Eigenvalue Problems

EigenvalueProblem and EigenvalueTarget are SciMLBase types re-exported by LinearSolve.jl. Their docstrings are maintained in SciMLBase and rendered in the SciMLBase problem interface documentation; this page summarizes how LinearSolve.jl uses them. See the eigenvalue tutorial for worked examples and the eigenvalue solvers page for the available algorithms.

Mathematical Specification

The standard problem finds pairs $(\lambda, v)$ with

\[A v = \lambda v\]

and, when a second operator B is given, the generalized problem

\[A v = \lambda B v\]

Eigenvectors follow the type of u0 when supplied, otherwise the dense vector type matching a row of A. Eigenvalues have eltype(A) when real and Complex{eltype(A)} when a general real A yields conjugate pairs.

Constructor

EigenvalueProblem(A, B = nothing, p = NullParameters();
    num_eigenpairs = nothing, eigentarget = EigenvalueTarget.LargestMagnitude,
    shift = nothing, u0 = nothing, kwargs...)

• num_eigenpairs: how many eigenpairs to compute; nothing requests all of them from the dense solver or a solver-chosen default from the iterative backends.
• eigentarget: which part of the spectrum to return, an EigenvalueTarget (below).
• shift: when supplied, the eigenvalues nearest this value are returned (shift-and-invert).
• u0: an optional starting vector for the iterative backends.
• kwargs: extra keyword arguments passed on to the solver.

EigenvalueTarget

EigenvalueTarget is an EnumX enum selecting the part of the spectrum returned when only a subset of the eigenpairs is requested:

Not every backend supports every target directly; the eigenvalue solvers page lists each algorithm's restrictions.

Example

using LinearSolve

A = [2.0 1.0; 1.0 3.0]
prob = EigenvalueProblem(A; num_eigenpairs = 1,
    eigentarget = EigenvalueTarget.LargestMagnitude)
sol = solve(prob)
sol.values, sol.vectors