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Re: Speeding up Replacement Rules

  • To: mathgroup at
  • Subject: [mg24572] Re: Speeding up Replacement Rules
  • From: Jens-Peer Kuska <kuska at>
  • Date: Tue, 25 Jul 2000 00:56:05 -0400 (EDT)
  • Organization: Universitaet Leipzig
  • References: <8lgqqm$>
  • Sender: owner-wri-mathgroup at


> Of course, if someone has Mathematica code which takes a vector
> valued function and generates a Compiled gradient or hessian
> function, I will accept it gratefully.

Here is the code that take a system of differential equations
and make a compiled function and a compiled jacobian. It is not
exactly the same but it may give you  a hint.


(* We must wrap a Hold[] around the Part[new, _] to  
   prevent that the part of a symbol is serached *)
dirkMakeVars[y_List, new_Symbol] :=
  Module[{x, i},
    Thread[y -> Table[Hold[new[[x]]] /. x -> i, {i, Length[y]}]]

DIRKNDSolve::noexpl = "No explicit solution found for equations `1`."

dirkMakeFun[eqn : {_Equal ..}, y_List, t_Symbol, compile_:True] :=
  Module[{sol, deqn, f, jac, rules, newsym, fj},
    deqn = Select[eqn, ! FreeQ[#, t] &];
    sol = Solve[deqn, D[#, t] & /@ y];
   If[sol == {} || Head[sol] === Solve,
     Message[DIRKNDSolve::noexpl, eqn];
   f = D[y, t] /. sol; (* explicit solution *)
   jac = Outer[D[#1, #2] &, #, y] & /@ f;
   newsym = Unique[];
   rules = dirkMakeVars[y, newsym];
   fj = Transpose[{f, jac}] /.  rules;
    (* at first we add the function arguments for compile
       with an evaluated newsym, than Hold[] is removed
       and the Hold[part[_]] are removed with the last Replace[] *)
    fj = Map[Hold, 
           Map[{{{t, _Real}, {newsym, _Real, 1}}, #} & , fj, {2}] , {2}]
             Hold[a_Part] :> a;
    (* Now we can start with compile. To prevent a early
       evaluation of the Part[] function in the Apply[] call an
       is needed *)
    fj /. Hold[l_] :> Apply[Compile, Unevaluated[l]]
   (* Else *),
    ReleaseHold[Map[Function[Evaluate[{t, newsym}], #]  &, fj, {2}]]

dirkMakeFun[] generate for the variables in y a vector newsym so that
y={x[t],q[t],z[t]} the new vector ist newsym[[1]], newsym[[2]] and
The tricky part is to prevent  the evaluation of newsym[[i]]  before
it is in the Compile[] command. 
dirkMakeVars will return something like


than I make the argument for Compile[]  but with a hold arround the full
arguments and get 
Hold[{{{newsym, _Real,1}},theExpressionWithManyHoldParts}}

since a Hold[] is around the expression I can remove the Hold[a_Part]
inside the expression. Finaly I have to Apply[] the Compile command 
*and* to remove the Hold[]

    fj /. Hold[l_] :> Apply[Compile, Unevaluated[l]]

and I have my pair of function/jacobian compiled functions for
the right hand side of my systems of differential equations.

Hope that helps

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