scipy.sparse.linalg.eigsh() doesn't give out the same result as Matlab's eigs(), why?

Question

I'm using scipy.sparse.linalg.eigsh() to solve the generalized eigenvalue problem. I wanna use eigsh() because I'm manipulating some large sparse matrix. The problem is I cannot get the right answers and the eigenvalues and eigenvectors output from eigsh() are totally different from what I've got from Matlab's eigs().

It looks like this: data:

a: 
   304.7179  103.1667   36.9583   61.3478   11.5724
    35.5242  111.4789   -9.8928    8.2586   -4.7405
    10.8358    4.3433  145.6586   26.5153   13.1871
    -1.1924   -2.5430    0.4322   43.1886   -0.6098
   -18.7751   -8.8031   -4.3962   -5.8791   17.6588
b: 
   736.9822  615.7946  587.6828  595.7169  545.1878
   615.7946  678.2142  575.7579  587.3469  524.7201
   587.6828  575.7579  698.6223  593.5402  534.3675
   595.7169  587.3469  593.5402  646.0410  530.1114
   545.1878  524.7201  534.3675  530.1114  590.1373

in python: a,b are numpy.ndarray

In [11]: import scipy.sparse.linalg as lg

In [14]: x,y=lg.eigsh(a,M=b,k=2,which='SM')

In [15]: x
Out[15]: array([ 0.01456738,  0.22578463])

In [16]: y
Out[16]: 
array([[ 0.00052614,  0.00807034],
       [ 0.00514091, -0.01593113],
       [ 0.00233622, -0.00429671],
       [ 0.01877451, -0.06259276],
       [ 0.01491696,  0.08002341]])

In [18]: a.dot(y[:,0])-x[0]*b.dot(y[:,0])
Out[18]: array([ 1.74827445,  0.30325634,  0.71299604,  0.42842245, -0.24724681])

In [19]: a.dot(y[:,1])-x[1]*b.dot(y[:,1])
Out[19]: array([-2.2463206 , -1.64704567, -0.80086734, -1.56796329, 0.03027861])

It could be seen that the eigenvalues and eigenvectors are not good enough to recomposing the original matrix.

However, in MATLAB it works well:

[y,x] = eigs(a,b,2,'sm');

y =

    0.0037   -0.0141
   -0.0056    0.0151
    0.0015    0.0079
   -0.0117    0.0666
   -0.0298   -0.0753
x =

    0.0202         0
         0    0.3499
a*x(:,1)-y(1,1)*b*x(:,1)

ans =

   1.0e-14 *

   -0.3775
    0.0777
    0.0777
    0.0555
    0.0666

Plus, data b is positive definite:

In [24]: np.linalg.eigvals(b)
Out[24]: 
array([ 2951.07297125,   137.81545217,    90.40223937,   107.04818229,
          63.65818086])

Anybody could explain why I cannot get the right answer in python?

---

Using lg.eigs()we do get the same outputs as in MATLAB. But...problem occurs when matrix becomes large like this:

test_eigs.mat

in MATLAB we've got things like this:

>> [x,y] = eigs(A,B,4,'sm');
y =

0.0001         0         0         0
     0    0.0543         0         0
     0         0    0.1177         0
     0         0         0    0.1350

while in python(python3.5.2,scipy1.0.0) using lg.eigs(A,M=B,k=4,which='SM') it results in eigenvalues as:

array([  4.43277284e+51 +0.00000000e+00j,
     1.04797857e+48 +8.30096152e+47j,
     1.04797857e+48 -8.30096152e+47j,  -1.45582240e+31 +0.00000000e+00j])

Answer 1

As Paul Panzer said, "h" in "eigsh" stands for Hermitian, which your matrix A is not. (Also, having positive eigenvalues does not imply being positive definite; this is only true if the matrix is Hermitian to begin with.) The method eigsh does not check the input for being Hermitian; it just follows a process assuming it is; so the output is incorrect when the assumption fails.

Using eigs method produces the same results as Matlab:

x, y = lg.eigs(a,M=b,k=2,which='SM') 
np.real(x), np.real(y)  # x and y have tiny imaginary parts due to float math errors

(array([ 0.02022333,  0.34993346]), 
 array([[-0.00368007, -0.0140898 ],
    [ 0.0056435 ,  0.01509067],
    [-0.00154725,  0.00790518],
    [ 0.01170563,  0.06664118],
    [ 0.02981777, -0.07528778]]))

Of course, eigs takes a lot longer to run than eigsh.

---

Your second example is a 34 by 34 dense matrix, it has no zeros at all. Using sparse linear algebra on it is not reasonable; and there is a warning saying that the method did not converge. The regular linear algebra module works fine.

import scipy.linalg as la
sorted_eigenvals = np.sort(np.real(la.eigvals(Am, Bm)))

This returns

5.90947734e-05, 5.42521180e-02, 1.17669899e-01, 1.34952286e-01, ...

in agreement with MATLAB's output that you quoted (except Matlab rounds the numbers)

0.0001, 0.0543, 0.1177, 0.1350