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I am trying to speed up my Mathematica code, so instead of testing and calculating the same thing again and again I want to save the result in a SparseArray. The goal is to generate a matrix which I can later use e.g. in a Matrix-Vector product. I am having trouble constructing the SparseArray - when trying to pass the indeces to a function I get a bunch of errors.

Rough description of the SparseArray $S$: The value at each position depends on the indexes of that position, $S$ looks somewhat like this:

SparseArray[{{jj_, kk_} /; Some test depending on jj and kk -> some value depending on jj and kk, {n, n}]

What works: My test is somewhat complicated, but it works fine (see below for the complete minimal example). I can generate a matrix like so:

SparseArray[{{jj_, kk_} /; Some test depending on jj and kk ->  jj + kk, {n, n}]

e.g. for n=6 the MatrixForm looks like this:

$\begin{bmatrix}4&0&0&7&0&9 \\ 0&6&0&0&0&0\\ 0&0&8&0&0&0\\ 7&0&0&10&0&12\\ 0&0&0&0&12&0\\ 9&0&0&12&0&14\end{bmatrix}$

What does not work: Instead of just adding the two indexes $jj$ and $kk$ I want to pass them to a function getSval and use the value that this function returns, i.e.:

SparseArray[{{jj_, kk_} /; Some test depending on jj and kk -> getSval[degree,jj,kk], {n, n}]

getSval works fine when I call it outside the SparseArray definition, e.g. getSval[2, 4, 6] evaluates to $4\sqrt{2}\pi^{3/2}$. But using it in SparseArray throws a bunch of errors.

Here is the minimal working example:

(*Define IDX, essentially a list of indexes*)
Do[
 IDX[n] = 
   Flatten[Table[
     Table[{n - ii, ii - jj, jj}, {jj, 0, ii}], {ii, 0, n}], 1];
 , {n, 0, 40}]

(*define the function getSval*)
getSval[degree_, j_, k_] := Block[{a, b, c, idx1, idx2},
  idx1 := IDX[degree][[j]];
  idx2 := IDX[degree][[k]];
  
  a = 1/2 (idx1[[1]] + idx2[[1]]);
  b = 1/2 (idx1[[2]] + idx2[[2]]);
  c = 1/2 (idx1[[3]] + idx2[[3]]);
  \[Pi]^(3/2) 2^(3/2)(a+b+c)
  ]

(*choose some setup-parameters for S*)
degree = 2;
length = Length[IDX[degree]]

(*try out the SparseArray function*)
MatrixForm[
 SparseArray[{{jj_, kk_} /; 
     EvenQ[IDX[degree][[jj]][[1]] + IDX[degree][[kk]][[1]]] && 
      EvenQ[IDX[degree][[jj]][[2]] + IDX[degree][[kk]][[2]]] && 
      EvenQ[IDX[degree][[jj]][[3]] + IDX[degree][[kk]][[3]]] 
  -> 
    degree + jj + kk}, {length, length}]] (*this works just fine*)

(*trying to use SparseArray with getSval*)
MatrixForm[
 SparseArray[{{jj_, kk_} /; 
     EvenQ[IDX[degree][[jj]][[1]] + IDX[degree][[kk]][[1]]] && 
      EvenQ[IDX[degree][[jj]][[2]] + IDX[degree][[kk]][[2]]] && 
      EvenQ[IDX[degree][[jj]][[3]] + IDX[degree][[kk]][[3]]] 
  -> 
    getSval[degree, jj, kk]}, {length, length}]] (*this crashes*)

I have tried several different things, e.g. including the calculation of getSval directly into the SparseArray, but so far nothing works. My impression is that $jj$ and $kk$ are handled differently after the arrow ->. For example

MatrixForm[
 SparseArray[{{jj_, kk_} /; 
     EvenQ[IDX[degree][[jj]][[1]] + IDX[degree][[kk]][[1]]] && 
      EvenQ[IDX[degree][[jj]][[2]] + IDX[degree][[kk]][[2]]] && 
      EvenQ[IDX[degree][[jj]][[3]] + IDX[degree][[kk]][[3]]] -> 
    IDX[degree][[jj]][[1]] }, {length, length}]]

crashes with the complaint, that the value specified by the rule should not be a list, even though for example IDX[degree][[2]][[1]] evaluates to a number (in this case $1$).

What is the reason for this and is there some way to fix this?

Edit: I found the following work-around, but I am not happy with it - it certainly is not efficient:

  • Step 1: Hold the evaluation when setting up the SparseArray:
    S = SparseArray[{{jj_, kk_} /; 
          EvenQ[IDX[degree][[jj]][[1]] + IDX[degree][[kk]][[1]]] && 
           EvenQ[IDX[degree][[jj]][[2]] + IDX[degree][[kk]][[2]]] && 
           EvenQ[IDX[degree][[jj]][[3]] + IDX[degree][[kk]][[3]]] -> 
         Hold[getSval[degree, jj, kk]]}, {length, length}];
  • Step 2: ReleaseHold of the Hold-expressions in $S$. Unfortunately ReleaseHold[S] does not work (why?). However, I can parse $S$ as a list, ReleaseHold of that list and then parse the result back into a SparseArray:
SparseArray[ReleaseHold[Normal[S]]]

Obviously, this takes quite some time - is there a more efficient way?

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    $\begingroup$ What happens if you use RuleDelayed (:>) instead of Rule (->)? $\endgroup$
    – Carl Woll
    Jul 7, 2020 at 17:27
  • $\begingroup$ Ahhh - yes, this solved the issue. I honestly wasn't aware of RuleDelayed, thank you :-) $\endgroup$
    – Andrea
    Jul 7, 2020 at 18:34

1 Answer 1

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As Carl pointed out, RuleDelayed :> is your friend in this case and fixes the issue -

MatrixForm[SparseArray[{{jj_, kk_} /; 
          EvenQ[IDX[degree][[jj]][[1]] + IDX[degree][[kk]][[1]]] && 
           EvenQ[IDX[degree][[jj]][[2]] + IDX[degree][[kk]][[2]]] && 
           EvenQ[IDX[degree][[jj]][[3]] + IDX[degree][[kk]][[3]]] :> 
         getSval[degree, jj, kk]}, {length, length}]]

gives the wanted result.

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