Using ARPACK solving eigenvalueproblem, but getting inconsistent results with Matlab

问题

I'm new to ARPACK, I downloaded a script like the following

import time
import numpy as np
from scipy.linalg import eigh
from scipy.sparse.linalg import eigs
np.set_printoptions(suppress=True)

n=30
rstart=0
rend=n

A=np.zeros(shape=(n,n))

# first row
if rstart == 0:
    A[0, :2] = [2, -1]
    rstart += 1
# last row
if rend == n:
    A[n-1, -2:] = [-1, 2]
    rend -= 1
# other rows
for i in range(rstart, rend):
    A[i, i-1:i+2] = [-1, 2, -1]

A[0,8]=30

start_time = time.time()

evals_large, evecs_large = eigs(A, 10, sigma=3.6766133, which='LM')
print evals_large

end_time=time.time()-start_time
print(" Elapsed time: %12f seconds " % end_time)

It solves a very simple eigenvalue problem (The matrix A there is not symmetric, I set A[0,8] to be 30). The 3 eigenvalues closest to 3.6766133 (sigma=3.6766133 in the setting) according to the ARPACK results are

[ 3.68402411+0.j  3.82005897+0.j  3.51120293+0.j]

Then I go to MATLAB, and solve the same eigenvalue problem, and the results are

4.144524409923138 + 0.000000000000000i 
3.642801014184622 + 0.497479798520641i
3.642801014184622 - 0.497479798520641i
2.372392770347609 + 0.762183281789166i
2.372392770347609 - 0.762183281789166i
3.979221766266502 + 0.000000000000000i
3.918541441830947 + 0.000000000000000i
3.820058967057387 + 0.000000000000000i 
3.684024113506185 + 0.000000000000000i
3.511202932803536 + 0.000000000000000i
3.307439963195127 + 0.000000000000000i
3.080265978640102 + 0.000000000000000i
2.832849552917550 + 0.000000000000000i
2.565972630556613 + 0.000000000000000i
2.283744793210587 + 0.000000000000000i
1.996972474451519 + 0.000000000000000i
0.927737801889518 + 0.670252740725955i
0.927737801889518 - 0.670252740725955i
1.714561796881689 + 0.000000000000000i
-0.015193770830045 + 0.264703483268519i
-0.015193770830045 - 0.264703483268519i
1.438919271663752 + 0.000000000000000i
0.019951101383019 + 0.000000000000000i
0.080534338862828 + 0.000000000000000i 
0.181591307101504 + 0.000000000000000i
0.318955140475174 + 0.000000000000000i
0.488231021129767 + 0.000000000000000i
0.688030188040126 + 0.000000000000000i
1.171318650526539 + 0.000000000000000i
0.917612528393044 + 0.000000000000000i

Apparently, the second mode 3.642801014184622 + 0.497479798520641i is more close to sigma=3.6766133, but ARPACK didn't pick it out.

What could be the problem? Could you help me figure this out? Thanks a lot.

回答1:

A few things first about the MATLAB functions:

Eigenvalues returned by eig are NOT sorted. In [V,D] = eig(A) we are only guaranteed that the columns of V are the corresponding right eigenvectors to the eigenvalues in D(i,i). On the other hand, svd returns singular values sorted in decreasing order.
d = eigs(A,k) return the k largest-magnitude eigenvalues. However it is intended for large and sparse matrices, and generally is not a substitute for:
```
d = eig(full(A));
d = sort(d, 'descend');
d = d(1:k);
```
(eigs is based on ARPACK, while eig uses LAPACK routines).
There is no natural ordering of complex numbers. The convention is that the sort function sorts complex elements first by magnitude (i.e. abs(x)), then by phase angle on [-pi,pi] interval (i.e. angle(x)) if magnitudes are equal.

MATLAB

With that in mind, consider the following MATLAB code:

% create the same banded matrix you're using
n = 30;
A = spdiags(ones(n,1)*[-1,2,-1], [-1 0 1], n, n);
A(1,9) = 30;
%A = full(A);

% k eigenvalues closest to sigma
k = 10; sigma = 3.6766133;
D = eigs(A, k, sigma);

% lets check they are indeed sorted by distance to sigma
dist = abs(D-sigma);
issorted(dist)

I get:

>> D
D =
  3.684024113506185 + 0.000000000000000i
  3.820058967057386 + 0.000000000000000i
  3.511202932803535 + 0.000000000000000i
  3.918541441830945 + 0.000000000000000i
  3.979221766266508 + 0.000000000000000i
  3.307439963195125 + 0.000000000000000i
  4.144524409923134 + 0.000000000000000i
  3.642801014184618 + 0.497479798520640i
  3.642801014184618 - 0.497479798520640i
  3.080265978640096 + 0.000000000000000i

>> dist
dist =
   0.007410813506185
   0.143445667057386
   0.165410367196465
   0.241928141830945
   0.302608466266508
   0.369173336804875
   0.467911109923134
   0.498627536953383
   0.498627536953383
   0.596347321359904

You can try to get similar results using dense eig:

% closest k eigenvalues to sigma
ev = eig(full(A));
[~,idx] = sort(ev - sigma);
ev = ev(idx(1:k))

% compare against eigs
norm(D - ev)

The difference is acceptably small (close to machine epsilon):

>> norm(ev-D)
ans =
     1.257079405021441e-14

Python

Similarly in Python:

import numpy as np
from scipy.sparse import spdiags
from scipy.sparse.linalg import eigs

# create banded matrix
n = 30
A = spdiags((np.ones((n,1))*[-1,2,-1]).T, [-1,0,1], n, n).todense()
A[0,8] = 30

# EIGS: k closest eigenvalues to sigma
k = 10
sigma = 3.6766133
D = eigs(A, k, sigma=sigma, which='LM', return_eigenvectors=False)
D = D[::-1]
for x in D:
    print '{:.16f}'.format(x)

# EIG
ev,_ = np.linalg.eig(A)
idx = np.argsort(np.abs(ev - sigma))
ev = ev[idx[:k]]
for x in ev:
    print '{:.16f}'.format(x)

with similar results:

# EIGS
3.6840241135061853+0.0000000000000000j
3.8200589670573866+0.0000000000000000j
3.5112029328035343+0.0000000000000000j
3.9185414418309441+0.0000000000000000j
3.9792217662665070+0.0000000000000000j
3.3074399631951246+0.0000000000000000j
4.1445244099231351+0.0000000000000000j
3.6428010141846170+0.4974797985206380j
3.6428010141846170-0.4974797985206380j
3.0802659786400950+0.0000000000000000j

# EIG
3.6840241135061880+0.0000000000000000j
3.8200589670573906+0.0000000000000000j
3.5112029328035339+0.0000000000000000j
3.9185414418309468+0.0000000000000000j
3.9792217662665008+0.0000000000000000j
3.3074399631951201+0.0000000000000000j
4.1445244099231271+0.0000000000000000j
3.6428010141846201+0.4974797985206384j
3.6428010141846201-0.4974797985206384j
3.0802659786400906+0.0000000000000000j

回答2:

The results are consistent between NumPy and Matlab if you use the eigs function:

>> format long
>> A = diag(-ones(n-1,1),-1) + diag(2*ones(n,1)) + diag(-ones(n-1,1),+1);A(1,9)=30;
>> eigs(A,3,3.6766133)'
ans =
   3.684024113506185   3.820058967057386   3.511202932803534

As for why the true closest eigenvalue isn't selected, I think that has to do with convergence to complex eigenvalues of real matrices and the choice of a real shift. I don't know how ARPACK calculates its iterates, but I remember being told that a real A with a real σ cannot by default converge to a complex conjugate pair since their ratio in absolute value is 1 (for Inverse Power Iteration). Since ARPACK will generate the complex eigenvalues at the 8^th and 9^th iteration (+/- ordering is random), I'm guessing their is some fix for this that I've forgotten about or never knew:

>> ev = eigs(A,9,3.6766133);ev(8:9)
ans =
  3.642801014184617 - 0.497479798520639i
  3.642801014184617 + 0.497479798520639i

I'm not sure if their is a general work around for this other than guessing a complex part for the shift or just grabbing extra eigenvalues until the conjugate pair falls into the ball of convergence for the ARPACK method.

来源：https://stackoverflow.com/questions/26520447/using-arpack-solving-eigenvalueproblem-but-getting-inconsistent-results-with-ma

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python

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eigenvalue

arpack