# How to solve an eigensystem faster?

I have a module that I need to call 1-10 million times in my program. Currently, it is taking several hours to run so I am hoping that I can cut down some runtime with your help.

r = RandomReal[NormalDistribution[0., 1./2.], 6];

es = Eigensystem[H0[ω0, r[[1]], r[[2]], r[[3]], r[[4]], r[[5]], r[[6]] ];

ε = es[[1]];
v = es[[2]];
vS = Conjugate[v];

(*elements of v and vS are called later; v[[1]], vS[[1]] etc...*)


H0 is a compiled function which sped things up a little. It looks like this:

.

In copy-paste form,

 H0={{0, (ωz1 - ωz2)/
2, (-ωx1 + ωx2)/(2 Sqrt[2]) - (I ωy1)/(
2 Sqrt[2]) + (I ωy2)/(
2 Sqrt[2]), (ωx1 - ωx2)/(2 Sqrt[2]) - (
I ωy1)/(2 Sqrt[2]) + (I ωy2)/(
2 Sqrt[2])}, {(ωz1 - ωz2)/2,
0, (ωx1 + ωx2)/(2 Sqrt[2]) + (I ωy1)/(
2 Sqrt[2]) + (I ωy2)/(
2 Sqrt[2]), (ωx1 + ωx2)/(2 Sqrt[2]) - (
I ωy1)/(2 Sqrt[2]) - (I ωy2)/(
2 Sqrt[2])}, {(-ωx1 + ωx2)/(2 Sqrt[2]) + (
I ωy1)/(2 Sqrt[2]) - (I ωy2)/(
2 Sqrt[2]), (ωx1 + ωx2)/(2 Sqrt[2]) - (
I ωy1)/(2 Sqrt[2]) - (I ωy2)/(
2 Sqrt[2]), ω0 + (ωz1 + ωz2)/2,
0}, {(ωx1 - ωx2)/(2 Sqrt[2]) + (I ωy1)/(
2 Sqrt[2]) - (I ωy2)/(
2 Sqrt[2]), (ωx1 + ωx2)/(2 Sqrt[2]) + (
I ωy1)/(2 Sqrt[2]) + (I ωy2)/(2 Sqrt[2]),
0, -ω0 + 1/2 (-ωz1 - ωz2)}}


Is there anything else that can be optimized here?

• Without you saying more on H0[], there's really nothing else to say, except that the computation of an eigensystem can be simplified a bit, if the matrix given to it has some structure (e.g. symmetry, sparseness). Oct 22, 2012 at 2:24
• Just to check - is whatever loop you have running 10 mil times parallellized?
– VF1
Oct 22, 2012 at 2:43
• @J. M. H0[] is 4x4 Hermitian. Four elements of the matrix are 0. Oct 22, 2012 at 3:06
• "Four elements of the matrix are 0." - which ones? You'll have to be more explicit than that for answerers to get anywhere... Oct 22, 2012 at 3:53
• @BeauGeste Are you calculating the eigensystem for each choice of random parameters (rather than doing it once symbolically and then substitute specific values) ? Oct 22, 2012 at 7:43

This is too long for a comment and honestly, to give a real answer, there is more information required in your question. Isn't it possible, that you give a working example, so that we see what takes long and how you implemented it?

If you are calling Eigensystem for many different input values which are know, there is still some place for speed-up. Since your expressions are very lengthy, please find the initialization in an extra section.

First we measure how long it takes to calculate the Eigenvectors of H0 for 1 Million random values

data = RandomReal[{-1, 1}, {1000000, 7}];
First@AbsoluteTiming[Eigenvectors[H0[#]] & /@ data]


This took 44.4 sec here. The next thing you can try is to distribute H0 over parallel kernels and use ParallelMap

DistributeDefinitions[H0];
First@AbsoluteTiming[ParallelMap[Eigenvectors[H0[#]] &, data]]


This took 25.3 sec with 4 subkernels. Let's test the compiled code. First when we apply it non-parallel

First@AbsoluteTiming[evectors @@@ data]


This took 2.4 sec which is almost 20 times faster then the initial version. Let's see what we can get if we call it parallel

First@AbsoluteTiming[evectors @@ Transpose[data]]


This took only 0.24 sec. If this scales well, that it means I can run 10 million samples in about 2.5 seconds. An indeed, a test with $10^7$ runs required 2.75 sec.

Now you might ask, whaat??, why is evectors @@@ data a serial call while evectors @@ Transpose[data] is parallel? It's because of the Listable attribute in the Compile-call and since we turn on Parallelization. Sure is

evectors @@@ data == evectors @@ Transpose[data]

(* Out[21]= True *)


## Initialization

Compiled parallel "C" versions of Eigenvalues and Eigenvectors

{evalues, evectors} =
Compile[{{ω0, _Complex, 0}, {ωx1, _Complex,
0}, {ωx2, _Complex, 0}, {ωy1, _Complex,
0}, {ωy2, _Complex, 0}, {ωz1, _Complex,
0}, {ωz2, _Complex, 0}}, #, Parallelization -> True,
CompilationTarget -> "C", RuntimeAttributes -> {Listable}] & /@
Eigensystem[{{0, (ωz1 - ωz2)/
2, (-ωx1 + ωx2)/(2 Sqrt[
2]) - (I ωy1)/(2 Sqrt[2]) + (I ωy2)/(2 Sqrt[
2]), (ωx1 - ωx2)/(2 Sqrt[
2]) - (I ωy1)/(2 Sqrt[2]) + (I ωy2)/(2 Sqrt[
2])}, {(ωz1 - ωz2)/2,
0, (ωx1 + ωx2)/(2 Sqrt[
2]) + (I ωy1)/(2 Sqrt[2]) + (I ωy2)/(2 Sqrt[
2]), (ωx1 + ωx2)/(2 Sqrt[
2]) - (I ωy1)/(2 Sqrt[2]) - (I ωy2)/(2 Sqrt[
2])}, {(-ωx1 + ωx2)/(2 Sqrt[
2]) + (I ωy1)/(2 Sqrt[2]) - (I ωy2)/(2 Sqrt[
2]), (ωx1 + ωx2)/(2 Sqrt[
2]) - (I ωy1)/(2 Sqrt[2]) - (I ωy2)/(2 Sqrt[
2]), ω0 + (ωz1 + ωz2)/2,
0}, {(ωx1 - ωx2)/(2 Sqrt[
2]) + (I ωy1)/(2 Sqrt[2]) - (I ωy2)/(2 Sqrt[
2]), (ωx1 + ωx2)/(2 Sqrt[
2]) + (I ωy1)/(2 Sqrt[2]) + (I ωy2)/(2 Sqrt[
2]), 0, -ω0 + 1/2 (-ωz1 - ωz2)}}];


Furthermore, I try to copy your approach by defining H0

H0[{ω0_, ωx1_, ωx2_, ωy1_, ωy2_,
ωz1_, ωz2_}] =
N[{{0, (ωz1 - ωz2)/
2, (-ωx1 + ωx2)/(2 Sqrt[
2]) - (I ωy1)/(2 Sqrt[2]) + (I ωy2)/(2 Sqrt[
2]), (ωx1 - ωx2)/(2 Sqrt[
2]) - (I ωy1)/(2 Sqrt[2]) + (I ωy2)/(2 Sqrt[
2])}, {(ωz1 - ωz2)/2,
0, (ωx1 + ωx2)/(2 Sqrt[
2]) + (I ωy1)/(2 Sqrt[2]) + (I ωy2)/(2 Sqrt[
2]), (ωx1 + ωx2)/(2 Sqrt[
2]) - (I ωy1)/(2 Sqrt[2]) - (I ωy2)/(2 Sqrt[
2])}, {(-ωx1 + ωx2)/(2 Sqrt[
2]) + (I ωy1)/(2 Sqrt[2]) - (I ωy2)/(2 Sqrt[
2]), (ωx1 + ωx2)/(2 Sqrt[
2]) - (I ωy1)/(2 Sqrt[2]) - (I ωy2)/(2 Sqrt[
2]), ω0 + (ωz1 + ωz2)/2,
0}, {(ωx1 - ωx2)/(2 Sqrt[
2]) + (I ωy1)/(2 Sqrt[2]) - (I ωy2)/(2 Sqrt[
2]), (ωx1 + ωx2)/(2 Sqrt[
2]) + (I ωy1)/(2 Sqrt[2]) + (I ωy2)/(2 Sqrt[
2]), 0, -ω0 + 1/2 (-ωz1 - ωz2)}}];

• thanks for you analysis. The times look very promising. I need the eigenvectors to be normalized. Is it possible to get normalized vectors out of Eigensystem? Oct 23, 2012 at 3:59
• @Beau, look up Normalize[] and Orthogonalize[]. Oct 23, 2012 at 4:05
• @halirutan This code is still of great help to me(for fast processing). Just one query(which will solve my problem of asking similar question in MMSE), in intialization {evalues, evectors}, you wrote whole matrix again inside Eigensystem, is it required. As in my case, matrix is insanely large and its size changes(by changing size parameter), do I need to copy my matrix again. Error I am gettingo,compiledFunction1,{{Real,0,Constant},{Real,0,Constant},{Real,0,\ Constant}},Real]]] are not the same shape
– L.K.
Jan 30, 2017 at 10:22
• The main reason for the speed up here is that the OP wanted eigenvalues and eigenvectors of the same matrix for different matrix elements. So we calculated the Eigensystem only one time and built a compiled function from it. When your matrix changes on each call, this method won't work for you. Do you call {evalues, evectors} for a specific matrix several times or only exactly once? Jan 30, 2017 at 11:48
• @halirutan Thanks. I am calling for particular size only once. Ex: If I have a 11X11 then I am only evaluating(evectors and evalues) for that size once. Storing them inside some matrix, to say. Will this work?
– L.K.
Jan 30, 2017 at 12:29

You may want to consider simplifying your matrix by substituting the [Omega] pairs that occur often with a dummy. This will simplify your matrix as there are only very few variables left. All you need to do is design a pattern to replace the most common [Omega] pairs (in that order). After the Eigensystem calculation, you can substitute them back.

I did something similar where several component were frequently occuring however were nested in divisions on several levels. Standard Simplify did not work, so I wrote my own which substantially increased performance of the simplified matrix.

The pattern I used was the simple Power[x_,y__] How you identify what combination to substitute depends on what's underlying your system, I am not really up to speed on Fourier transformations but this seems to be underlying what you are analysing here.

Below example gives your a transformastionRules fir the matrix and a reverseTransformation after the Eigensystem is calculated.

dummycount=1;
Position[HO,(*pattern*)];
H0[[#/.List->Sequence]]&/@Position[H0,(*pattern*)];
Sort[Cases[Tally[%],Except[{_,1}]],#1[[2]]>#2[[2]]&];
transformationRules=Rule[#[[1]],dummy[dummycount++]]&/@%;
reverseTransformation=transformationRules/.Rule[x_,y_]->Rule[y,x];


@halirutan : Thanks for the excellent example, I achieved several orders of magnitude speed-up for a program I was working on; however when I tested the execution of the compiled code without CompilationTarget->"C" something changes:

evectors@@@data==Eigenvectors[H0[#]]&/@data
(* False *)
Eigenvectors[H0[#]]&/@data==ParallelMap[Eigenvectors[H0[#]]&,data]
(* True *)
ParallelMap[Eigenvectors[H0[#]]&,data]==evectors@@Transpose[data]
(* False *)
evectors@@@data==evectors@@Transpose[data]
(* True *)


It seems to me the compiled functions behave differently from the non-compiled but I cannot discover where the difference is. I did create a trivial (2 dimension) example which simply swapped output and hence was easily corrected by

Reverse/@(evectors@@Transpose[data])


but for higher dimensions it is unclear what causes it to change its output.

Also, removing the RuntimeAttributes -> {Listable} attribute did not change anything. The problem seems to be that parameters are passed on to EigenSystem in listable manner and Eigensystem seems not to work in Listable mode; Eigensystem::matsq : "Argument {…} is not a non-empty square matrix.

However rewriting the code in what I believe to be functional identical by:

cpfnc = Compile[{{ω0,ωx1,ωx2,ωy1,ωy2,ωz1,ωz2},
Module[{eval,evec},{eval,evec} = Eigensystem[{{0, (ωz1 - ωz2)/
2, (-ωx1 + ωx2)/(2 Sqrt[
2]) - (I ωy1)/(2 Sqrt[2]) + (I ωy2)/(2 Sqrt[
2]), (ωx1 - ωx2)/(2 Sqrt[
2]) - (I ωy1)/(2 Sqrt[2]) + (I ωy2)/(2 Sqrt[
2])}, {(ωz1 - ωz2)/2,
0, (ωx1 + ωx2)/(2 Sqrt[
2]) + (I ωy1)/(2 Sqrt[2]) + (I ωy2)/(2 Sqrt[
2]), (ωx1 + ωx2)/(2 Sqrt[
2]) - (I ωy1)/(2 Sqrt[2]) - (I ωy2)/(2 Sqrt[
2])}, {(-ωx1 + ωx2)/(2 Sqrt[
2]) + (I ωy1)/(2 Sqrt[2]) - (I ωy2)/(2 Sqrt[
2]), (ωx1 + ωx2)/(2 Sqrt[
2]) - (I ωy1)/(2 Sqrt[2]) - (I ωy2)/(2 Sqrt[
2]), ω0 + (ωz1 + ωz2)/2,
0}, {(ωx1 - ωx2)/(2 Sqrt[
2]) + (I ωy1)/(2 Sqrt[2]) - (I ωy2)/(2 Sqrt[
2]), (ωx1 + ωx2)/(2 Sqrt[
2]) + (I ωy1)/(2 Sqrt[2]) + (I ωy2)/(2 Sqrt[
2]), 0, -ω0 + 1/2 (-ωz1 - ωz2)}}];
{eval,evec}],
RuntimeAttributes -> {Listable}],Parallelization -> True];


does give the correct output, however again Listable provides problems for EigenSystem[] once parallel input is provided.

• I can't test this in version 7, but the difference in behavior that you observe may be caused by RuntimeAttributes -> {Listable}. Have you considered that? May 13, 2014 at 12:01
If Length[r]==6 the following will be faster.
es = Apply[Eigensystem[H0[\[Omega]0,##]]&,r];

If Length[r]>6, then change ## to (#1,#2,#3,#4,#5,#6).