[math-fun] Gamma(p/q) values
Joerg Arndt: Just as quick copy and paste (suggest to start with the last one): J.\ M.\ Borwein, I.\ J.\ Zucker: {Fast evaluation of the gamma function for small rational fractions using complete elliptic integrals of the first kind}, IMA Journal of Numerical Analysis, vol.12, no.4, pp.519-526, \bdate{1992}.} Greg Martin: {A product of Gamma function values at fractions with the same denominator}, arXiv:0907.4384v1 [math.CA], \bdate{24-July-2009}. URL: \url{http://arxiv.org/abs/0907.4384}.} Albert Nijenhuis: {Small Gamma Products with Simple Values}, arXiv:0907.1689v1 [math.CA], \bdate{9-July-2009}. URL: \url{http://arxiv.org/abs/0907.1689}.} Raimundas Vid\={u}nas: {Expressions for values of the gamma function}, arXiv:math.CA/0403510, \bdate{30-March-2004}. URL: \url{http://arxiv.org/abs/math/0403510}.} ----WDS: Vidunas expresses all the Gamma(p/q) with q=60 or q=24 in terms of algebraic numbers and the Gamma values at just these ten p/q: 1/4, 1/8, 1/3, 1/15, 1/20, 1/24, 1/5, 2/5, 1/60, 7/60. (*) But he then notes that the first 6 among those 10 (i.e the first line) can be expressed in terms of the AGM. He also says all Gamma(p/120) can be expressed in terms of the following six: 1/40, 3/40, 7/40, 1/120, 7/120, 11/120. (**) So... focus your Riesian efforts on the Gamma(p/q) with p/q in the (*) and (**) lines! Vidunas at end also notes that Gamma(1/5) and Gamma(2/5) can be expressed in terms of two hyperelliptic integrals... but is there any fast AGM-like scheme for evaluating them? My earlier mathfun post on this topic was 25 Nov http://mailman.xmission.com/cgi-bin/mailman/private/math-fun/2011-November/0...
Noticing 1 and 11 form a multiplicative subgroup mod 120, I tried Ries on (11/120)! * (1/120)! = 0.95023643380439452766545957659649988150532 and (11/120)! / (1/120)! = 0.95931292205678601103043910565064773096559 but it failed to find anything I liked. A certain proportion of Ries formulas seem to be "tasteless" and the sort of thing that are never going to happen, like using (e^pi) or 1/ln(pi) or e or sin(1) in an exponent...
On Wed, Dec 28, 2011 at 14:58, Warren Smith <warren.wds@gmail.com> wrote:
A certain proportion of Ries formulas seem to be "tasteless" and the sort of thing that are never going to happen, like using (e^pi) or 1/ln(pi) or e or sin(1) in an exponent...
Yes, definitely. As Bill Gosper just pointed out to me, sin(1) anywhere is practically useless. And e^pi is a curio at best ("Gelfond's Constant" [5]). If you look at the webpage I wrote about ries [1], most of which is dedicated to "Stupid Math Tricks", perhaps the idea behind ries will be more clear. I actually started by looking at all the stuff in Plouffe's Inverter, consisting mostly of long expressions using obscure and undefined functions, peppered with the occasional fraction like "2087/1457" which I could have figured out on my own. Then, by chance I got a couple emails in a row from members of the Cult of 137 [3], and I decided to prove to myself (and to them) that it is easy to make all sorts of meaningless formulas for any number you choose. More to your point, the stuff on Plouffe's Inverter [4] is clearly all meaningful to someone, but most of it is of no meaning to any one person in particular, so we have a severe target marketing problem. Armed with these hundreds of higher-math functions, ries would be as bad, if not much worse. It will probably never have much serious application. Nevertheless, it sure is a lot of fun, and fun is a large part of the meaning of life. So there, xyzzy! - Robert. [1] http://mrob.com/pub/ries/index.html [2] http://pi.lacim.uqam.ca/ [3] http://mrob.com/pub/num/n-b137_035.html#cult_137 [4] http://bootes.math.uqam.ca/cgi-bin/ipcgi/lookup.pl?Submit=GO+&number=1.43239... [5] http://en.wikipedia.org/wiki/Gelfond's_constant -- Robert Munafo -- mrob.com Follow me at: gplus.to/mrob - fb.com/mrob27 - twitter.com/mrob_27 - mrob27.wordpress.com - youtube.com/user/mrob143 - rilybot.blogspot.com
Hello, I agree with Robert Munafo, cracking real numbers is hard, as far as I know there are 4-5 ways to do that. 1) Use my inverter with all the imperfections and these huge tables. Some findings have been made based on the version in Vancouver. The current one has 5.213 billion entries, some is available here http://pictor.math.uqam.ca/~plouffe/pi/ip/ in raw format, beware there are 9000 files totalling 657 gigabytes of data. 2) Use the Maple routine called identify(); well it gives neat examples in some cases but misses many others, with the option 'all' it can find things sometimes. 3) Use the RIES program, very fast yes, can find genuine approximations in a jiffy, better than any other methods but if we follow Steven Finch approach to the problem there are many improvements we could make by using named constants, there are hundreds of them like the Madelung constants, the Riemann zeros, the parking constant, the sofa constant, the Feigenbaum constants, etc. 4) Use a version of a generalized expansion program, it can find some continued fractions, some exotic expansions, but falls short on most real numbers and produces too much usable data. I made many versions of this idea and came out with more than 500 ways to expand a real number into a sequence. Some findings were made using this idea. It could be generalized even more but a good questions is , yes but in what direction ?? 5) The LLL-PSLQ algorithm, excellent for narrow cases but cannot find any of the neat examples suggested here recently like this approximation within 13 digits of a factorial expression found by the RIES program. In some cases like finding an algebraic expression the LLL-PSLQ program is the best on this planet but with compound GAMMA values mixed with algebraic numbers, yes it can find some of the genuine Bill Gosper-like identities but for this you need more than brute force approach, you need a clue on the possible expressions because we are dealing here with non-linear expressions even if we take the log, unless you are mr Gosper itself, but mr Gosper is not a method it is a human being.!, ... There is so far no global answer to this problem, the best is a mix of all this I presume ? Best Regards to all, Simon Plouffe Le 29/12/2011 13:32, Robert Munafo a écrit :
On Wed, Dec 28, 2011 at 14:58, Warren Smith<warren.wds@gmail.com> wrote:
A certain proportion of Ries formulas seem to be "tasteless" and the sort of thing that are never going to happen, like using (e^pi) or 1/ln(pi) or e or sin(1) in an exponent...
Yes, definitely. As Bill Gosper just pointed out to me, sin(1) anywhere is practically useless. And e^pi is a curio at best ("Gelfond's Constant" [5]).
If you look at the webpage I wrote about ries [1], most of which is dedicated to "Stupid Math Tricks", perhaps the idea behind ries will be more clear.
I actually started by looking at all the stuff in Plouffe's Inverter, consisting mostly of long expressions using obscure and undefined functions, peppered with the occasional fraction like "2087/1457" which I could have figured out on my own. Then, by chance I got a couple emails in a row from members of the Cult of 137 [3], and I decided to prove to myself (and to them) that it is easy to make all sorts of meaningless formulas for any number you choose.
More to your point, the stuff on Plouffe's Inverter [4] is clearly all meaningful to someone, but most of it is of no meaning to any one person in particular, so we have a severe target marketing problem. Armed with these hundreds of higher-math functions, ries would be as bad, if not much worse. It will probably never have much serious application.
Nevertheless, it sure is a lot of fun, and fun is a large part of the meaning of life. So there, xyzzy!
- Robert.
[1] http://mrob.com/pub/ries/index.html
[3] http://mrob.com/pub/num/n-b137_035.html#cult_137
[4] http://bootes.math.uqam.ca/cgi-bin/ipcgi/lookup.pl?Submit=GO+&number=1.43239...
Simon, I'm glad you've weighed in on this issue. I've felt bad for disrespecting your Inverter just because it doesn't meet my simple and shallow needs. I'm not qualified to appreciate its benefits or even to understand most of the functions (I've tried learning many of them but my eyes glaze over). I wonder if you get as much mail from the dedicated, but somewhat fanatical and misguided, members of the "Cult of 137". I refer to those people, including none less than Sir Arthur Stanley Eddington [1], who were/are convinced that there is profound truth about the Universe hidden in the simplest of formulas involving elementary functions. It would be all right, if they didn't keep nagging me to put their findings on my website. Constants are actually pretty easy to add to ries and it's in my plans for the near future. The symbol and rule tables were designed with the idea in mind from the start. - Robert On Thu, Dec 29, 2011 at 09:42, Simon Plouffe <simon.plouffe@gmail.com> wrote:
[...] we are dealing here with non-linear expressions even if we take the log, unless you are mr Gosper itself, but mr Gosper is not a method it is a human being.!, ...
Sounds like "The Prisoner"... I am not a number! I am a free man! (see [2]) [1] http://en.wikipedia.org/wiki/Eddington_number [2] http://en.wikipedia.org/wiki/The_Prisoner On Thu, Dec 29, 2011 at 09:42, Simon Plouffe <simon.plouffe@gmail.com>wrote:
Hello,
I agree with Robert Munafo, cracking real numbers is hard, as far as I know there are 4-5 ways to do that.
[...] 3) Use the RIES program, very fast yes, can find genuine approximations in a jiffy, better than any other methods but if we follow Steven Finch approach to the problem there are many improvements we could make by using named constants, there are hundreds of them like the Madelung constants, the Riemann zeros, the parking constant, the sofa constant, the Feigenbaum constants, etc.
[...]
There is so far no global answer to this problem, the best is a mix of all this I presume ?
Best Regards to all, Simon Plouffe
-- Robert Munafo -- mrob.com Follow me at: gplus.to/mrob - fb.com/mrob27 - twitter.com/mrob_27 - mrob27.wordpress.com - youtube.com/user/mrob143 - rilybot.blogspot.com
More wish list: Supply the Inverter as a program (like ries): It would need to include code to calculate the special functions to moderate precision, 100D or so. Cost: Loss of control by authors. Benefit: Smaller data transfer problems. User can choose to run for a month to try harder. My cycles, not yours. User can add own functions. +-: Private searching. (NJAS might have some comment here.) Allow for some user control of the generation priorities. You might assign points to each function based on likelihoods and arrange the search to generate low-scoring numbers first. 355/113 should be tried before quotients of ten digit numbers. Add a few more specific search algorithms: create numbers related to the original X, like e^X, X^2, etc. as 'reach out', and then try 'reach back' by expanding continued fractions, and using LLL to find linear relationships. It would be nice to automatically compare generated continued fractions (&c) against OEIS sequences, but that requires a local copy of OEIS. There's a simple 'two-list' scheme that can find splits like X = A+B where A and B are supplied as sorted lists. (Simultaneous scan through values X-A, and B, looking for a match.) A slightly more complicated version 'four-lists' can look for X = A+B+C+D. This reduces memory to the sum of the list sizes, but still needs time (size A)(size B)+(size C)(size D). The combining operator need not be +. Allow several numbers to be input, perhaps to find a relationship. Automatically check for rational numbers with continued fraction, and for algebraic numbers with LLL. Can anything be done with complex numbers? Expand the language for defining special functions, beyond the named set. Include numerical integrals, ODEs, infinite series, and iteration limits like AGM. Before being too hasty to deprecate e^pi and sin 1, remember that sin 1 has a similar series to e, and that e^pi is another name for (-1)^i. Both tan 1 = sin 1 / cos 1, and Bessel I0(2)/I1(2) have nice continued fractions. User control of the search priority is double edged: The upside is that your personal knowledge of the number's origins is a big clue to what expressions might be relevant. (We don't need no stinkin' Bessel functions.) The downside is that you lose the benefit of a fresh point of view, since every number and special function is an input from the math community at large. The tautology problem will become worse. If we can supply the target number as a program (to calculate more digits), then we can auto-discard some mismatches. The expected yield of this general approach is low, but not quite zero. So user cost & convenience is a consideration. Having a 'set & forget' option becomes interesting; or a background searcher. Even 'middle- click the number to do a quick check', a sort of google-for-numbers. Rich ---- Quoting Simon Plouffe <simon.plouffe@gmail.com>:
Hello,
I agree with Robert Munafo, cracking real numbers is hard,
as far as I know there are 4-5 ways to do that.
1) Use my inverter with all the imperfections and these huge tables. Some findings have been made based on the version in Vancouver. The current one has 5.213 billion entries, some is available here http://pictor.math.uqam.ca/~plouffe/pi/ip/ in raw format, beware there are 9000 files totalling 657 gigabytes of data.
2) Use the Maple routine called identify(); well it gives neat examples in some cases but misses many others, with the option 'all' it can find things sometimes.
3) Use the RIES program, very fast yes, can find genuine approximations in a jiffy, better than any other methods but if we follow Steven Finch approach to the problem there are many improvements we could make by using named constants, there are hundreds of them like the Madelung constants, the Riemann zeros, the parking constant, the sofa constant, the Feigenbaum constants, etc.
4) Use a version of a generalized expansion program, it can find some continued fractions, some exotic expansions, but falls short on most real numbers and produces too much usable data. I made many versions of this idea and came out with more than 500 ways to expand a real number into a sequence. Some findings were made using this idea. It could be generalized even more but a good questions is , yes but in what direction ??
5) The LLL-PSLQ algorithm, excellent for narrow cases but cannot find any of the neat examples suggested here recently like this approximation within 13 digits of a factorial expression found by the RIES program. In some cases like finding an algebraic expression the LLL-PSLQ program is the best on this planet but with compound GAMMA values mixed with algebraic numbers, yes it can find some of the genuine Bill Gosper-like identities but for this you need more than brute force approach, you need a clue on the possible expressions because we are dealing here with non-linear expressions even if we take the log, unless you are mr Gosper itself, but mr Gosper is not a method it is a human being.!, ...
There is so far no global answer to this problem, the best is a mix of all this I presume ?
Best Regards to all, Simon Plouffe
Le 29/12/2011 13:32, Robert Munafo a écrit :
On Wed, Dec 28, 2011 at 14:58, Warren Smith<warren.wds@gmail.com> wrote:
A certain proportion of Ries formulas seem to be "tasteless" and the sort of thing that are never going to happen, like using (e^pi) or 1/ln(pi) or e or sin(1) in an exponent...
Yes, definitely. As Bill Gosper just pointed out to me, sin(1) anywhere is practically useless. And e^pi is a curio at best ("Gelfond's Constant" [5]).
If you look at the webpage I wrote about ries [1], most of which is dedicated to "Stupid Math Tricks", perhaps the idea behind ries will be more clear.
I actually started by looking at all the stuff in Plouffe's Inverter, consisting mostly of long expressions using obscure and undefined functions, peppered with the occasional fraction like "2087/1457" which I could have figured out on my own. Then, by chance I got a couple emails in a row from members of the Cult of 137 [3], and I decided to prove to myself (and to them) that it is easy to make all sorts of meaningless formulas for any number you choose.
More to your point, the stuff on Plouffe's Inverter [4] is clearly all meaningful to someone, but most of it is of no meaning to any one person in particular, so we have a severe target marketing problem. Armed with these hundreds of higher-math functions, ries would be as bad, if not much worse. It will probably never have much serious application.
Nevertheless, it sure is a lot of fun, and fun is a large part of the meaning of life. So there, xyzzy!
- Robert.
[1] http://mrob.com/pub/ries/index.html
[3] http://mrob.com/pub/num/n-b137_035.html#cult_137
[4] http://bootes.math.uqam.ca/cgi-bin/ipcgi/lookup.pl?Submit=GO+&number=1.43239...
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Users clamoring for improvement are among the sincerest form of flattery... I think Simon has already done a lot to address this: His offering of that 657 gigabytes(!) of raw data seems meant to encourage the development of third-party search tools. It seems for example, that one could parse the ASCII text expressions to tag all values, indexed by what functions they're made of, thus allowing custom weight functions to sort the expressions by relative value, according to the user's peculiar tastes. So, not the full source code solution, but definitely a thumping good start. Personally, I was happy when the online Inverse Symbolic Calculator started sorting their results by length of expression (try [1] for example). Tautologies are evident, but when you get 'em, the simplest one is first. My wishes for PI (as opposed to Log[-1]/I) are fairly simple: let's please list X separately from 10X and X/10. See [1] again and note that log(35027/27791) = 2.314069653887953... is listed right after exp(Pi) = 23.14069263277926... Is a factor of 10 really that insignificant? (-: - Robert [1] http://bootes.math.uqam.ca/cgi-bin/ipcgi/lookup.pl?Submit=GO+&number=23.1406... On Thu, Dec 29, 2011 at 14:52, <rcs@xmission.com> wrote:
More wish list:
Supply the Inverter as a program [...] Allow for some user control of the generation priorities. [...] Add a few more specific search algorithms: [...] Allow several numbers to be input, perhaps to find a relationship. [...]
[...] User control of the search priority is double edged: The upside is
that your personal knowledge of the number's origins is a big clue to what expressions might be relevant. (We don't need no stinkin' Bessel functions.) The downside is that you lose the benefit of a fresh point of view, since every number and special function is an input from the math community at large.
The tautology problem will become worse. [...]
[...] -- Robert Munafo -- mrob.com Follow me at: gplus.to/mrob - fb.com/mrob27 - twitter.com/mrob_27 - mrob27.wordpress.com - youtube.com/user/mrob143 - rilybot.blogspot.com
Ries found pi*( (11/120)! * (1/120)!)^10 = 1 + exp( cos(phi-5) / 8 ) accurate to 6*10^(-13). Ridiculous :)
I recall seeing an AGM-like method for evaluating genus 2 hyperelliptic integrals. I don't remember where. IIRC, it involved 3 linked AGM iterations, on several state variables. Rich ---- Quoting Warren Smith <warren.wds@gmail.com>:
Vidunas at end also notes that Gamma(1/5) and Gamma(2/5) can be expressed in terms of two hyperelliptic integrals... but is there any fast AGM-like scheme for evaluating them?
This may be more than you want: Carls' thesis about generalized AGM's: http://www.math.leidenuniv.nl/scripties/carls.pdf The "ordinary" AGM can be viewed as a process on elliptic curves using 2-isogenies. The AGM in genus 2 uses so-called "Richelot Isogenies". Carls talks about all that in the introduction. Victor On Wed, Dec 28, 2011 at 3:50 PM, <rcs@xmission.com> wrote:
I recall seeing an AGM-like method for evaluating genus 2 hyperelliptic integrals. I don't remember where. IIRC, it involved 3 linked AGM iterations, on several state variables.
Rich
----
Quoting Warren Smith <warren.wds@gmail.com>:
Vidunas at end also notes that Gamma(1/5) and Gamma(2/5) can be expressed in terms of two hyperelliptic integrals... but is there any fast AGM-like scheme for evaluating them?
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Interleaved among messages in this thread, I got an advertisement that said "Insider specials end 12/31!" Took me a second to realize they weren't making a weird claim about (12/31)!. On Wed, Dec 28, 2011 at 5:57 PM, Victor Miller <victorsmiller@gmail.com>wrote:
This may be more than you want: Carls' thesis about generalized AGM's: http://www.math.leidenuniv.nl/scripties/carls.pdf
The "ordinary" AGM can be viewed as a process on elliptic curves using 2-isogenies. The AGM in genus 2 uses so-called "Richelot Isogenies". Carls talks about all that in the introduction.
Victor
On Wed, Dec 28, 2011 at 3:50 PM, <rcs@xmission.com> wrote:
I recall seeing an AGM-like method for evaluating genus 2 hyperelliptic integrals. I don't remember where. IIRC, it involved 3 linked AGM iterations, on several state variables.
Rich
----
Quoting Warren Smith <warren.wds@gmail.com>:
Vidunas at end also notes that Gamma(1/5) and Gamma(2/5) can be expressed in terms of two hyperelliptic integrals... but is there any fast AGM-like scheme for evaluating them?
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participants (6)
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Allan Wechsler -
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Robert Munafo -
Simon Plouffe -
Victor Miller -
Warren Smith