Generalized Eigen Decomposition

For the two-input generalized eigensolution version,

eigs(A, B; nev=6, ncv=max(20,2*nev+1), which=:LM, tol=0.0, maxiter=300, sigma=nothing, ritzvec=true, v0=zeros((0,))) -> (d,[v,],nconv,niter,nmult,resid)

the following keyword arguments are supported:

  • nev: Number of eigenvalues
  • ncv: Number of Krylov vectors used in the computation; should satisfy nev+1 <= ncv <= n for real symmetric problems and nev+2 <= ncv <= n for other problems, where n is the size of the input matrices A and B. The default is ncv = max(20,2*nev+1). Note that these restrictions limit the input matrix A to be of dimension at least 2.
  • which: type of eigenvalues to compute. See the note below.
whichtype of eigenvalues
:LMeigenvalues of largest magnitude (default)
:SMeigenvalues of smallest magnitude
:LReigenvalues of largest real part
:SReigenvalues of smallest real part
:LIeigenvalues of largest imaginary part (nonsymmetric or complex A only)
:SIeigenvalues of smallest imaginary part (nonsymmetric or complex A only)
:BEcompute half of the eigenvalues from each end of the spectrum, biased in favor of the high end. (real symmetric A only)
  • tol: relative tolerance used in the convergence criterion for eigenvalues, similar to tol in the eigs(A) method for the ordinary eigenvalue problem, but effectively for the eigenvalues of $B^{-1} A$ instead of $A$. See the documentation for the ordinary eigenvalue problem in eigs(A) and the accompanying note about tol.
  • maxiter: Maximum number of iterations (default = 300)
  • sigma: Specifies the level shift used in inverse iteration. If nothing (default), defaults to ordinary (forward) iterations. Otherwise, find eigenvalues close to sigma using shift and invert iterations.
  • ritzvec: Returns the Ritz vectors v (eigenvectors) if true
  • v0: starting vector from which to start the iterations

eigs returns the nev requested eigenvalues in d, the corresponding Ritz vectors v (only if ritzvec=true), the number of converged eigenvalues nconv, the number of iterations niter and the number of matrix vector multiplications nmult, as well as the final residual vector resid.

We can see the various keywords in action in the following examples:

julia> A = sparse(1.0I, 4, 4); B = Diagonal(1:4);

julia> λ, ϕ = eigs(A, B, nev = 2);

julia> λ
2-element Array{Float64,1}:

julia> A = Diagonal([1, -2im, 3, 4im]); B = sparse(1.0I, 4, 4);

julia> λ, ϕ = eigs(A, B, nev=1, which=:SI);

julia> λ
1-element Array{Complex{Float64},1}:
 -1.5720931501039814e-16 - 1.9999999999999984im

julia> λ, ϕ = eigs(A, B, nev=1, which=:LI);

julia> λ
1-element Array{Complex{Float64},1}:
 0.0 + 4.000000000000002im

The sigma and which keywords interact: the description of eigenvalues searched for by which do not necessarily refer to the eigenvalue problem $Av = Bv\lambda$, but rather the linear operator constructed by the specification of the iteration mode implied by sigma.

sigmaiteration modewhich refers to the problem
nothingordinary (forward)$Av = Bv\lambda$
real or complexinverse with level shift sigma$(A - \sigma B )^{-1}B = v\nu$