Given a big square matrix and some eigenvalues, how to find the corresponding eigenvectors?

My computer configuration: CPU: 8 cores 4.6GHz, 16GB memory.

I have a big matrix A with size(A) = (3072,3072)

I can get the eigenvalues by doing

v = eig(A)

But my computer dies if I do

[V,D] = eig(A)

I need the corresponding eigenvectors of some of the eigenvalues in v, saying I need 180 eigenvectors, and the corresponding eigenvalues are biggest among v. How to do it?

[Bruno Luong]

A matrix 3072 x 3072 is by no mean BIG to my standard. I'm surprise your PC stall.

Mine return results in few seconds.

I'm afraid EIGS (as @the cyclist suggests) will NOT able to output 180 eigen vectors accurately.

[the cyclist]

@Bruno Luong, can you explain a bit more (or point to documentation) about why one would not expect

eigs(A,180)

would be expected to give less accurate eigenvectors than the first 180 eigenvectors (sorted by magnitude) of

eig(A)

?

I'm not doubting you! It's just surprising to me, and I'd like to understand that better.

[Bruno Luong]

The eigs command is based on Krylov subspace and it is quite poor and fragile numerically. This kind of warning happens all the time

A=rand(1000);

[V,D]=eig(A,'vector');
[~,is]=sort(abs(D),'descend');
Vs=V(:,is);
Ds=D(is);

[Veigs,Deigs]=eigs(A,10);

Warning: 7 of the 10 requested eigenvalues converged. Eigenvalues that did not converge are NaN.

Deigs=diag(Deigs);
for k=1:size(Veigs,2)
    errorD = abs((Ds(k)-Deigs(k))/Ds(k));
    errorV = norm(dot(Veigs(:,k),Vs(:,k))/(norm(Veigs(:,k).*norm(Vs(:,k)))))-1;
    fprintf('k=%d, errorD = %f, errorV = %f\n', k, errorD, errorV)
end

k=1, errorD = 0.000000, errorV = 0.000000
k=2, errorD = 0.000000, errorV = 0.000000
k=3, errorD = 0.000000, errorV = 0.000000
k=4, errorD = NaN, errorV = 0.000000
k=5, errorD = NaN, errorV = 0.000000
k=6, errorD = 1.229746, errorV = -0.974433
k=7, errorD = 1.940090, errorV = -0.971715
k=8, errorD = 1.995754, errorV = -0.972432
k=9, errorD = 1.965161, errorV = -0.930798
k=10, errorD = NaN, errorV = -0.973482

eig works just fine.

That happens more on the middle of the spectrum when the eigenvalues are dense, creating converging issue of the iterative method.

You can run my code several time, some time eigs works ùost of the time it's not.

Answer 1

[Christine Tobler]

I wouldn't expect a 3072-by-3072 matrix to be a problem on the machine you describe. Could you try to run the following on your machine?

A = randn(3072, 3072);

% A = A + A'; % if your original matrix is symmetric, please also symmetrize here

[U, D] = eig(A);

I'd like to find out if it's any matrix of this size that's causing an issue, or just your particular one.

[yi yang]

I tried your code. My computer died.

Bluescreen with stop code:

WHEA_UNCORRECTABLE_ERROR

[Walter Roberson]

@yi yang

That WHEA_UNCORRECTABLE_ERROR error message appears to be associated with overclocking to the point where your system becomes unstable.

By extension it could also probably happen if your system is getting overheated even without overclocking.

If this is a desktop or tower machine, be sure to clean out any dust that may have accumulated inside of it.

[yi yang]

Confirmed. It is my memory problem.

My memory was running at 3200MHz, which is recommended by the manufacture. bluescreen, freezing

Now it is running at 2400MHz, which is my system default. No bluescreen, no freezing.

Thanks guys for helping

Answer 2

[the cyclist]

You could try to use eigs instead of eig. It allows you to get the largest N eigenvalues and the corresponding eigenvectors.

Answer 3

[John D'Errico]

A = magic(8)

A = 8×8

64 2 3 61 60 6 7 57 9 55 54 12 13 51 50 16 17 47 46 20 21 43 42 24 40 26 27 37 36 30 31 33 32 34 35 29 28 38 39 25 41 23 22 44 45 19 18 48 49 15 14 52 53 11 10 56 8 58 59 5 4 62 63 1

format long g
[V,D] = eig(A); d = diag(D)

d = 
                        260 +                     0i
           51.8459255872628 +                     0i
          -51.8459255872629 +                     0i
       3.30788760574535e-15 +    1.120272933028e-14i
       3.30788760574535e-15 -    1.120272933028e-14i
       6.18932227044616e-15 +                     0i
       9.94987338090162e-16 +                     0i
      -4.24981561804611e-15 +                     0i

Now we wish to solve for the eigenvector, corresponding to one of the eigenvalues, as if we did not know the eigenvector already. In this case, we know it, but we need to check how well we did.

eigenvec = @(A,e,n) [A - e*eye(n);[1,zeros(1,n-1)]]\[zeros(n,1);1];

All I did there was to enforce that the first element of the eigenvector is 1. This arbitrarily scales the eigenvector. But except for rare occasions, when the first element of the corresponding eigenvector would have been exactly zero, this will make the system non-singular.

n = size(A,1);
V1 = eigenvec(A,d(1),n);V1 = V1/norm(V1);
[V(:,1),V1]

ans = 8×2

0.353553390593274 0.353553390593274 0.353553390593273 0.353553390593274 0.353553390593274 0.353553390593274 0.353553390593274 0.353553390593274 0.353553390593274 0.353553390593274 0.353553390593274 0.353553390593274 0.353553390593274 0.353553390593274 0.353553390593274 0.353553390593274

In some cases, the two vectors will differ by a factor of -1, but that should be all.

Honestly what I'm not sure I know is why your computer is failing to compute the eigenvectors. 3024 is a pretty small system these days. My computer did not even make a small hiccup when I try that.