Use a part specification as the argument to a compiled function

Question

I have some code that needs to generate a large matrix based on some complicated combinations of smaller matrix elements. I find that if I write a function for the element of the larger matrix, and then compile that function, I can get the larger matrix much faster. But I get an error when I compile, and I'd like to find out the right way to do this.

This simpler code reproduces the effect.

matrix = RandomReal[{1, 2}, {25, 25}];
tupleslist = RandomInteger[{1, 25}, {1000000, 2}];

matsubtractA[i_Integer, j_Integer] := matrix[[i, j]] - matrix[[j, i]];

Total[matsubtractA @@@ tupleslist] // AbsoluteTiming

matsubtractB = Compile[{{i, _Integer}, {j, _Integer}},
                       matrix[[i, j]] - matrix[[j, i]]
                       ];

Total[matsubtractB @@@ tupleslist] // AbsoluteTiming

matsubtractC = Compile[{{i, _Integer}, {j, _Integer}},
                       Evaluate[
                               matrix[[i, j]] - matrix[[j, i]]
                               ]
                       ];

Total[matsubtractC @@@ tupleslist] // AbsoluteTiming

The result is

All three methods give the same answer, but with much different execution times. The first attempt at Compile is apparently worthless, since it increases the time. The second attempt, where I use Evaluate within compile, greatly improves the execution time but gives a Part::pkspec1 error when compiling: "The expression i cannot be used as a part specification."

In my actual code, the speedup is even more drastic, so I'll use it even with the error. But I'd love to know how to fix it. I even tried using the mysterious third argument to Compile,

matsubtractC2 = Compile[{{i, _Integer}, {j, _Integer}},
        Evaluate[
            mat = matrix;
            mat[[i, j]] - mat[[j, i]]
            ], 
      {{mat, _Real, 2}}];

Total[matsubtractC @@@ tupleslist] // AbsoluteTiming

But I got the same error.

Answer 1

The problem here is more simple: it's the same error you'd get if you tried matrix[[i, j]]. You told Mathematica to Evaluate the expression matrix[[i,j]] - matrix[[j,i]] before doing the Compile, so it reduces to that problem. The problem is that the symbol i must be an integer to be a valid part specification: Part[{1, 2, 3}, i] throws an error.

Of course, Mathematica keeps the expression unevaluated, so you can subsequently substitute i -> 1 to get the answer 1. You can prevent the error from being thrown using Quiet.