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compile - Speeding up this recursive function



I have a recursive function defined like this:


Clear[MyFnc];
MyFnc[X_] := Block[{n, val, Xm, Xmn},
    n = Length[X]; If[n == 1, Return[{2, 1}]];
    Xm[mm_] := X[[1 ;; mm]];
    Xmn[mm_, nn_] := X[[mm + 1 ;; nn]];
    val = Table[Block[{XmT = Total[Xm[m]], XmnT = Total[Xmn[m, n]]},
          (XmT.XmnT)/(XmT.XmT)*{m^2, 2 m}*(MyFnc[Xm[m]][[1]])*(MyFnc[Xmn[m,n]][[2]])],
          {m, 1, n - 1}] // Total;
    Return[val/{n^2, n}]];


Function takes a list of vectors of arbitrary length n, e.g. for n=5 and 3D vectors;


arg := Table[RandomReal[{-1, 1}, 3], {5}]

and gives a list of two numbers, i.e. MyFnc[arg] gives for example {0.24443, 1.10547}.


Since this function is to be used as an integrand in numerical integration, it would need to be called many times. So evaluation time is important,


(Table[MyFnc[arg], {10^4}] // AbsoluteTiming)[[1]]  

(* ==> 18.5959 *)


Is there a way to significantly speed up this function? Can such recursive functions be compiled efficiently?



Answer



You can get it through Compile as below. Note that I have not tested for correctness.


myFncC = Compile[{{X, _Real, 2}}, Block[
    {n, val},
    n = Length[X];
    If[n == 1, Return[{2., 1.}]];
    val = Total[
      Table[
       Block[

        {XmT = Total[X[[1 ;; m]]], XmnT = Total[X[[m + 1 ;; n]]], 
         mFm = myFncC[X[[1 ;; m]]], mFmn = myFncC[X[[m + 1 ;; n]]]},
        (XmT.XmnT)/(XmT.XmT)*{m^2, 2 m}*mFm[[1]]*mFmn[[2]]], {m, 1, 
        n - 1}]];
    val/{n^2, n}]];

arg := Table[RandomReal[{-1, 1}, 3], {5}]
(Table[myFncC[arg], {10^4}] // AbsoluteTiming)[[1]]

(* Out[67]= 1.13199 *)


Adding "CompilationTarget" -> "C" brings it down a hair more, to .75 seconds.


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