eigs

Find a few eigenvalues and eigenvectors of a square large sparse matrix

Syntax

• d = eigs(A)
  d = eigs(A,B)
  d = eigs(A,k)
  d = eigs(A,B,k)
  d = eigs(A,k,sigma)
  d = eigs(A,B,k,sigma)
  d = eigs(A,k,sigma,options)
  d = eigs(A,B,k,sigma,options)
  d = eigs(Afun,n)
  d = eigs(Afun,n,B)
  d = eigs(Afun,n,k)
  d = eigs(Afun,n,B,k)
  d = eigs(Afun,n,k,sigma)
  d = eigs(Afun,n,B,k,sigma)
  d = eigs(Afun,n,k,sigma,options)
  d = eigs(Afun,n,B,k,sigma,options)
  d = eigs(Afun,n,k,sigma,options,p1,p2...)
  d = eigs(Afun,n,B,k,sigma,options,p1,p2...)
  [V,D] = eigs(A,...)
  [V,D] = eigs(Afun,n,...)
  [V,D,flag] = eigs(A,...)
  [V,D,flag] = eigs(Afun,n,...)
  

Description

d = eigs(A) returns a vector of A's six largest magnitude eigenvalues.

[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:

scalar (real or complex, including 0)	The eigenvalues closest to sigma. If A is a function, Afun must return Y = (A-sigma*B)\x (i.e., Y = A\x when sigma = 0). Note, B need only be symmetric (Hermitian) positive semi-definite.
'lm'	Largest magnitude (default).
'sm'	Smallest magnitude. Same as sigma = 0. If A is a function, Afun must return Y = A\x. Note, B need only be symmetric (Hermitian) positive semi-definite.
For real symmetric problems, the following are also options:
'la'	Largest algebraic ('lr' in MATLAB 5)
'sa'	Smallest algebraic ('sr' in MATLAB 5)
'be'	Both ends (one more from high end if k is odd)
For nonsymmetric and complex problems, the following are also options:
'lr'	Largest real part
'sr'	Smallest real part
'li'	Largest imaginary part
'si'	Smallest imaginary part

Note The MATLAB 5 value sigma = 'be' is obsolete for nonsymmetric and complex problems.

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

Parameter	Description	Values
options.issym	1 if A or A-sigma*B represented by Afun is symmetric, 0 otherwise.	[{0} | 1]
options.isreal	1 if A or A-sigma*B represented by Afun is real, 0 otherwise.	[0 | {1}]
options.tol	Convergence: Ritz estimate residual <= tol*norm(A).	[scalar | {eps}]
options.maxit	Maximum number of iterations.	[integer | {300}]
options.p	Number of basis vectors. p >= 2k (p >= 2k+1 real nonsymmetric) advised. Note: p must satisfy k <> for real symmetric, k+1 <> otherwise.	[integer | 2*k]
options.v0	Starting vector.	Randomly generated by ARPACK
options.disp	Diagnostic information display level.	[0 | {1} | 2]
options.cholB	1 if B is really its Cholesky factor chol(B), 0 otherwise.	[{0} | 1]
options.permB	Permutation vector permB if sparse B is really chol(B(permB,permB)).	[permB | {1:n}]

Note MATLAB 5 options stagtol and cheb are no longer allowed.

eigs(Afun,n,...) accepts the function Afun instead of the matrix A. y = Afun(x) should return:

A*x	if sigma is not specified, or is a string other than 'sm'
A\x	if sigma is 0 or 'sm'
(A-sigma*I)\x	if sigma is a nonzero scalar (standard eigenvalue problem). I is an identity matrix of the same size as A.
(A-sigma*B)\x	if sigma is a nonzero scalar (generalized eigenvalue problem)

n is the size of A. 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,...).

eigs(Afun,n,k,sigma,opts,p1,p2,...) and eigs(Afun,n,B,k,sigma,opts,p1,p2,...) provide for additional arguments which are passed to Afun(x,p1,p2,...).

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: This example shows the use of function handles.

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

Equivalently, if dnRk is the following one-line function:

• function y = dnRk(x,R,k)
  y = (delsq(numgrid(R,k))) \ x;
  

then pass dnRk's additional arguments, 'C' and 15, to eigs.

• n = size(A,1); 
  opts.issym = 1; 
  d2 = eigs(@dnRk,n,5,'sm',opts,'C',15);
  

Example 2: 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 3: 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.