si(x) in Maxima (Was: Mathematica vs. Lisp)

From: Albert Reiner (areiner_at_tph.tuwien.ac.at)
Date: 06/22/04 

Date: 22 Jun 2004 11:45:22 +0200
To: Richard Fateman <rfateman@sbcglobal.net>

Richard Fateman <rfateman@sbcglobal.net> writes:

> There is a tradeoff in computer algebra systems "The good of the
> one outweighs the good of the many." to reverse Spock's quote.
> Do you want a/b to be (quotient a b) or do you want to minimize the
> proliferation of "kernels" like quotient, and use (times a (expt b
> -1)) ? which is wordier, but re-uses kernels that you can't really
> get rid of.
> Si(x), the sine integral is one of those kernels that you can pretty
> much get rid of: just use the integral. In some cases you want to
> pander to the common usage. e.g. sine and cosine could be expressed
> in terms of each other or in terms of complex exponentials, but are
> so familiar, they must all coexist somehow.

Of course, but in my case si and ci (or actually some related
functions called, I think, f and g in Abramowitz-Stegun) are just as
basic as sin and cos, simply because I have an efficient numerical
implementation for them to use in later processing.

> But I think the original question was really about the
> programming language issues, not the mathematical capabilities
> of lisp...

Actually I am no longer talking about the OP's question but rather
about something related to my own work (involving 3-dimensional
Fourier transforms of interaction potentials in simple liquids with
cutoffs in both k- and r-space), where the symbolic end result I need
to get should be in terms of si, ci (or the f and g mentioned above)
and I also need the small-k series expansions of those expressions,
all of these written in a way suitable for numerical evaluation. The
types of expressions that arise are pretty limited, and their
numerical properties are well understood, so one basically knows what
form one needs to produce.

In the past I just did all of this by hand, with my good old
Gradstein-Ryzhik by my side, but this is barely feasible with the
scheme to be considered now. Also, I would like to be able to have
those manipulations automated for the standard potentials in liquid
state physics; note that I do not really need integrate() to do those
things automatically, I only need to arrive at a procedure that will
yield the end result in a usable form, checking the validity of the
transformations on the way, and telling me if I need to expand that
procedure.

If I were doing this in Mathematica, I wouldn't rely on Integrate
either (branch cuts!) but rather work with replacement rules; but then
again, Mathematica is terrible for scripting, and it seems I cannot
use mash, <http://ai.eecs.umich.edu/people/dreeves/mash/>, at this
place.

My idea was that by using Maxima I could get the basics of those
manipulations, simple simplifications in particular, for free, that I
would be able to do what I need with replacements (I see subst and
substpart etc. in the manual), and that it would integrate well into
the rest of the software if written in Common Lisp; actually, this was
the primary reason for considering CL at all.

BTW, does anyone have an example of using a Maxima result for a lambda
body? I am thinking of a function that returns a function for
evaluating the maxima result numerically. E.g., given a
representation of cos(x), such a function should compute the
derivative and return

    '(lambda (x)
        (declare (type long-float x))
        (- (sin x)))

On a related note, does anyone know how close the Maxima
representation of algebraic expressions is to that of CL or Fortran?

Albert.

Relevant Pages

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