NetNews Usenet Archive 1992 #27/NN_1992_27.iso/spool/sci/math/symbolic/3074

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From: [e-mail] (Mark Stankus)
Newsgroups: sci.math.symbolic
Subject: Re: an example of comparative CAS programming
Message-ID: <[e-mail]>
Date: 22 Nov 92 04:23:06 GMT
References: <[e-mail]>
Sender: [e-mail]
Organization: Mathematics @ UCSD
Lines: 274
Nntp-Posting-Host: oba.ucsd.edu

In article <[e-mail]> [e-mail] (Jeffrey P. Golden) writes:
>Reply-To: [e-mail]
>
>Given all the recent messages about comparative programming in
>the various CAS, I thought the following might be elucidating.
>
>Barry Simon, "Symbolic Math: Problems and Solutions",
>Notices of the AMS, Vol. 39, No. 7, pp. 700-710, Sept. 1992
>has the following problem:
>
>  9.  Rule based algebra
>  Consider the Clifford algebra in 10 variables, that is the
>  complex algebra with ten generators, s0,...,s9 obeying
>          si sj + sj si = 0 if i is different from j
>          si si = 1
>  That is multiplication is NOT commutative (but is assumed
>  associative).  Compute (s0+s1+....+s9)^5
>  NOTE: This must be done with general methods - it is cheating
>  to use the invariance of the Clifford algebra under orthogonal
>  transformations.
>
>The article gives the following solutions provided by the
>respective CAS vendors:
>
>-------------------------------------------------------------------------
>
>Derive Problem 9.  A solution was provided with 55 lines of function
>definition of which the following is typical:
>ADD_AUX(u,v,j,k):=~
> IF(j=0,HEAD(v,k),~
>    IF(k=0,HEAD(u,j),~
>       IF(SIMILAR(ELEMENT(u,j),ELEMENT(v,k)),~
>          APPEND(ADD_AUX(u,v,j-1,k-1),~
>  ADDSIMILAR(ELEMENT(ELEMENT(u,j),1)+ELEMENT(ELEMENT(v,k),1),ELEMENT(u,j))),~
>          IF(BEFORE(ELEMENT(u,j),ELEMENT(v,k)),~
>             APPEND(ADD_AUX(u,v,j,k-1),[ELEMENT(v,k)]),~
>             APPEND(ADD_AUX(u,v,j-1,k),[ELEMENT(u,j)])))))
>Anyone who thinks you can't program with just an IF statement
>should look at this.  It's awkward but certainly doable!  The
>answer, by the way is 100 times the sum of the sigmas.
>
>Maple Problem 9: Here's the code to set up the rule based algebra:
>   s.(0..9):
>   readlib(commutat):
>   for i from 0 to 9 do
>    for j from i+1 to 9 do
>     &*(s.j,s.i) := -&*(s.i,s.j);
>    od;
>    &*(s.i,s.i) := 1;
>   od:
>
>   &^ := proc(a,n) local t;
>   t := 1;
>   to n do
>    if type(t,constant) then t := t*a
>    else t := expand(t&*a)
>    fi
>   od;
>   t
>   end:
>
>   q9 := sum('s.k',k=0..9);
>   a9:=q9&^5;
>
>Mathematica Problem 9
>  (* rulesfor  Clifford algebra *)
>  ncm[s[i_],s[j_]] := -ncm[s[j], s[i]]/;i > j
>  ncm[s[i_], s[i_]] := 1
>  (* rules for non-commutative multiplication *)
>  (* - behavior of multiplication of integers *)
>  ncm[a_?NumberQ, b_?NumberQ, c___] := ncm[a b, c]
>  (* - distributivity of addition *)
>  ncm[m___, a_Plus,b_Plus, n___] :=
>          ncm[m, Distribute[tmp[a,b],Plus]/.tmp -> ncm, n]
>  (*  - associativity *)
>  ncm[a___, ncm[r___], b___] := ncm[a, r, b]
>  (* here's the expression *)
>  expr = Plus @@ Map[s, Range[1,9]]
>  (* and here's the power.  Note the output form involves ncm; this could
>   be formatted to look cleaner for one's individual purposes. *)
>  expr^5/.
>        x_^n_ :> Fold[ncm, x, Table[x, {n - 1}]]
>
>Reduce Problem 9: Shows simple elegance of the Reduce language.
>   operator s;
>   noncom s;
>   for all i,j such that i>j let s(i)*s(j)=-s(j)*s(i);
>   for all i let s(i)*s(i) = 1;
>   xxx := for i:=0:9 sum s(i);
>   xxx^2;
>   xxx*ws^2;
>
>-------------------------------------------------------------------------
>
>Macsyma (from Macsyma Inc.) has a built in package ATENSOR for
>working with Clifford algebras, but here's an alternative solution
>done with Macsyma 417, going back to basic principles and using
>pattern matching:
>
>
>(c1) matchdeclare([i,j],integerp)$
>
>(c2) tellsimpafter(s[i]^^2,1)$
>
>(c3) tellsimpafter(s[i].s[j],-s[j].s[i],i>j)$
>
>(c4) /* optional: */  compile_rule(all)$
>
>(c5) factor(expand(sum(s[i],i,0,9)^^5));
>
>(d5)    100 (s  + s  + s  + s  + s  + s  + s  + s  + s  + s )
>              9    8    7    6    5    4    3    2    1    0
>
>-------------------------------------------------------------------------
>
>
>You should of course make your own judgments but I don't find the
>Derive, Maple, or Mathematica solutions to be elegant.
>
>One easily sees the price one pays with Maple due to its lack of
>pattern matching - notice all the enumerated assignments that are
>being made!  I don't know if a better solution is possible with
>Maple VR2.
>
>And notice all the extra stuff that's required in the Mathematica
>solution!
>
>I think the Reduce solution would be the most elegant if it weren't
>for the     xxx^2;  xxx*ws^2;   .  Does the simpler  xxx^5;
>not work, or is this circumlocution done just for the sake of
>efficiency?
>
>But I also think the Macsyma solution is pretty elegant too.
>
>
>From: Jeffrey P. Golden <[e-mail]>
>Organization: Macsyma Inc.
>Reply-To: [e-mail]

It is true that the Mathematica solution required some code. On the
other hand, we have developed an add-on to mathematica (which
is long, but does a significant amount) which is available
through anonymous ftp (instructions below).

If you get a copy of this program, please send us your name
so that we can keep track of our users for bug reports,
modifications to the program, etc.

Mark Stankus

(****************************************************************************)


                               NCALGEBRA

                              Version 0.1



(****************************************************************************)

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          % ftp 128.54.18.1
            -------------------

          Connected to 128.54.18.1.

          220 osiris FTP server (SunOS 4.0) ready.


          Name (128.54.18.1:dhurst): anonymous
                                     ---------

          331 Guest login ok, send ident as password.

          Password: "Any thing will do."
                    --------------------

          230 Guest login ok, access restrictions apply.

          ftp> cd pub/ncalg
               -------------

          250 CWD command successful.

          ftp> mget *
               ------

          mget CEEP?  "You just hit the <Return> key."
                      --------------------------------

          200 PORT command successful.

          150 ASCII data connection for CEEP (128.54.18.8,3053) (453 bytes).

          226 ASCII Transfer complete.

          local: CEEP remote: CEEP

          474 bytes received in 0.0043 seconds (1.1e+02 Kbytes/s)

          mget CONTENTS? "Just keep on hitting that <Return> key until ... "
                         ---------------------------------------------------
          ftp>

          ftp> bye
               ---
          221 Goodbye.



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mstankus
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