MATLAB Functions

Find a few eigenvalues and eigenvectors

[V,D] = eigs(A)
[V,D] = eigs('Afun',n)
[V,D] = eigs(A,B,k,sigma,options)
[V,D] = eigs('Afun',n,B,k,sigma,options)

Description

eigs solves the eigenvalue problem A*v = lambda*v or the generalized eigenvalue problem A*v = lambda*B*v. Only a few selected eigenvalues, or eigenvalues and eigenvectors, are computed.

[V,D] = eigs(A) or [V,D] = eigs('Afun',n)

solves the eigenvalue problem where the first input argument is either a square matrix (which can be full or sparse, symmetric or nonsymmetric, real or complex), or a string containing the name of an M-file which applies a linear operator to the columns of a given matrix. In the latter case, the second input argument must be n, the order of the problem. For example, eigs('fft', ...) is much faster than
eigs(F, ...) where F is the explicit FFT matrix.

With one output argument, D is a vector containing k eigenvalues.With two output arguments, D is a k-by-k diagonal matrix and V is a matrix with k columns so that A*V = V*D or A*V = B*V*D.

The remaining input arguments are optional and can be given in practically any order:


`Argument`	`Value`
`B`	A matrix the same size as `A`. If `B` is not specified, `B = eye(size(A))` is used.
`k`	An integer, the number of eigenvalues desired. If `k` is not specified, `k = 6` eigenvalues are computed.
`sigma`	A scalar shift or a two letter string. If `sigma` is not specified, the `k`-th eigenvalues largest in magnitude are computed. If `sigma` is `0`, the `k`-th eigenvalues smallest in magnitude are computed. If `sigma` is a real or complex scalar, the shift, the `k`-th eigenvalues nearest `sigma`, are computed. If `sigma` is one of the following strings, it specifies the desired eigenvalues: `'lm'` Largest Magnitude (the default) `'sm'` Smallest Magnitude (same as sigma = 0) `'lr'` Largest Real part `'sr'` Smallest Real part `'be'` Both Ends. Computes `k/2` eigenvalues from each end of the spectrum (one more from the high end if `k` is odd.)

The options structure specifies certain parameters in the algorithm.


`Parameter`	`Description`	`Value`
`options.tol`	Convergence tolerance `norm(AV-VD) <= tol`	`1e-10` (symmetric) `1e-6` (nonsymmetric)
`options.p`	Dimension of the Arnoldi basis	`2*k`
`options.maxit`	Maximum number of iterations	`300`
`options.disp`	Number of eigenvalues displayed at each iteration. Set to `0` for no intermediate output.	`20`
`options.issym`	Positive if `Afun` is symmetric	`0`
`options.cheb`	Positive if `A` is a string, `sigma` is `'lr'`,`'sr'`, or a shift, and polynomial acceleration should be applied.	`0`
`options.v0`	Starting vector for the Arnoldi factorization	`rand(n,1)-.5`