MATLAB®

eigs - Largest eigenvalues and eigenvectors of a matrix

Syntax

d = eigs(A)
[V,D] = eigs(A)
[V,D,flag] = eigs(A)
eigs(A,B)
eigs(A,k)
eigs(A,B,k)
eigs(A,k,sigma)
eigs(A,B,k,sigma)
eigs(A,K,sigma,opts)
eigs(A,B,k,sigma,opts)
eigs(Afun,n,...)

Description

d = eigs(A) returns a vector of A's six largest magnitude eigenvalues. A must be a square matrix. A should be large and sparse, though eigs will work on full matrices as well. See Remarks below.

[V,D] = eigs(A) returns a diagonal matrix D of A's six largest magnitude eigenvalues and a matrix V whose columns are the corresponding eigenvectors.

[V,D,flag] = eigs(A) also returns a convergence flag. If flag is 0 then all the eigenvalues converged; otherwise not all converged.

eigs(A,B) solves the generalized eigenvalue problem A*V == B*V*D. B must be symmetric (or Hermitian) positive definite and the same size as A. eigs(A,[],...) indicates the standard eigenvalue problem A*V == V*D.

eigs(A,k) and eigs(A,B,k) return the k largest magnitude eigenvalues.

eigs(A,k,sigma) and eigs(A,B,k,sigma) return k eigenvalues based on sigma, which can take any of the following values:

Note   The syntax eigs(A,k,...) is not valid when A is scalar. To pass a value for k, you must specify B as the second argument and k as the third (eigs(A,B,k,...)). If necessary, you can set B equal to [], the default.

eigs(A,K,sigma,opts) and eigs(A,B,k,sigma,opts) specify an options structure. Default values are shown in brackets ({}).

eigs(Afun,n,...) accepts the function handle Afun instead of the matrix A. See Function Handles in the MATLAB Programming documentation for more information. Afun must accept an input vector of size n.

y = Afun(x) should return:

Parametrizing Functions in the MATLAB Mathematics documentation, explains how to provide additional parameters to the function Afun, if necessary.

The matrix A, A-sigma*I or A-sigma*B represented by Afun is assumed to be real and nonsymmetric unless specified otherwise by opts.isreal and opts.issym. In all the eigs syntaxes, eigs(A,...) can be replaced by eigs(Afun,n,...).

Remarks

d = eigs(A,k) is not a substitute for

d = eig(full(A))
d = sort(d)
d = d(end-k+1:end)

but is most appropriate for large sparse matrices. If the problem fits into memory, it may be quicker to use eig(full(A)).

Algorithm

eigs provides the reverse communication required by the Fortran library ARPACK, namely the routines DSAUPD, DSEUPD, DNAUPD, DNEUPD, ZNAUPD, and ZNEUPD.

Examples

Example 1

A = delsq(numgrid('C',15));  
d1 = eigs(A,5,'sm')

returns

Iteration 1: a few Ritz values of the 20-by-20 matrix:
     0
     0
     0
     0
     0

Iteration 2: a few Ritz values of the 20-by-20 matrix:
    1.8117
    2.0889
    2.8827
    3.7374
    7.4954

Iteration 3: a few Ritz values of the 20-by-20 matrix:
    1.8117
    2.0889
    2.8827
    3.7374
    7.4954



d1 =

    0.5520
    0.4787
    0.3469
    0.2676
    0.1334

Example 2

This example replaces the matrix A in example 1 with a handle to a function dnRk. The example is contained in an M-file run_eigs that

• Calls eigs with the function handle @dnRk as its first argument.

• Contains dnRk as a nested function, so that all variables in run_eigs are available to dnRk.

The following shows the code for run_eigs:

function d2 = run_eigs
n = 139;  
opts.issym = 1;
R = 'C';
k = 15;
d2 = eigs(@dnRk,n,5,'sm',opts);
 
    function y = dnRk(x)
        y = (delsq(numgrid(R,k))) \ x;
    end
end

Example 3

west0479 is a real 479-by-479 sparse matrix with both real and pairs of complex conjugate eigenvalues. eig computes all 479 eigenvalues. eigs easily picks out the largest magnitude eigenvalues.

This plot shows the 8 largest magnitude eigenvalues of west0479 as computed by eig and eigs.

load west0479
d = eig(full(west0479))
dlm = eigs(west0479,8)
[dum,ind] = sort(abs(d));
plot(dlm,'k+') 
hold on 
plot(d(ind(end-7:end)),'ks') 
hold off 
legend('eigs(west0479,8)','eig(full(west0479))')

Example 4

A = delsq(numgrid('C',30)) is a symmetric positive definite matrix of size 632 with eigenvalues reasonably well-distributed in the interval (0 8), but with 18 eigenvalues repeated at 4. The eig function computes all 632 eigenvalues. It computes and plots the six largest and smallest magnitude eigenvalues of A successfully with:

A = delsq(numgrid('C',30)); 
d = eig(full(A)); 
[dum,ind] = sort(abs(d)); 
dlm = eigs(A); 
dsm = eigs(A,6,'sm');

subplot(2,1,1) 
plot(dlm,'k+') 
hold on 
plot(d(ind(end:-1:end-5)),'ks') 
hold off 
legend('eigs(A)','eig(full(A))',3) 
set(gca,'XLim',[0.5 6.5])

subplot(2,1,2) 
plot(dsm,'k+') 
hold on 
plot(d(ind(1:6)),'ks') 
hold off 
legend('eigs(A,6,''sm'')','eig(full(A))',2) 
set(gca,'XLim',[0.5 6.5])

However, the repeated eigenvalue at 4 must be handled more carefully. The call eigs(A,18,4.0) to compute 18 eigenvalues near 4.0 tries to find eigenvalues of A - 4.0*I. This involves divisions of the form 1/(lambda - 4.0), where lambda is an estimate of an eigenvalue of A. As lambda gets closer to 4.0, eigs fails. We must use sigma near but not equal to 4 to find those 18 eigenvalues.

sigma = 4 - 1e-6
[V,D] = eigs(A,18,sigma)

The plot shows the 20 eigenvalues closest to 4 that were computed by eig, along with the 18 eigenvalues closest to 4 - 1e-6 that were computed by eigs.

See Also

eig, svds, function_handle (@)

References

[1] Lehoucq, R.B. and D.C. Sorensen, "Deflation Techniques for an Implicitly Re-Started Arnoldi Iteration," SIAM J. Matrix Analysis and Applications, Vol. 17, 1996, pp. 789-821.

[2] Lehoucq, R.B., D.C. Sorensen, and C. Yang, ARPACK Users' Guide: Solution of Large-Scale Eigenvalue Problems with Implicitly Restarted Arnoldi Methods, SIAM Publications, Philadelphia, 1998.

[3] Sorensen, D.C., "Implicit Application of Polynomial Filters in a k-Step Arnoldi Method," SIAM J. Matrix Analysis and Applications, Vol. 13, 1992, pp.357-385.

eig

ellipj