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MathGroup Archive 1996

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inert wrapping function

  • To: mathgroup at smc.vnet.net
  • Subject: [mg5592] inert wrapping function
  • From: "w.meeussen" <w.meeussen at solair4b.eunet.be>
  • Date: Fri, 27 Dec 1996 01:59:03 -0500
  • Sender: owner-wri-mathgroup at wolfram.com

hi,

I need to develop and refine an inert wrapping function. I found to my
surprise that it takes quite some thought. With 'inert wrapping function' I
mean a label like 'fatty acid',  'fa[]', that behaves like a kind of Type
such as 'Integer' or 'Character'.

Is there a way to make fa[] Flat, not only over sequences of arguments, but
over sums and (scalar) products ? Call it 'LinearlyFlat' or so.

The only thing it should do is to automatically Thread over sums or linear
combinations of fa[]'s :

fa[a fa[A] + b fa[B] + c fa[C] ] should evaluate to  a fa[A] + b fa[B] + c fa[C]

and,

fa[a fa[A] + b fa[ 
                        fa[ a fa[A] + b fa[B] + c fa[C] ]
                 ]     
should evaluate to :
   a fa[A] + b fa[  
                            a fa[A] + b fa[B] + c fa[C]  
                 ]   
and even further to, finally :
   a fa[A] + b a fa[A] + b^2 fa[B] + b c fa[C]   

for clarity, I chose capital lettres for the arguments of the innermost
fa[]'s. Those capital arguments should of course never loose their
fa[]-wrapper! Something like 'A+B' makes no sense, and neither does fa[a]
(since a is just a numeric, not a fatty acid identifier)

After some hours of clobbering 'round in the murk, I came up with a
temporary cludge :

In[224]:=
SetAttributes[fa,{Flat,OneIdentity}]
In[225]:=
fa[id_Plus]:=Thread[Hold[fa][id//Expand],Plus]//ReleaseHold
In[226]:=
fa[a___ fa[b_]]:=a fa[b]


several things make me feel uneasy :

fa[id_Plus]:=Thread[fa[id],Plus] doesn't work (infinite recursion ?!), but

fa[id_Plus]:=Thread[dummy[id],Plus]/.dummy->fa  is ok at first sight.
the extra 'Expand' is needed in case a term like .. + p( q fa[] + r fa[] )
appears.

I'm not shure if wrapping a 'Hold' around the head 'fa' is orthodox practice
to avoid the unexpected infinite recursion in line 225. The trick with the
dummy function is not exactly what is needed either, since fa[] is meant to
Thread over any number of nested levels of fa[]'s.

I have a   s t r o n g    feeling of re-inventing the wheel here. The
nagging feeling persists that, somewhere in 'The Book', all this is already
discussed at length. But I can not find it. The use of a function (purely as
head) seems to be evident, but to work it out this deep just to make it
behave linear feels like Overkill.



------------------------------------------
Please, check any suggestions against the following nasty test that creates
a nested list of fa[]'s :
All fA, ... fL should end up inside a wrapper fa[], end all a ... l outside
of it.



a1={fA,fB,fC,fD,fE,fF}
a2={a,b,c,d,e,f};  a2=a2/Plus@@a2
b1={fG,fH,fI,fJ,fK,fL}
b2={g,h,i,j,k,l};  b2=b2/Plus@@b2

(fb/@a1) . a2
%/.fb[fC]->(fb/@b1) . b2
Nest[#/.fb[fJ]->(fb/@b1) . b2 &,%,3]//fb

  (*
 now, replacing fb[] with fa[] should be linear in the fa[]'s : 
  *)
Timing[final=%//.fb->fa]
   (*
since it's linear, no big deal to collect the fa[]'s :
   *)
Collect[final,_fa]//Timing



Suggestions or pointers to The Book warmely wellcomed.

Tanx for reading this far, 
Dr. Wouter L. J. MEEUSSEN
w.meeussen at solair4b.eunet.be



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