What do you think?

bobbym · 2011-03-17 03:31:47

Hi gAr;

That is what I am talking about. They have the same command Maple does, identify(). Once you get the 2 values I will show you how to use that and the PSLQ to figure the answer.

Basically inspired by Ramanujans ability to take a number and show that it could be represented in terms of other known constants.
This has become a powerful technique to do sums, integrals, recurrences etc.

gAr · 2011-03-17 03:41:34

It's amazing.
I never knew such an algorithm existed. Wow!

bobbym · 2011-03-17 03:54:13

It also solves diophantine equations, does products and much more.

For instance supposing a integral that you did numerically produced this answer.

3.4936551040529309771

How could you tell what that number equals in terms of known constants?

gAr · 2011-03-17 04:43:39

Hi bobbym,

What constants did you get for your example: 3.4936551040529309771 ?

I used pi, e and phi as base, here is the list:

((77/100) + (1/50)*e + (69/100)*pi + (31/100)*phi)
e/((-7/36) + (7/12)*e + (11/36)*pi + (-35/36)*phi)
sqrt((413 + 40*e + (-138)*pi + (-34)*phi))/sqrt(e)
exp(((-47/11) + 4*e + (-101/33)*pi + (29/11)*phi))
exp(((-43/38) + 1*e + (5/19)*pi + (-8/19)*phi))/phi
((-37/23) + (88/23)*e + (51/23)*pi + (-89/23)*phi)/e
((-7/9) + (10/7)*e + (11/21)*pi + (-65/63)*phi)**2/e
1/log(((-147/59) + 2*e + (-37/59)*pi + (13/59)*phi))
log(phi*((-67) + (78/5)*e + (54/5)*pi + (34/5)*phi))
exp(((-36/11) + 4*e + (-101/33)*pi + (29/11)*phi))/e
e*exp(((-58/11) + 4*e + (-101/33)*pi + (29/11)*phi))
(7**(35/8)*e**(5/2))/(2*3**4*5**(9/4)*pi**(1/2)*phi)
((178/21) + (-5/21)*e + (37/21)*pi + (-31/21)*phi)/pi
((21/20) + (41/20)*e + (17/10)*pi + (-39/10)*phi)/phi
pi*((-12/7) + (104/63)*e + (-19/21)*pi + (46/63)*phi)
((-160/99) + (32/99)*e + (43/99)*pi + (76/99)*phi)**2
log(((-7/3) + (77/3)*e + (49/3)*pi + (-19/2)*phi)/pi)
phi/((124/99) + (40/33)*e + (-4/11)*pi + (-20/11)*phi)
pi/((-91/45) + (361/45)*e + (-5)*pi + (-29/15)*phi)**2
1/exp(((47/11) + (-4)*e + (101/33)*pi + (-29/11)*phi))
e/exp(((58/11) + (-4)*e + (101/33)*pi + (-29/11)*phi))
e*((-18/71) + (160/213)*e + (4/213)*pi + (-74/213)*phi)
exp(e/((37/52) + (-7/12)*e + (19/78)*pi + (55/39)*phi))
phi*((73/59) + (-67/59)*e + (134/59)*pi + (-114/59)*phi)
e/((87/344) + (-1/172)*e + (15/344)*pi + (27/86)*phi)**2
log(pi*((-163/12) + 12*e + (-197/12)*pi + (319/12)*phi))
log(pi/((141/40) + (1/8)*e + (-7/20)*pi + (-33/20)*phi))
exp(((-61/85) + (67/85)*e + (-1/5)*pi + (84/85)*phi))/pi
1/((22/131) + (-54/131)*e + (61/131)*pi + (2/131)*phi)**2
phi*log(((7/16) + (137/64)*e + (13/64)*pi + (35/32)*phi))
pi/log(((107/11) + (2/11)*e + (-38/11)*pi + (21/11)*phi))
phi*exp(((7/15) + (-24/25)*e + (17/75)*pi + (34/25)*phi))
exp(((-144/43) + (63/43)*e + (-9/43)*pi + (91/43)*phi)/e)
exp(1/((-3/7) + (71/245)*e + (23/245)*pi + (22/245)*phi))
exp(phi/((-37/75) + (13/75)*e + (26/75)*pi + (7/50)*phi))
e*sqrt(((-151/20) + (-9/20)*e + (-49/20)*pi + (56/5)*phi))
((43/552) + (149/276)*e + (77/138)*pi + (5/552)*phi)**2/pi
phi*((-116/237) + (1/3)*e + (65/237)*pi + (28/237)*phi)**2
phi/((53/20) + (63/20)*e + (-151/20)*pi + (163/20)*phi)**2
e**2/((-37/68) + (-87/136)*e + (4/17)*pi + (63/34)*phi)**2
log(e/((-61/20) + (81/20)*e + (-159/40)*pi + (57/20)*phi))
log(phi/((-66/65) + (-7/65)*e + (-9/65)*pi + (72/65)*phi))
exp(phi*((-156/5) + (-13/5)*e + (-11/5)*pi + (142/5)*phi))
exp(pi/((-69/124) + (11/62)*e + (33/62)*pi + (35/62)*phi))
1/((-51/211) + (-121/211)*e + (152/211)*pi + (-23/211)*phi)
((-23/40) + (-18/5)*e + (173/40)*pi + (-1/8)*phi)**2/phi**2
log(((53/7) + (123/14)*e + (145/14)*pi + (-93/14)*phi)/phi)
phi/exp(((-7/15) + (24/25)*e + (-17/75)*pi + (-34/25)*phi))
sqrt(((-171/158) + (160/79)*e + (223/79)*pi + (-53/79)*phi))
sqrt(e)/sqrt(((8/25) + (7/50)*e + (-2/25)*pi + (-7/50)*phi))
phi**2*((-28/39) + (4/13)*e + (-17/26)*pi + (149/78)*phi)**2
phi**2/((93/7) + (-41/14)*e + (-127/14)*pi + (104/7)*phi)**2
log(e*((-192/13) + (-42/13)*e + (163/13)*pi + (-30/13)*phi))
log(1/((-352/87) + (173/87)*e + (-35/87)*pi + (-7/174)*phi))
sqrt(((-58/19) + (-65/19)*e + (235/19)*pi + (64/19)*phi))/phi
pi/sqrt(((220/53) + (-18/53)*e + (-13/53)*pi + (-54/53)*phi))
sqrt(pi)/sqrt(((-52/5) + (56/5)*e + (-66/5)*pi + (67/5)*phi))
pi**2*((88/153) + (79/153)*e + (-6/17)*pi + (-26/153)*phi)**2
pi**2/((113/48) + (7/24)*e + (-43/144)*pi + (-47/144)*phi)**2
pi*log(((-187/58) + (160/29)*e + (-205/58)*pi + (85/58)*phi))
phi/log(((-51/182) + (123/182)*e + (1/28)*pi + (-9/182)*phi))
exp(((345/103) + (-5/103)*e + (-6/103)*pi + (57/103)*phi)/pi)
exp(e*((-43/122) + (-11/61)*e + (133/122)*pi + (-80/61)*phi))
1/sqrt(((81/215) + (121/215)*e + (-19/43)*pi + (-58/215)*phi))
phi/sqrt(((-205/32) + (11/4)*e + (-447/32)*pi + (851/32)*phi))
sqrt(((-123/2) + (37/2)*e + (-28)*pi + (147/2)*phi))/sqrt(phi)
pi*((244/307) + (146/307)*e + (-120/307)*pi + (37/307)*phi)**2
e*log(((97/151) + (194/151)*e + (-95/151)*pi + (136/151)*phi))
e/log(((-381/574) + (53/287)*e + (79/287)*pi + (523/574)*phi))
sqrt(e)*sqrt(((13/9) + (-46/9)*e + (139/36)*pi + (107/36)*phi))
e*((-914/151) + (443/151)*e + (215/151)*pi + (-491/151)*phi)**2
pi*exp(((-241/364) + (97/364)*e + (-135/364)*pi + (68/91)*phi))
pi/exp(((241/364) + (-97/364)*e + (135/364)*pi + (-68/91)*phi))
exp(pi*((1/149) + (316/149)*e + (-414/149)*pi + (309/149)*phi))
pi*sqrt(((-71/143) + (-175/143)*e + (134/143)*pi + (17/13)*phi))
sqrt(phi)*sqrt(((-475/54) + (14/3)*e + (26/27)*pi + (7/18)*phi))
((10/137) + (-195/137)*e + (332/137)*pi + (107/137)*phi)**2/e**2
exp(((-13/320) + (-49/64)*e + (129/160)*pi + (319/320)*phi)/phi)
phi*sqrt(((11/267) + (-287/267)*e + (485/267)*pi + (101/89)*phi))
e/sqrt(((313/545) + (404/545)*e + (-578/545)*pi + (454/545)*phi))
e**2*((256/181) + (-441/181)*e + (384/181)*pi + (-86/181)*phi)**2
sqrt(phi)/sqrt(((199/84) + (-43/42)*e + (15/28)*pi + (-59/84)*phi))
((-427/166) + (635/166)*e + (299/166)*pi + (-781/166)*phi)**2/pi**2
sqrt(pi)*sqrt(((-548/117) + (-2/117)*e + (223/117)*pi + (190/117)*phi))

bobbym · 2011-03-17 10:41:18

Hi gAr;

You obviously used a PSLQ. Usually they are used with a bigger base than that. Try using identify(). If that does not work then there is a little trick that will.

gAr · 2011-03-17 14:54:15

Hi bobbym,

I used identify() itself, with syntax like this: identify(3.4936551040529309771,['pi','e','phi'])
It did not show anything for tolerance greater than 1e-15.
But identify calls pslq, isn't it? How can I go for any level of accuracy?

bobbym · 2011-03-17 15:05:57

Hi gAr;

Looking at the documentation for that command.

mpmath.identify(ctx, x, constants=[], tol=None, maxcoeff=1000, full=False, verbose=False)

You might have two choses tol = something other than none, try 10^-30. Also experiment with maxcoeff to see if it will get a smaller answer.

All packages have environment variables. This might be one.

mp.dps = 15 try setting that higher.

gAr · 2011-03-17 15:12:24

Okay, thanks.
I'll look into it.

bobbym · 2011-03-17 17:24:40

Looking at it again that is an environment variable. The one that controls the digits of precision. Also you do not need to specify a vector of known constants. That is an optional parameter. You could just try indentify( your value ).

Also check out this page. The father of all this.

http://oldweb.cecm.sfu.ca/projects/ISC/ISCmain.html

gAr · 2011-03-17 18:22:00

Thank you.
That's a very good link.

gAr · 2011-03-17 18:40:09

I got the answer after providing log(2) and sqrt(2) as base.
It would have been better if it tries the combinations itself by default.

bobbym · 2011-03-17 18:45:23

Did you try just identify(3.4936551040529309771)

and with what precision?

gAr · 2011-03-17 18:47:54

After I provided the base, a precision of 15 was sufficient.
I tried: identify(3.4936551040529309771,['log(2)','sqrt(2)'])

Wolframalpha is able to give the answer, but doesn't seem to be having a mathematica command.

Last edited by gAr (2011-03-17 18:58:34)

bobbym · 2011-03-17 19:11:04

You got no answer at all with just identify?

According to the documentation that could be a sign of not enough precision.

Try this:

>>> from mpmath import *
>>> mp.dps = 50; mp.pretty = True
>>> identify(sqrt(2)+ 3 *log(2))

See if raising mp.dps helps.

There is no direct Mathematica command like identify in Maple. Wolfram alpha and Mathematica are 2 different things.

gAr · 2011-03-17 19:26:22

>>> from mpmath import *
>>> mp.dps = 50; mp.pretty = True
>>> identify(sqrt(2)+ 3 *log(2))

Did not identify for any precision, unless the base is provided.

bobbym · 2011-03-17 20:26:35

Hi gAr;

Rule #4: On that page

http://mpmath.googlecode.com/svn/trunk/ … ation.html

seems to indicate that it should have. Sometimes you just are stuck waiting for another version of a package.

gAr · 2011-03-17 20:36:02

Hi bobbym,

I'm using it from sage, it uses v0.15, so that may be the reason,

bobbym · 2011-03-17 20:42:29

Are you using the latest Sage and can you update the mpmath package?

gAr · 2011-03-17 20:55:38

Yes, I'm using the latest version of sage, v4.6.2.
I'm not sure how to update individual packages. But it's possible.

bobbym · 2011-03-18 01:33:13

Hi gAr;

I forgot to thank for your explanation on the recurrence in the other thread.

bobbym · 2011-03-19 16:02:06

New problem!

What is the largest determinant that a 7 x 7 matrix of 1's and 0's can have?

A says) That is easy. The answer is 32.
B says) There you go still blurting out the wrong answer. Did you choose that because it is a power of 2? The correct answer is 11.
C says) Caught you B, that is wrong also.
D says) B is usually wrong. Stick with A, just examine his record. He has been right 99% of the time.
A says) I agree that B is usually wrong but only 99% ? I am a lot higher than that.

gAr · 2011-03-20 00:14:38

Hi bobbym,

bobbym · 2011-03-20 00:35:41

Hi gAr;

gAr · 2011-03-20 00:54:30

Hi bobbym,

Interchanging two rows of the matrix with highest determinant would give the minimum I guess. It may not be possible to go below that value.
I can't prove it though.

bobbym · 2011-03-20 00:59:33

Hi gAr;

I had the same problem. I found a page that uses the formula for the hadamard maximum without the abs value, so I guess we do not have to worry about it.

Math Is Fun Forum

#726 2011-03-17 03:31:47

Re: What do you think?

#727 2011-03-17 03:41:34

Re: What do you think?

#728 2011-03-17 03:54:13

Re: What do you think?

#729 2011-03-17 04:43:39

Re: What do you think?

#730 2011-03-17 10:41:18

Re: What do you think?

#731 2011-03-17 14:54:15

Re: What do you think?

#732 2011-03-17 15:05:57

Re: What do you think?

#733 2011-03-17 15:12:24

Re: What do you think?

#734 2011-03-17 17:24:40

Re: What do you think?

#735 2011-03-17 18:22:00

Re: What do you think?

#736 2011-03-17 18:40:09

Re: What do you think?

#737 2011-03-17 18:45:23

Re: What do you think?

#738 2011-03-17 18:47:54

Re: What do you think?

#739 2011-03-17 19:11:04

Re: What do you think?

#740 2011-03-17 19:26:22

Re: What do you think?

#741 2011-03-17 20:26:35

Re: What do you think?

#742 2011-03-17 20:36:02

Re: What do you think?

#743 2011-03-17 20:42:29

Re: What do you think?

#744 2011-03-17 20:55:38

Re: What do you think?

#745 2011-03-18 01:33:13

Re: What do you think?

#746 2011-03-19 16:02:06

Re: What do you think?

#747 2011-03-20 00:14:38

Re: What do you think?

#748 2011-03-20 00:35:41

Re: What do you think?

#749 2011-03-20 00:54:30

Re: What do you think?

#750 2011-03-20 00:59:33

Re: What do you think?

Board footer