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Take two complex matrices depending on one variable, for example:

m1 = {{1. + 1. I, 1. - 1. I}, {a (1. + .2 I) + 2. I, 0. + 3. I}};
m2 = {{1. - 3. I, 1. - 2. I}, {a (- 1. - .2 I) + 5. I, 0. - 2. I}};

Using LinearSolve we can find the matrix m3 satisfying m1.m3 == m2:

m3 = Block[{a = 1. - .5 I}, LinearSolve[m1, m2]];
Block[{a = 1. - .5 I}, m1.m3 - m2] // Chop
(* {{0,0},{0,0}} *)

I want to turn this LinearSolve into a compiled function of a, for example as:

linSol = Hold[Compile][{{a, _Complex, 0}}, Hold[LinearSolve][m1, m2]] /. Hold[x_] :> x;

(The Hold's are because I don't want LinearSolve to evaluate symbolically)

Now linSol[1.-.5I] does give the correct answer, but it does not compile properly, giving the error message 

CompiledFunction::cfex: Could not complete external evaluation at instruction 7; proceeding with uncompiled evaluation. >>

Furthermore, if we either make both matrices real, or replace the second matrix by a vector, the problem disappears, i.e. if we define,

v1 = m2[[1]];
linSol2 = Hold[Compile][{{a, _Complex, 0}}, Hold[LinearSolve][m1, v1]] /. Hold[x_] :> x;

m1Re = Assuming[{a > 0}, m1 // Re // Simplify];
m2Re = Assuming[{a > 0}, m2 // Re // Simplify];
linSol3 = Hold[Compile][{{a, _Complex, 0}}, Hold[LinearSolve][m1Re, m2Re]] /. Hold[x_] :> x;

Then linSol2 and linSol3 work without any problems.

Am I missing something, or is this a bug? I took care that each numeric entry in m1 and m2 is a machine precision complex number.

I get this in version 10.4.0 on a mac with OS X.

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  • $\begingroup$ Have you seen this? $\endgroup$
    – J. M.'s missing motivation ♦
    Commented Jun 14, 2016 at 15:55
  • $\begingroup$ Are you referring to the fact that LinearSolve can't be compiled without a call to MainEvaluate? This I know, but I find compiling still speeds it up, i.e. in linSol3 which does compile, if i make it a pure function instead it is 10x slower for 40x40 matrices. But I'm not able to get this speed increase from compiling in the complex case. $\endgroup$
    – Jansen
    Commented Jun 14, 2016 at 16:25
  • $\begingroup$ @Jansen Perhaps you could include your last comment and an example showcasing the speedup you get with Compile in your original question. Those seem important points to make. $\endgroup$
    – MarcoB
    Commented Jun 15, 2016 at 17:26
  • $\begingroup$ @MarcoB, Hmm it seems I was wrong in my previous comment, I can't replicate the speedup anymore. The issue is still there though, one can imagine it being part of a larger piece of code which needs to be compiled. From this question it seems the compiler can only determine the output type for arguments {matrix,vector}, but actually it also works for {matrix,matrix} as long as they are both real. Seems there are some definitions missing? $\endgroup$
    – Jansen
    Commented Jun 16, 2016 at 8:27

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