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Internals

Data structures

Pseudospectra.Portrait — Type
Portrait

structure representing a spectral portrait; includes a mesh for z=x+iy, values of the resolvent norm on the mesh, suitable contour levels, and some other metadata.

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Arnoldi iteration

Pseudospectra.xeigs — Function
xeigs(A, B, channel=nothing; nev=6, ncv=max(20,2*nev+1), which=:LM,
      tol=0.0, maxiter=300, sigma=nothing, ritzvec=true, v0=zeros((0,)),
      wantH=true, options=nothing)

Compute (generalized) eigenpairs of A x=λ B x and projections.

Modified version of eigs() (from the Arpack package, q.v.) which optionally

  1. provides the projection matrices,
  2. provides intermediate states (typically for plotting).

For intermediates, caller must provide a Channel argument; in this case xeigs is implemented as a producer which fills channel. When finished, it puts (:finale, d,[v,],nconv,niter,nmult,resid[,H,V]) to channel.

The type of A must provide methods for mul!, issymmetric, eltype and size. Some variants are not implemented for the case where A is not a factorable matrix.

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Pseudospectra.ArpackOptions — Type
ArpackOptions{T}(; nev=6, ncv=0, which=:LM,
                                       tol=zero(T),maxiter=300,
                                       v0=Vector{T}(0), have_v0=false,
                                       sigma=nothing)

constructs an object to manage the Arnoldi scheme; see the documentation for eigs in the Arpack package for the meaning of fields.

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Algorithm selection

IRAM means implicitly restarted Arnoldi method, using the ARPACK implementation. "Large" and "small" are determined by constants which might be made configurable someday.

Dense square matrix

  • If large and not opts[:direct], project via IRAM and go to rectangular branch.
  • If small, use SVD.
  • Otherwise, use inverse Lanczos.
    • If a Schur decomposition is available, use that to simplify the solvers.

Sparse square matrix

  • If small, and not opts[:keep_sparse], convert to dense then use above logic.
  • If opts[:direct], use inverse Lanczos. This involves a sparse factorization for every grid point, so is slow.
  • Otherwise project via IRAM and go to rectangular branch.

Dense rectangular matrix

  • If not Hessenberg, factor first.
    • If very tall, factor by QR, otherwise QZ (generalized Schur).
  • Use QR for resolvents, then inverse Lanczos.
    • Use a simplified QR for large Hessenberg.