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Re: what actually is in the WRI "functions" database?
On 8/6/04, Richard Fateman wrote: >Why should I believe Michael Trott's assertion that some standard-form formula >is correct? Who does he think he is, Stephen Wolfram? :) Michael should answer this. However, I would expect that:  the formulae are compared to other tables;  self-consistency checks are made (using Mathematica) within and between formulae;  feedback from people who have used the functions website (such as myself) is used to identify errors. And at http://functions.wolfram.com/About/mathematica.html it says that "A wide variety of methods were used to derive and verify the identities listed on this site. Many standard techniques were used to calculate indefinite and definite integrals. Examples of other frequently used methods include symbolic high-order series expansions and matching coefficients (for instance, for deriving contiguous relations for generalized hypergeometric functions and modular equations for modular function), polynomialization and application of elimination techniques (for instance, for deriving differential equations for elliptic integrals and functions)." >And the Mathematica formulas that he knows to be wrong are not even >marked as wrong (or inadequately encoded). Which formulas are you talking about? I'm not aware of obviously wrong StandardForm formulae, though there are coding issues such as the use of ellipsis. >This is, to me, quite unacceptable. Excuses after the fact like >"this will have to do until a better encoding comes along in a >future Mathematica"... I assume you're referring to the ellipsis example? These issues could be resolved by coding appropriate notations using the current version of Mathematica. >...or "MathML doesn't have the expressive capacity for this" are not >adequate rationals to provide known erroneous formulas. I wonder if erroneous formulas of this type can be spotted by the comparing the input and output of the cycle StandardForm -> MathML -> StandardForm ? And whether this can be used as a MathML testbed? >I don't understand how this could even be controversial. If >something is semantically meaningless, you don't just hand it to >someone without mentioning this fact. A caveat to this effect could be added in such cases. I have no idea how many formulae have such problems. >If Lozier and his DLMF team say a formula is correct, it could still >be wrong, but at least we have some idea that it has been checked >over by an official "team" of experts. Users of the functions website, including experts, have been testing/checking the website for over two years now and many will have supplied feedback. While I respect the DLMF experts, until DLMF is live no external testing is possible. >Yet I have expressed my disappointment with the DLMF project to >Lozier and others, not only on the time scale, but the general lack >of ambition. I strongly agree here. >I have pointed out to them the possibility of having more of a >"live" aspect, and in a talk at NIST (years ago) I had slides of >pages from the functions.wolfram site. If NIST were to do something >like this, it would have more authority, being "vendor independent" >and might be optionally attached to one (or more) CAS. Some might >even be on-line, live, and free. I suggested to Hans Volkmer that, rather than hyperlinks to references and code, it would be nice for the DLMF to be able to directly call functions via, e.g., web form interface (in particular, I was thinking of using webMathematica to call Peter Falloon's SpheroidalHarmonics package). However, it appears that this sort of functionality was never intended to be part of the DLMF, which is disappointing. >G&R has been checked over by many people. It may still have >mistakes, but there have been many eyes looking at it. For a long >time. And it has cross references. I see that there is a more complete errata at http://www.mathtable.com/gr/. I agree that GR is useful, as is A&S, partially because both are well tested, and I refer to both regularly. However, I now find that I go to the functions website first because usually I want to compute something and I can get the required formula very quickly. >And the latest version, edited by Dan Zwillinger, was generated from >a macro language with an expansion into live CAS language >(Mathematica, I think.) I was not aware of this. It is not mentioned at the book website but I, and I'm sure many others, would be interested to know more about such a capability. Especially since the functions website is restricted, at present, to functions built-in to Mathematica. >While the concept, the typography, and the scope of >functions.wolfram is admirable, the content seems to be >untrustworthy, so what's the point? I think that you are exaggerating how untrustworthy it is, extrapolating from two examples. I still cannot see your problem with the first of these -- the use of a symbol in the typeset representation and the use of Element[n,Integers] && n > 0 in the code -- and the second formulae, involving ellipsis, is not incorrect, only incomplete. The point is that this site, even accepting the problems you've identified, is still more useful (at least to someone who does sanity checks on the formulae) than DLMF or ESF. >[You may say, just as G&R accept reports of errors, so can the web >site --- yes, this is an important positive aspect of the on-line >repository. Will Michael Trott fix the broken formulas that I have >reported? Michael Trott certainly has fixed all the problems that I have reported. >Will he go through all formulas with ellipsis and mark the InputForm >somehow? I hope so.] That is up to Michael and the functions team. Cheers, Paul