MatrixExp[m, v] not always faster than MatrixExp[m].v

Question

For non sparse matrix m, is MatrixExp[m, v] supposed to be faster than MatrixExp[m].v? This seems to be true only if m is purely real or imaginary.

Block[{n = 500},
v = RandomComplex[1. + I, n];
s = RandomReal[{-1., 1.}, {n, n}];
s1 = RandomComplex[1. + I, {n, n}];
]

MatrixExp[s].v; // AbsoluteTiming

{0.12628, Null}

MatrixExp[s, v]; // AbsoluteTiming

{0.0379219, Null}

MatrixExp[s1].v; // AbsoluteTiming

{0.365637, Null}

MatrixExp[s1, v]; // AbsoluteTiming

{5.17002, Null}

MatrixExp[s1, v, Method -> "Krylov"]; // AbsoluteTiming

{4.93712, Null}

MatrixExp[s1, v, Method -> "Pade"]; // AbsoluteTiming

{0.416058, Null}

I can't reproduce that. Try quitting kernel between calculations. I'm getting MatrixExp[s1, v]; // AbsoluteTiming``{5.*10^-6, Null} — Feyre, Jul 27 '16 at 21:18
@Feyre Hi, I tried quitting kernels between calculations and got the same results. What version of Mathematica are you using? — user64620, Jul 27 '16 at 21:35
I'm voting to close because I cannot reproduce the issue. If others manage to reproduce this reliably then I'm happy to take back that vote. — Daniel Lichtblau, Jul 27 '16 at 23:11
I get very similar results to what the OP shows running his code on V10.4.1 on OS X 10.10.2. — m_goldberg, Jul 28 '16 at 00:20
I do not get similar results: http://i.stack.imgur.com/Oyk1B.png (Mac OSX 10.11.5, V10.4.1) — Michael E2, Jul 28 '16 at 00:54
Okay, I retracted my vote since this appears to be platform dependent. Will see if we can reproduce it here. — Daniel Lichtblau, Jul 28 '16 at 14:26

score 2 · Answer 1 · edited Aug 01 '16 at 08:54

2

Mr.Wizard

Timing results in Mathematica 10.1.0 under Windows 7 x64:

{0.0660777, Null}
{0.0303608, Null}
{0.190943, Null}
{1.46382, Null}
{1.41635, Null}
{0.183233, Null}

So I confirm MatrixExp[s1, v] and MatrixExp[s1, v, Method -> "Krylov"] as being slower on my system.

Alexey Popkov

Timing results with Mathematica 10.4.1 under Windows 7 x64 (CPU with 2 physical cores):

{1.26094, Null}
{0.638639, Null}
{1.26434, Null}
{19.7368, Null}
{18.708, Null}
{1.89468, Null}

Mariusz Iwaniuk

Timing results with Mathematica 10.2.0 under Windows 8.1 x64 (CPU with 2 physical cores):

{1.03687, Null}
{0.128066, Null}
{3.9396, Null}
{21.1016, Null}
{21.0798, Null}
{3.95479, Null}

( Use this Community Wiki to share any other timing results of interest. )

edited Aug 01 '16 at 08:54

Mariusz Iwaniuk

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answered Jul 28 '16 at 01:20

Mr.Wizard

271,378
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"Krylov" is really only good for large, sparse matrices, and mostly intended for use with the action form of the exponential on a vector. – J. M.'s missing motivation Jul 28 '16 at 01:28
1

@J.M. One wonders then why MatrixExp[s1, v] seemingly defaults to this method on my system? – Mr.Wizard Jul 28 '16 at 01:32
1

My point was that it is not fair to "Krylov" to be using a dense matrix as the argument; of course it will then be slow. Have you also tried Method -> "BlockDecomposition"? – J. M.'s missing motivation Jul 28 '16 at 01:40
@J.M. BlockDecomposition is even slower when I tested it. The slowdown is also present for sparse matrix m. – user64620 Jul 28 '16 at 14:28
@Alexey, can you confirm the OP's observations regarding "BlockDecomposition"? – J. M.'s missing motivation Aug 01 '16 at 06:48
@J.M. Yes, definitely: I wasn't able to measure timing for "BlockDecomposition" because it crashed my laptop due to CPU overheating, but it was more than 1 minute... – Alexey Popkov Aug 01 '16 at 06:52
I have the same order of timings as Mr.Wizard using 10.4.1 for Linux x86 (64-bit). @J.M. "BlockDecomposition" was around 70 times slower then "Krylov" for the complex matrix. – sebhofer Aug 01 '16 at 08:44

MatrixExp[m, v] not always faster than MatrixExp[m].v

1 Answers1

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