^{144}</math>
⚫	~~289x~~^{240} - ~~9460000x~~^{216} - ~~2392500000x~~^{192} - ~~190781250000x~~^{168} +
	~~1127929687500x^{144}~~ - ~~249084472656250x~~^{120} + ~~12817382812500000*x~~^{96} -		::<math>- 249084472656250x^{120} + 12817382812500000x^{96} -
			457763671875000000x^{72}</math>
	457763671875000000x^{72} + 10728836059570312500x^{48} - 149011611938476562500*x^{24} + 931322574615478515625
			::<math>+ 10728836059570312500x^{48} - 149011611938476562500x^{24} + 931322574615478515625
⚫	~~</math>~~ (very ugly .. especially considering you are not done yet and the next matrix to compute will be quite large)
		⚫	</math>

		⚫	::(very ugly .. especially considering you are not done yet and the next matrix to compute will be quite large)

	This is how far I got so far. The program itself will do any polynomials just fine (I tweaked it a lot to get the final poly, so it may still have some bugs, however it works). The problem is working with such large numbers. The reducing part will multiply with 289 each time a <math>x^{240}</math> is reduced, so we could potentially get coefficients of the order <math>289^{964 - 240}</math>, which is very large indeed. On my computer, I get about 30% done, when it starts running out of memory and I had to stop. This is why the program just generates the matrices for the last part (and by the way, Maxima crashed when trying to load the matrix to invert, so I will have to find another program that will handle such large sets of data)		This is how far I got so far. The program itself will do any polynomials just fine (I tweaked it a lot to get the final poly, so it may still have some bugs, however it works). The problem is working with such large numbers. The reducing part will multiply with 289 each time a <math>x^{240}</math> is reduced, so we could potentially get coefficients of the order <math>289^{964 - 240}</math>, which is very large indeed. On my computer, I get about 30% done, when it starts running out of memory and I had to stop. This is why the program just generates the matrices for the last part (and by the way, Maxima crashed when trying to load the matrix to invert, so I will have to find another program that will handle such large sets of data)

			'''Comments by {{Dror}}.''' Great work! Your grade for the class is bounded below by 80; the exams and HW will determine which fraction of the other 20 you will get. To raise your lower bound to 85, you'll have to post the programs (either here, as text files, or elsewhere with a link from here) along with a brief manual page that will allow anybody else to run them. To raise your lower bound to 90, you'll also have to tell me, I don't care how, what is the second largest coefficient that occurs in the solution to the overall problem. The largest is
			:<math>718501723791605854129828161114970610468631182092721527047411632461591465663364\backslash</math>
			::<math>100118652346710314482731150749632703657021113788752749525948078496866540654\backslash</math>
			:::<math>8665959513826347108430640406618155029520076750443339789185589187423680</math>

Andrei new at 00:25, 26 March 2007

2007-03-26T00:25:55Z

← Older revision		Revision as of 20:25, 25 March 2007
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	This is how far I got so far. The program itself will do ~~others~~ just ~~file~~ (I ~~fiddled with~~ it a lot, so it may still have some bugs, however it works). The problem is working with such large numbers. The reducing part will multiply with 289 each time a <math>x^{240}</math> is reduced, so we could potentially get coefficients of the order <math>289^{964 - 240}</math>, which is very large indeed. On my computer, I get about 30% done, when it starts running out of memory and I had to stop. This is why the program just generates the matrices for the last part (and by the way, Maxima crashed when trying to load the matrix to invert, so I will have to find another program that will handle such large sets of data)		This is how far I got so far. The program itself will do any polynomials just fine (I tweaked it a lot to get the final poly, so it may still have some bugs, however it works). The problem is working with such large numbers. The reducing part will multiply with 289 each time a <math>x^{240}</math> is reduced, so we could potentially get coefficients of the order <math>289^{964 - 240}</math>, which is very large indeed. On my computer, I get about 30% done, when it starts running out of memory and I had to stop. This is why the program just generates the matrices for the last part (and by the way, Maxima crashed when trying to load the matrix to invert, so I will have to find another program that will handle such large sets of data)

76.64.136.153 at 00:03, 26 March 2007

2007-03-26T00:03:59Z

New page

Here is what I have so far in terms of polynomials:

Polynomial with root <math>\sqrt{2}</math>:

<math>{x}^{2} - 2</math> (duh)

Polynomial with root <math>\sqrt[5]{7}</math>:

<math>x^5 - 7</math> (duh as well)

Polynomial with root <math>\sqrt{2} + \sqrt[5]{7}</math>:

<math>x^{10} - 10*x^8 + 40*x^6 - 14*x^5 - 80*x^4 - 280*x^3 + 80*x^2 - 280*x + 17
</math> (more interesting!)

Polynomial with root <math>\sqrt[3]{\sqrt{2} + \sqrt[5]{7}}</math>:

<math>x^{30} - 10*x^{24} + 40*x^{18} - 14*x^{15} - 80*x^{12} - 280*x^9 + 80*x^6 - 280*x^3 + 17
</math> (similar to the above)

Polynomial with root <math> {1} \over {\sqrt[3]{\sqrt{2} + \sqrt[5]{7}}}</math>:

<math>
17*x^{30} - 280*x^{27} + 80*x^{24} - 280*x^{21} - 80*x^{18} - 14*x^{15} + 40*x^{12} - 10*x^6 + 1
</math> (just invert the above .. quite the same as the above 2)

Polynomial with root <math> {\sqrt{5}} \over {\sqrt[3]{\sqrt{2} + \sqrt[5]{7}}}</math>:

<math>
289*x^{60} - 9460000*x^{54} - 2392500000*x^{48} - 190781250000*x^{42} + 1127929687500*x^{36} - 249084472656250*x^{30} + 12817382812500000*x^{24} - 457763671875000000*x^{18} + 10728836059570312500*x^{12} - 149011611938476562500*x^6 + 931322574615478515625
</math> (ok .. this is ugly)

Polynomial with root <math> \sqrt[4]{{\sqrt{5}} \over {\sqrt[3]{\sqrt{2} + \sqrt[5]{7}}}}</math>:

<math>
289*x^{240} - 9460000*x^{216} - 2392500000*x^{192} - 190781250000*x^{168} +
1127929687500*x^{144} - 249084472656250*x^{120} + 12817382812500000*x^{96} -
457763671875000000*x^{72} + 10728836059570312500*x^{48} - 149011611938476562500*x^{24} + 931322574615478515625
</math> (very ugly .. especially considering you are not done yet and the next matrix to compute will be quite large)

This is how far I got so far. The program itself will do others just file (I fiddled with it a lot, so it may still have some bugs, however it works). The problem is working with such large numbers. The reducing part will multiply with 289 each time a <math>x^{240}</math> is reduced, so we could potentially get coefficients of the order <math>289^{964 - 240}</math>, which is very large indeed. On my computer, I get about 30% done, when it starts running out of memory and I had to stop. This is why the program just generates the matrices for the last part (and by the way, Maxima crashed when trying to load the matrix to invert, so I will have to find another program that will handle such large sets of data)