How does Randall do these?

bobbym · 2014-04-22 23:48:06

 pslq[l_,dig_]:=Module[{a},
     a=IdentityMatrix[Length[l]];
     a=Append[a,10^dig*N[l,dig]];
     a=Transpose[a];
     a=Rationalize[a,10^-dig];
     a=LatticeReduce[a];
     Take[a,All,{1,Length[l]}]
     ];

Agnishom · 2014-04-22 23:53:36

Then?

bobbym · 2014-04-22 23:56:06

Try one:

pslq[{-16.69947371922907049618724347541020677037,1, \[Pi], \[Pi]^2, \[Pi]^3, \[Pi]^4, \[Pi]^5, \[Pi]^6, \[Pi]^7}, 25]

and then

pslq[{-16.6994737192290704961872434007314678413017917428814446693080866964921,1, \[Pi], \[Pi]^2, \[Pi]^3, \[Pi]^4, \[Pi]^5, \[Pi]^6, \[Pi]^7},50]

ShivamS · 2014-04-23 00:33:45

Agnishom wrote:

Can you teach me that?

You need some linear algebra.

bobbym · 2014-04-23 00:42:06

Also, he could use numerical analysis and a good grasp of M commands.

Agnishom · 2014-04-23 02:16:29

What is linear algebra like?

ShivamS · 2014-04-23 02:32:47

One of my friends describe it as "The hand wavy stuff."

It deals with potentially any aspect of maths that is algebraic, so involves operations of addition and multiplication, and such that these operations are linear. Something earns the title linear if it has to do with lines, planes and so on. Such things can be given by equations that are linear in the sense of they only involve x,y etc and NOT x^2, y^2 or higher powers.

It is a vast subject, and it is very easy to learn, hard to describe because there is so much of it. Generally it deals with: vectors, vector spaces, linear maps (matrices), solving linear equations in more than one unknown. It might go on to talk about determinants, eigenvectors, eigenspaces.

It is practical, and has a very elegant theoretical aspect as well, but not difficult.

Last edited by ShivamS (2014-04-23 02:33:00)

bobbym · 2014-04-23 03:51:19

Did you try the example yet?

Also, you should keep in mind that some results are misleading.

When Randall says that the 4th root of 9^2 + 19^2/22 is pi he is of course incorrect!

Here is the correct use of M.

NSolve[x^4 == N[Solve[x^4 == 9^2 + 19^2/22]], WorkingPrecision -> 25]

N[Pi, 25]

Pi is a transcendental number that means it is not a root of a non-zero polynomial equation with rational (or integer) coefficients.

anonimnystefy · 2014-04-23 07:13:23

If you had read the comic in post #1 you would know that he does not mean that seriously.

bobbym · 2014-04-23 07:23:57

If I know Randall, he is serious.

anonimnystefy · 2014-04-23 10:08:05

Well, then you don't.

I have a question about PSLQ, how would you get what 10.34159265358979 is with it?

bobbym · 2014-04-23 10:28:03

Can you come up with more digits easily to test?

anonimnystefy · 2014-04-23 11:18:14

Well, that is the one I had in mind, but, what does PSLQ actually give out for that?

bobbym · 2014-04-23 11:24:41

It would depend on what your basis vector was.

pslq[{10.341592653589792, 1, \[Pi], \[Pi]^2, \[Pi]^3, \[Pi]^4}, 10]

See the top row.

anonimnystefy · 2014-04-23 13:12:02

Hm, it seems to get worse to more arguments it has.

bobbym · 2014-04-23 13:18:56

The top row is the exact answer!

anonimnystefy · 2014-04-23 13:21:11

Yes, yes, that's okay. What I'm saying is that it gets worse at finding the relations the more constants I add.

bobbym · 2014-04-23 13:22:25

Again, that depends.Maple and the ISC have solved the problem succesfully I have not.

anonimnystefy · 2014-04-23 13:27:05

Hm?

bobbym · 2014-04-23 13:29:15

Do not be like a mathematician, Do not worry about the times the algorithm does not get there. Rejoice in the fact that it often does!

Agnishom · 2014-04-23 14:19:05

Who is the ISC?

bobbym · 2014-04-23 14:23:03

One of the mightiest sites on the web.

gAr · 2014-04-26 19:13:44

Try one:

pslq[{-16.69947371922907049618724347541020677037,1, \[Pi], \[Pi]^2, \[Pi]^3, \[Pi]^4, \[Pi]^5, \[Pi]^6, \[Pi]^7}, 25]

and then

pslq[{-16.6994737192290704961872434007314678413017917428814446693080866964921,1, \[Pi], \[Pi]^2, \[Pi]^3, \[Pi]^4, \[Pi]^5, \[Pi]^6, \[Pi]^7},50]

What is that number supposed to be, there's some mismatch.

16.69947371922907049618724347541020677037
16.6994737192290704961872434007314678413017917428814446693080866964921

bobbym · 2014-04-26 20:33:42

Probably to higher precision.

That is the integral in

http://www.mathisfunforum.com/viewtopic.php?id=15243

since it is an approximation each version of M gives a different answer although the pslq still gets there!

gAr · 2014-04-26 21:49:59

Ah, okay!
Thanks for reminding that, I had almost lost touch with the pslq.

This is the higher precision output by D. H. Bailey's "experimental mathematician's toolkit": -16.69947371922907049618724340073146784130179174288144470245664281170485

Math Is Fun Forum

#26 2014-04-22 23:48:06

Re: How does Randall do these?

#27 2014-04-22 23:53:36

Re: How does Randall do these?

#28 2014-04-22 23:56:06

Re: How does Randall do these?

#29 2014-04-23 00:33:45

Re: How does Randall do these?

#30 2014-04-23 00:42:06

Re: How does Randall do these?

#31 2014-04-23 02:16:29

Re: How does Randall do these?

#32 2014-04-23 02:32:47

Re: How does Randall do these?

#33 2014-04-23 03:51:19

Re: How does Randall do these?

#34 2014-04-23 07:13:23

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#35 2014-04-23 07:23:57

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#36 2014-04-23 10:08:05

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#37 2014-04-23 10:28:03

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#38 2014-04-23 11:18:14

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#39 2014-04-23 11:24:41

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#40 2014-04-23 13:12:02

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#41 2014-04-23 13:18:56

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#42 2014-04-23 13:21:11

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#43 2014-04-23 13:22:25

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#44 2014-04-23 13:27:05

Re: How does Randall do these?

#45 2014-04-23 13:29:15

Re: How does Randall do these?

#46 2014-04-23 14:19:05

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#47 2014-04-23 14:23:03

Re: How does Randall do these?

#48 2014-04-26 19:13:44

Re: How does Randall do these?

#49 2014-04-26 20:33:42

Re: How does Randall do these?

#50 2014-04-26 21:49:59

Re: How does Randall do these?

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