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### Highlights from Eigenshuffle

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# Eigenshuffle

### John D'Errico (view profile)

04 Feb 2009 (Updated )

Consistently sorted eigenvalue and eigenvector sequences

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Description

A problem that I've seen many times on the newsgroup is how eig returns its eigenvalues and eigenvectors. By itself, eig returns an arbitrary order for the eigenvalues and eigenvectors. They are often nearly sorted in order, but this is not assured. The other issue is the eigenvectors can have an arbitrary sign applied to them.

Worse, when you have a sequence of eigenvalue problems, the eigenvalues can sometimes cross over. One would like to sort the eigenvalues/eigenvectors so the sequence is consistent.

I've designed eigenshuffle.m to do exactly that. It takes a pxpxn array, where each page of the array is one matrix where we wish to compute the eigenvalues. Eigenshuffle tries to permute the eigenvalues and eigenvectors to be maximally consistent from one step in the sequence to the next. Eigenshuffle also chooses the sign to be applied to each eigenvector to be maximally consistent with the the vectors prior to it in the sequence of eigenproblems.

As an example, try this simple matrix function of a parameter t.

Efun = @(t) [1 2*t+1 t^2 t^3;2*t+1 2-t t^2 1-t^3; ...
t^2 t^2 3-2*t t^2;t^3 1-t^3 t^2 4-3*t];

Aseq = zeros(4,4,21);
for i = 1:21
Aseq(:,:,i) = Efun((i-11)/10);
end
[Vseq,Dseq] = eigenshuffle(Aseq);

To see that eigenshuffle has done its work correctly,
look at the eigenvalues in sequence after the shuffle.

t = (-1:.1:1)';
[t,Dseq']
ans =
-1 8.4535 5 2.3447 0.20181
-0.9 7.8121 4.7687 2.3728 0.44644
-0.8 7.2481 4.56 2.3413 0.65054
-0.7 6.7524 4.3648 2.2709 0.8118
-0.6 6.3156 4.1751 2.1857 0.92364
-0.5 5.9283 3.9855 2.1118 0.97445
-0.4 5.5816 3.7931 2.0727 0.95254
-0.3 5.2676 3.5976 2.0768 0.858
-0.2 4.9791 3.3995 2.1156 0.70581
-0.1 4.7109 3.2 2.1742 0.51494
0 4.4605 3 2.2391 0.30037
0.1 4.2302 2.8 2.2971 0.072689
0.2 4.0303 2.5997 2.3303 -0.16034
0.3 3.8817 2.4047 2.3064 -0.39272
0.4 3.8108 2.1464 2.2628 -0.62001
0.5 3.8302 1.8986 2.1111 -0.83992
0.6 3.9301 1.5937 1.9298 -1.0537
0.7 4.0927 1.2308 1.745 -1.2685
0.8 4.3042 0.82515 1.5729 -1.5023
0.9 4.5572 0.40389 1.4272 -1.7883
1 4.8482 -8.0012e-16 1.3273 -2.1755

Here, columns 2:5 are the shuffled eigenvalues. See that the second eigenvalue goes to zero, but the third eigenvalue remains positive. We can plot eigenvalues and see that they have crossed, near t = 0.35 in Efun.

plot(-1:.1:1,Dseq')

For a better appreciation of what eigenshuffle did, compare the result of eig directly on Efun(.3) and Efun(.4). Thus:

[V3,D3] = eig(Efun(.3))
V3 =
-0.74139 0.53464 -0.23551 0.3302
0.64781 0.4706 -0.16256 0.57659
0.0086542 -0.44236 -0.89119 0.10006
-0.17496 -0.54498 0.35197 0.74061

D3 =
-0.39272 0 0 0
0 2.3064 0 0
0 0 2.4047 0
0 0 0 3.8817

[V4,D4] = eig(Efun(.4))
V4 =
-0.73026 0.19752 0.49743 0.42459
0.66202 0.21373 0.35297 0.62567
0.013412 -0.95225 0.25513 0.16717
-0.16815 -0.092308 -0.75026 0.63271

D4 =
-0.62001 0 0 0
0 2.1464 0 0
0 0 2.2628 0
0 0 0 3.8108

With no sort or shuffle applied, look at V3(:,3). See that it is really closest to V4(:,2), but with a sign flip. Since the signs on the eigenvectors are arbitrary, the sign is changed, and the most consistent sequence will be chosen. By way of comparison, see how the eigenvectors in Vseq have been shuffled, the signs swapped appropriately.

Vseq(:,:,14)
ans =
0.3302 0.23551 -0.53464 0.74139
0.57659 0.16256 -0.4706 -0.64781
0.10006 0.89119 0.44236 -0.0086542
0.74061 -0.35197 0.54498 0.17496

Vseq(:,:,15)
ans =
0.42459 -0.19752 -0.49743 0.73026
0.62567 -0.21373 -0.35297 -0.66202
0.16717 0.95225 -0.25513 -0.013412
0.63271 0.092308 0.75026 0.16815

With many thanks to Yi Cao, I've included munkres by permission as a subfunction here.

Acknowledgements

Hungarian Algorithm For Linear Assignment Problems (V2.3) inspired this file.

MATLAB release MATLAB 7.5 (R2007b)
Other requirements eigenshuffle uses nothing sophisticated or new, so it should run on virtually any release of MATLAB.
11 Dec 2014 William

### William (view profile)

Works very well. Not only performs the eig function over arrays of matrices, but sorts the eigenvalues and eigenvectors in a rational way.

21 Jan 2014 aslan

### aslan (view profile)

It was great, Thanks.

09 Apr 2013 Laura

### Laura (view profile)

This is exactly what I needed. Unfortunately, it doesn't seem to work on eigenvalue problems where the crossings happen over a wide range, or where they cross and then cross back. For my system, this really only worked for small crossings.

04 Dec 2012 Sandor Toth

### Sandor Toth (view profile)

Great code!

20 Mar 2011 pushkarini

### pushkarini (view profile)

20 Mar 2011 pushkarini

comfort.

16 Jun 2010 A

### A (view profile)

13 Dec 2009 Louie Lu

### Louie Lu (view profile)

13 Dec 2009 Louie Lu

### Louie Lu (view profile)

A minor suggestion: could put the output argument Dseq in front of Vseq, so that Dseq=eigenshuffle(A) can be used when Vseq is not wanted.

14 Nov 2009 Maurizio De Pitta'

### Maurizio De Pitta' (view profile)

04 Sep 2009 Ondrej

### Ondrej (view profile)

I thought so, that it couldn't be any harder than eigenvalues, but then I looked here
http://www.math.uu.nl/publications/preprints/1180.ps

Actually the code there seems to work pretty well. So schurly, no need to reinvent schurshuffle

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03 Sep 2009 John D'Errico

### John D'Errico (view profile)

For schur, I'd like to help. But schurly, then I would have to call the solution the schurshuffle.

Seriously, it seems the same methodology would work. And with a name like the schurshuffle, I'd hate to pass up the opportunity.

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03 Sep 2009 Ondrej

### Ondrej (view profile)

Is it possible to write the same functionality but for schur decomposition? Schur function in matlab returns the eigenvalues on diagonal in same order as eig function..which is not always nice. Thanks.

25 Mar 2009 Yi Cao

### Yi Cao (view profile)

John,

This is novel. A possible application is for MIMO frequency response. The following example needs control system toolbox:

w=logspace(-1,1,200);
H=freqresp(sys,w);
[V,D]=eigenshuffle(H);
semilogx(w,D)

Another thing you may consider is to alter the distance measure to the true distance between scaled eighenvectors of two permutations, i.e. define

y_i(k) = lambda_i(k) v_i(k) = lambda_i(k) A(k), for i = 1, ..., n

to pair the ith of A(k) with the jth of A(k-1), the distance

d_ij = |y_i(k)|^2 + |y_j(k-1)|^2 - 2 |y_i(k)| |y_j(k-1)| cos < y_i(k), y_j(k-1)

where
cos < y_i(k), y_j(k-1) = v_i(k)^T v_j(k-1)

Regards,
Yi

23 Mar 2009 Chris

### Chris (view profile)

Fantastic program which does exactly what I needed - thanks!

17 Mar 2009 DNF

### DNF (view profile)

If you check out the other ratings of this 'Xu Wings' you will see he has a thing in for D'Errico and possibly also Oliver Woodford. It seems possible that 'Xu Wings' is an alias for Shahab Anbarjafari.

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17 Mar 2009 Kenneth Eaton

### Kenneth Eaton (view profile)

Xu, I can only guess that you have very little experience with MATLAB and even less knowledge of whom you are criticizing.

17 Mar 2009 Xu Wings

### Xu Wings (view profile)

i found the level of the code very low, possibly a good high school boy in here can write such a programme!

09 Mar 2009 Jonas Lundgren

### Jonas Lundgren (view profile)

And the updated rating.

18 Feb 2009 John D'Errico

### John D'Errico (view profile)

Jonas is absolutely correct, and I had even thought to offer munkres (once I learned of its existence) as the underlying permutation engine in this code, with a test to see if the user had downloaded munkres. Yi Cao was gracious to allow me to include munkres as a subfunction however, so I have included his current version.

The munkres powered version of eigenshuffle is now about 4 times faster for small (4x4) systems. For larger systems on the order of 15x15, I'd expect nearly a 20-1 speedup. Larger systems that that will be even faster compared to the previous release of this code.

Even better, the use of munkres as the permutation engine may sometimes provide a better choice of permutation than did my own minimum trace scheme, so the user wins in two ways here.

Many thanks to Yi Cao for munkres, and to Jonas Lundgren for his comments on the code.

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18 Feb 2009 Jonas Lundgren

### Jonas Lundgren (view profile)

I missed the comment. Eigenshuffle works very well on all the test problems I have tried. The distance measure is clever! If you replace MINTRACE by MUNKRES of Yi Cao, it will be fast as well.

17 Feb 2009 Jonas Lundgren

### Jonas Lundgren (view profile)

05 Feb 2009 John D'Errico

### John D'Errico (view profile)

I've uploaded a new version that works better for complex problems.

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