On Tuesday, April 1, 2014 2:58:34 PM UTC-7, Ondřej Čertík wrote:
>
> On Mon, Mar 31, 2014 at 9:14 PM, Matthew Rocklin
> <mroc...@gmail.com<javascript:>>
> wrote:
> > http://www.evanmiller.org/mathematical-hacker.html
> >
> > I reference that blog post pretty often. I fully intend to reference it
> > again in my talk (if it is accepted).
>
That's unfortunate. I was totally unacquainted with Evan Miller. I read
this particular
blog entry. I did not follow any of the links, but I think that the essay
is a piece of crap.
It is uninformed, shallow, and, I daresay, just wrong. I hope you re-think
referring to it.
> >
> > The interesting thing about the Factorial / Gamma / loggamma example is
> that
> > to find the solution you need to find someone who knows both that n! =
> > Gamma(n+ 1) and who knows that a loggamma routine is commonly found in
> lower
> > level languages. Those bits of information are usually held by
> different
> > experts. Ondrej said "Of course, that's obvious" when I first reposted
> the
> > article on G+.
>
Well, let me quote from the article..
"
*if it is ever necessary to compute a factorial, a programmer should be
taught to take advantage of the system's log-gamma function, as the
following C code does: *
*
long int fac(unsigned long int n) {
return lround(exp(lgamma(n+1)));
}
*
* Again, no recursion is required as long as one...*
Here are a few things that are wrong. Rounding an approximate floating point
number to a integer may not get you the exact integer factorial you want.
The program will certainly not work for arbitrary unsigned long int n
because
the exp() will overflow. Say certainly by n=175.
If you want a floating-point approximation to any factorial that can be
represented as an IEEE double, you could do it by checking if N<175 and
then index into an array of length 175. [I'd check that 175 more carefully
if I were really doing this. And the number would change a little for
64-bit ints.
If you want exact integer factorials on a 32 bit computer, you only need an
array of 12, since
13! > 2^32.
So the program is stupid. Why? Because the author is mathematically
and computationally ignorant. (And seemingly quite full of himself.)
Say you want to solve the much more interesting problem of computing
factorials
of any size integer. Then you might see
http://www.luschny.de/math/factorial/FastFactorialFunctions.htm
which tells how it might be done better.
Here's a far superior factorial, in lisp, and yes it is recursive. I
suppose you
could write it iteratively, but why bother. A Lisp compiler will do that
for you.
(defun fact(n)(f n 1))
(defun f (n m) (if (<= n 1) 1 (* n (f (- n m) m))))
It is used in some computer algebra systems.
The snide characterizations that lisp programmers know more or less
mathematics
or that what they know is more or less relevant is just junk.
Of the 3 people quoted -- Raymond, Graham, Yegge...
I've never heard of Yegge; I think that Graham is smart, and Raymond just
says things
to be, uh, smart-ass.
>
> That's funny, I forgot that I said that and just had the same reaction.
>
> However, we are still missing rational function approximation in SymPy
> or mpmath.
> That's what is used to implement things like log_gamma or erf (error
> function) in
> Fortran or C. So I have just created an issue for it:
>
> https://github.com/sympy/sympy/issues/7359
>
> and I spent time explaining exactly how it works in it and giving
> examples,
> including for example the implementation of erf() in gfortran.
>
> Aaron, you were asking about examples of numerical cancellation. I
> worked out one in the
> issue as well.
>
> Ondrej
>
> >
> > You're right that this is similar to my last talk. The last one though
> was
> > mostly about an application (numerical linear algebra). I actually want
> to
> > talk a bit more about the philosophy and some of the more abstract tools
> > that people might actually use. Your first impression is a valuable one
> > though, I should go through my last talk and make sure that I'm not
> > repeating too much that shouldn't be repeated.
> >
> >
> > On Mon, Mar 31, 2014 at 6:03 PM, Aaron Meurer
> > <asme...@gmail.com<javascript:>>
> wrote:
> >>
> >> That's a good point. One of the nicest things about symbolics, when you
> >> can get it, is that it can make things drastically more efficient by
> doing
> >> mathematical simplifications. Evaluating integrals symbolically is a
> nice
> >> example of this (especially for SymPy, which has some pretty nice
> algorithms
> >> to compute definite integrals).
> >>
> >> I'm reminded of a popular blog post (I can't find a link right now)
> about
> >> how know math is important for programmers. It has the example of how
> all
> >> these programming languages show how they they compute factorial, and
> how
> >> tail recursion can make it linear or whatever, but the actual best way
> to
> >> compute it is to use loggamma, which gives the answer in constant time.
> >>
> >> Aaron Meurer
> >>
> >>
> >> On Mon, Mar 31, 2014 at 7:51 PM, Tim Lahey
> >> <tim....@gmail.com<javascript:>>
> wrote:
> >>>
> >>>
> >>>
> >>> On 31 Mar 2014, at 20:29, Aaron Meurer wrote:
> >>>
> >>>> On Mon, Mar 31, 2014 at 11:32 AM, Matthew Rocklin
> >>>> <mroc...@gmail.com <javascript:>>wrote:
> >>>>
> >>>>> I like that you emphasized the utility for numerics, I think that
> this
> >>>>> is
> >>>>> likely to be a selling point for the SciPy crowd.
> >>>>>
> >>>>
> >>>> Yes, this was very intentional. I may need some help gathering up
> some
> >>>> nice
> >>>> motivating examples if this is accepted.
> >>>
> >>>
> >>> One motivating example for me is the integration of products of
> functions
> >>> over areas and volumes. For finite elements, you'll get products of
> pairs of
> >>> trial functions (usually polynomials). It's even more useful for
> products of
> >>> trig functions. Performing the integration of any of theses is easy
> enough
> >>> with numerical integration, but it's much more efficient to calculate
> the
> >>> integrals symbolically and then perform the evaluation for each
> element.
> >>>
> >>> Cheers,
> >>>
> >>> Tim.
> >>>
> >>>
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