Experimental mathematics

bobbym · 2011-03-17 21:22:26

I have created a new thread for discussion of Computer Algebra and the PSLQ

Hi gAr;

Do not worry about that problem right now. It is typically for the user to provide the vector of constants. You are required to sort of know which constants apply for the problem you are working on.

Here is the example that is the beginning of my experience with the methods.

They begin with this example: evaluate the following integral in closed form.

No analytical methods can be used. Only the techniques of experimental math are allowed.

gAr · 2011-03-17 21:45:13

Hi bobbym,

The value I get is -16.699473719229... , but couldn't find a closed form.

bobbym · 2011-03-18 00:46:58

Hi gAr;

We get the integral by gaussian integration to 25 and then 50 places. We then PSLQ both values.

Look at the first row of both answers, you see the top vector. Both answers have the same one. That is the answer!

On mine I have it programmed to return the negative of the vector so we negate the top row to

[-24 , 0 , 0 , -120 , 0 , -140 , 0 , 15 , 0]

If you look at how the vector of constants is arranged in the pslq command then the answer is

Cleaning up and substituting we get:

It looks like we did the integral by some analytical method does it not? All done by integer relations. Of course the above is not a strict mathematical proof. But it is a strong conjecture because typically these are done to hundreds or thousands of places.

gAr · 2011-03-18 01:19:08

Hi bobbym,

That looks very good.
But it's so difficult to integrate!
I'll look around for more examples.

bobbym · 2011-03-18 01:26:00

Hi gAr;

I have other examples. The technique can be used for many different things. There are problems posted here that could not be answered. The OP thought that they were very tough but the truth is they are only tough when the classical methods are used. They break easily to the PSLQ.

gAr · 2011-03-18 01:39:28

Thanks!
I'll ask for more when I'm ready.

gAr · 2011-03-18 01:55:55

Hi bobbym,

I tried your example, mpmath's pslq didn't show anything, strange! I tried varying the precision, using the latest version directly.

Last edited by gAr (2011-03-18 01:56:19)

bobbym · 2011-03-18 02:09:09

Hi gAr;

It is due to the differences in the algorithms used maybe. Mine as I said is homegrown, it uses LLL or the lattice reduce algorithm. Mpmath is using the original Ferguson algorithm.

gAr · 2011-03-18 02:19:05

Oh, ok.
Experimental mathematics is cool!

bobbym · 2011-03-18 02:30:57

Did you use my 50 digit number and set your environment variable to around 50.

Here is a 100 digit answer

-16.69947371922907049618724340073146784130179174288144470245664281170485208378578722037304516803788261 so set your precision to around 100.

Also you can try removing the minus sign, or taking 16 away so that the value is less than 1.

If you think you can program in sage well enough I can take you through my program one line at a time ( there are only 6 commands ).

gAr · 2011-03-18 03:13:53

Hi,

I tried all the numbers you showed, still nothing.
I can try to program.

bobbym · 2011-03-18 03:16:40

Hi gAr;

Do you think it is better to try to copy my program or to try a few different examples of other type problems?
Think about it and I will see you later, I need a little rest.

gAr · 2011-03-18 03:35:14

Hi bobbym,

I think it's better to look into the program first.

gAr · 2011-03-18 16:05:51

Hi bobbym,

Got it! Your 25 digit value's sufficient.
I entered this:

pslq([mpf('-16.69947371922907049618724347541020677037'),1,pi,pi**2,pi**3,pi**4,pi**5,pi**6,pi**7], verbose=True, maxcoeff=10**30, maxsteps=300)

The vector I get is [24, 0, 0, 120, 0, 140, 0, -15, 0]

Without verbose I had no idea what was the reason, very helpful of it. It required 222 iterations. Also, I was entering without mpf('...').
The precision I set are:
mp.dps=30
mp.prec=103

Now, that was fun. Their documentation should have had examples like this.

I'm ready for more now!
But I do not understand the vector you get, why is there more than 9 values?

Last edited by gAr (2011-03-18 16:10:09)

bobbym · 2011-03-18 16:08:01

Hi gAr;

How are you today? I am glad to hear that. That mpmath is a very nice package. Give a couple of minutes to prepare a new example.

gAr · 2011-03-18 16:13:32

Hi bobbym,

I'm fine. How are you?
I edited my previous post when you were posting.

But I do not understand the vector you get, why is there more than 9 values?

bobbym · 2011-03-18 16:15:22

Hi gAr;

Which vector are we talking about?

gAr · 2011-03-18 16:19:29

This one:

I got only the first row, what are the rest?

bobbym · 2011-03-18 16:36:07

Oh, they are in some cases other answers, but usually just debris left over from the algorithm. I could have the program remove them and get one row vector but I like to see them.

Next problem:

http://www.mathisfunforum.com/message.php

post #333

If you look at phrontister's answer you will see he has produced what the 4 solutions are. The GF can only count them it cannot tell what type they are. He used a computer program let us see if we can use a PSLQ to get some of those answers.

If you PSLQ this vector [92,1,5,10,25] what do you get?

gAr · 2011-03-18 16:42:34

Hi bobbym,

Oh, okay.

I'm not able to follow that link. I get a blank message.
This page, right?: http://www.mathisfunforum.com/viewtopic.php?id=12832&p=14

Last edited by gAr (2011-03-18 16:46:30)

bobbym · 2011-03-18 16:48:56

I am sorry, please use this link instead:

http://www.mathisfunforum.com/viewtopic … 12832&p=17

and go to post #333.

gAr · 2011-03-18 17:06:50

Hi bobbym,

I get [0, 0, -1, -2, 1].
So should we try deleting some values, multiplying some values etc.?

bobbym · 2011-03-18 17:10:00

Keep tweaking and look for an answer like this:

[-1,-2,-1,-1,-3]

Which states:

92 = 2 p + 1 n + 1 d + 3 q.

I think I see the problem on mine the 92 goes into the vector on yours it does not!

gAr · 2011-03-18 17:21:22

I don't get that, is there anything I can do to force it to use all digits provided?
And the original problem stated there must be at least two of each denomination, can I enforce that condition?

bobbym · 2011-03-18 17:24:09

Hi gAr;

I have bent the rules on the problem a little as a first demonstration. Are you leaving the 92 out of your basis? In mine it goes in and yours it is not part of the vector.

Math Is Fun Forum

#1 2011-03-17 21:22:26

Experimental mathematics

#2 2011-03-17 21:45:13

Re: Experimental mathematics

#3 2011-03-18 00:46:58

Re: Experimental mathematics

#4 2011-03-18 01:19:08

Re: Experimental mathematics

#5 2011-03-18 01:26:00

Re: Experimental mathematics

#6 2011-03-18 01:39:28

Re: Experimental mathematics

#7 2011-03-18 01:55:55

Re: Experimental mathematics

#8 2011-03-18 02:09:09

Re: Experimental mathematics

#9 2011-03-18 02:19:05

Re: Experimental mathematics

#10 2011-03-18 02:30:57

Re: Experimental mathematics

#11 2011-03-18 03:13:53

Re: Experimental mathematics

#12 2011-03-18 03:16:40

Re: Experimental mathematics

#13 2011-03-18 03:35:14

Re: Experimental mathematics

#14 2011-03-18 16:05:51

Re: Experimental mathematics

#15 2011-03-18 16:08:01

Re: Experimental mathematics

#16 2011-03-18 16:13:32

Re: Experimental mathematics

#17 2011-03-18 16:15:22

Re: Experimental mathematics

#18 2011-03-18 16:19:29

Re: Experimental mathematics

#19 2011-03-18 16:36:07

Re: Experimental mathematics

#20 2011-03-18 16:42:34

Re: Experimental mathematics

#21 2011-03-18 16:48:56

Re: Experimental mathematics

#22 2011-03-18 17:06:50

Re: Experimental mathematics

#23 2011-03-18 17:10:00

Re: Experimental mathematics

#24 2011-03-18 17:21:22

Re: Experimental mathematics

#25 2011-03-18 17:24:09

Re: Experimental mathematics

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