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Re: A pattern matching problem

• To: mathgroup at smc.vnet.net
• Subject: [mg72546] Re: [mg72533] A pattern matching problem
• From: János <janos.lobb at yale.edu>
• Date: Sat, 6 Jan 2007 03:46:32 -0500 (EST)
• References: <200701050706.CAA10424@smc.vnet.net>

On Jan 5, 2007, at 2:06 AM, carlos at colorado.edu wrote:

> Here is an interesting challenge in pattern matching.  Suppose
> you are given an algebraic-differential expression exemplified by
>
> r = u[t+h]-2*u[t]+u[t-h]+a^2*u'[t+h/2]+4*u'[t-h/4]+
>      c*u''[t+alfa*h]/12;
>
> Here u[t] is a function of time t, assumed infinitely differentiable,
> h is a time interval, and primes denote derivatives wrt t.
> Relation  r==0 is called a delay-differential equation, and is the
> basic stuff in delayed automatic control (h is the signal "lag").
>
> The function name u and the lag h are always symbolic.
> Function u and its derivatives appear linearly in r, while
> h always appears linearly in arguments.
> Coefficients of h may be numeric or symbolic.
> Coefficients of u & derivatives may be numeric or symbolic.
>
> The challenge: given r,  get the coefficients of h as a 2D list,
> row-ordered by derivative order. Zero coefficients may be omitted.
> For the above r, it should  return
>
>    {{1,-1},{1/2,-1/4},{alfa}}
>
> Envisioned module invocation:   clist=LagCoefficients[r,u,t,h,m]
> with m=max u-derivative order to be considered.  Skeleton:
>
>         LagCoefficients[r_,u_,t_,h_,m_]:=Module[ {clist={}},
>             ??????
>         Return[clist]];
>
> Any ideas for ?????

Here is a newbie idea without too much pattern matching, because I  
still have trouble with that :)

In[1]:=
r = u[t + h] - 2*u[t] +
     u[t - h] +
     a^2*Derivative[1][u][
       t + h/2] +
     4*Derivative[1][u][
       t - h/4] +
     c*(Derivative[2][u][
        t + alfa*h]/12);

In[2]:=
(#1[[All,2]] & ) /@
   Split[Inner[List,
     Head /@ Extract[
       DeleteCases[List @@ r,
        x_ /; MemberQ[x, h,
           {0, Infinity}] ==
          False], Most /@
        Position[DeleteCases[
          List @@ r, x_ /;
           MemberQ[x, h,
           {0, Infinity}] ==
           False], u]],
     (Replace[#1, h ->
         1] & ) /@
      (First[#1] & ) /@
       Extract[r, Most /@
         Position[r, h]],
     List], #1[[1]] ==
      #2[[1]] & ]
Out[2]=
{{-1, 1}, {-(1/4), 1/2},
   {alfa}}

I am sure the gurus will give you a less convoluted one :)

János


----------------------------------------------
Trying to argue with a politician is like lifting up the head of a  
corpse.
(S. Lem: His Master Voice)

• References:
  • A pattern matching problem
    • From: carlos@colorado.edu