Finding eigenvalues and eigenvectors

eigsolve(A::AbstractMatrix, [howmany = 1, which = :LM, T = eltype(A)]; kwargs...)
eigsolve(f, n::Int, [howmany = 1, which = :LM, T = Float64]; kwargs...)
eigsolve(f, x₀, [howmany = 1, which = :LM]; kwargs...)
eigsolve(f, x₀, howmany, which, algorithm)

Compute howmany eigenvalues from the linear map encoded in the matrix A or by the function f. Return eigenvalues, eigenvectors and a ConvergenceInfo structure.

Arguments:

The linear map can be an AbstractMatrix (dense or sparse) or a general function or callable object. If an AbstractMatrix is used, a starting vector x₀ does not need to be provided, it is then chosen as rand(T, size(A,1)). If the linear map is encoded more generally as a a callable function or method, the best approach is to provide an explicit starting guess x₀. Note that x₀ does not need to be of type AbstractVector, any type that behaves as a vector and supports the required methods (see KrylovKit docs) is accepted. If instead of x₀ an integer n is specified, it is assumed that x₀ is a regular vector and it is initialized to rand(T,n), where the default value of T is Float64, unless specified differently.

The next arguments are optional, but should typically be specified. howmany specifies how many eigenvalues should be computed; which specifies which eigenvalues should be targetted. Valid specifications of which are given by

• LM: eigenvalues of largest magnitude
• LR: eigenvalues with largest (most positive) real part
• SR: eigenvalues with smallest (most negative) real part
• LI: eigenvalues with largest (most positive) imaginary part, only if T <: Complex
• SI: eigenvalues with smallest (most negative) imaginary part, only if T <: Complex
• ClosestTo(λ): eigenvalues closest to some number λ

The argument T acts as a hint in which Number type the computation should be performed, but is not restrictive. If the linear map automatically produces complex values, complex arithmetic will be used even though T<:Real was specified.

Return values:

The return value is always of the form vals, vecs, info = eigsolve(...) with

• vals: a Vector containing the eigenvalues, of length at least howmany, but could be longer if more eigenvalues were converged at the same cost. Eigenvalues will be real if Lanczos was used and complex if Arnoldi was used (see below).
• vecs: a Vector of corresponding eigenvectors, of the same length as vals. Note that eigenvectors are not returned as a matrix, as the linear map could act on any custom Julia type with vector like behavior, i.e. the elements of the list vecs are objects that are typically similar to the starting guess x₀, up to a possibly different eltype. In particular, for a general matrix (i.e. with Arnoldi) the eigenvectors are generally complex and are therefore always returned in a complex number format. When the linear map is a simple AbstractMatrix, vecs will be Vector{Vector{<:Number}}.
• info: an object of type [ConvergenceInfo], which has the following fields
  • info.converged::Int: indicates how many eigenvalues and eigenvectors were actually converged to the specified tolerance tol (see below under keyword arguments)
  • info.residual::Vector: a list of the same length as vals containing the residuals info.residual[i] = f(vecs[i]) - vals[i] * vecs[i]
  • info.normres::Vector{<:Real}: list of the same length as vals containing the norm of the residual info.normres[i] = norm(info.residual[i])
  • info.numops::Int: number of times the linear map was applied, i.e. number of times f was called, or a vector was multiplied with A
  • info.numiter::Int: number of times the Krylov subspace was restarted (see below)

Keyword arguments:

Keyword arguments and their default values are given by:

• krylovdim = 30: the maximum dimension of the Krylov subspace that will be constructed. Note that the dimension of the vector space is not known or checked, e.g. x₀ should not necessarily support the Base.length function. If you know the actual problem dimension is smaller than the default value, it is useful to reduce the value of krylovdim, though in principle this should be detected.
• tol = 1e-12: the requested accuracy (corresponding to the 2-norm of the residual for Schur vectors, not the eigenvectors). If you work in e.g. single precision (Float32), you should definitely change the default value.
• maxiter = 100: the number of times the Krylov subspace can be rebuilt; see below for further details on the algorithms.
• issymmetric: if the linear map is symmetric, only meaningful if T<:Real
• ishermitian: if the linear map is hermitian

The default value for the last two depends on the method. If an AbstractMatrix is used, issymmetric and ishermitian are checked for that matrix, ortherwise the default values are issymmetric = false and ishermitian = T <: Real && issymmetric.

Algorithm

The last method, without default values and keyword arguments, is the one that is finally called, and can also be used directly. Here, one specifies the algorithm explicitly as either Lanczos, for real symmetric or complex hermitian problems, or Arnoldi, for general problems. Note that these names refer to the process for building the Krylov subspace, but the actual algorithm is an implementation of the Krylov-Schur algorithm, which can dynamically shrink and grow the Krylov subspace, i.e. the restarts are so-called thick restarts where a part of the current Krylov subspace is kept.

schursolve(A, x₀, howmany, which, algorithm)

Compute a partial Schur decomposition containing howmany eigenvalues from the linear map encoded in the matrix or function A. Return the reduced Schur matrix, the basis of Schur vectors, the extracted eigenvalues and a ConvergenceInfo structure.

See also eigsolve to obtain the eigenvectors instead. For real symmetric or complex hermitian problems, the (partial) Schur decomposition is identical to the (partial) eigenvalue decomposition, and eigsolve should always be used.

Arguments:

The linear map can be an AbstractMatrix (dense or sparse) or a general function or callable object, that acts on vector like objects similar to x₀, which is the starting guess from which a Krylov subspace will be built. howmany specifies how many Schur vectors should be converged before the algorithm terminates; which specifies which eigenvalues should be targetted. Valid specifications of which are

• LM: eigenvalues of largest magnitude
• LR: eigenvalues with largest (most positive) real part
• SR: eigenvalues with smallest (most negative) real part
• LI: eigenvalues with largest (most positive) imaginary part, only if T <: Complex
• SI: eigenvalues with smallest (most negative) imaginary part, only if T <: Complex
• ClosestTo(λ): eigenvalues closest to some number λ

The final argument algorithm can currently only be an instance of Arnoldi, but should nevertheless be specified. Since schursolve is less commonly used as eigsolve, no convenient keyword syntax is currently available.

Return values:

The return value is always of the form T, vecs, vals, info = eigsolve(...) with

• T: a Matrix containing the partial Schur decomposition of the linear map, i.e. it's elements are given by T[i,j] = dot(vecs[i], f(vecs[j])). It is of Schur form, i.e. upper triangular in case of complex arithmetic, and block upper triangular (with at most 2x2 blocks) in case of real arithmetic.
• vecs: a Vector of corresponding Schur vectors, of the same length as vals. Note that Schur vectors are not returned as a matrix, as the linear map could act on any custom Julia type with vector like behavior, i.e. the elements of the list vecs are objects that are typically similar to the starting guess x₀, up to a possibly different eltype. When the linear map is a simple AbstractMatrix, vecs will be Vector{Vector{<:Number}}. Schur vectors are by definition orthogonal, i.e. dot(vecs[i],vecs[j]) = I[i,j]. Note that Schur vectors are real if the problem (i.e. the linear map and the initial guess) are real.
• vals: a Vector of eigenvalues, i.e. the diagonal elements of T in case of complex arithmetic, or extracted from the diagonal blocks in case of real arithmetic. Note that vals will always be complex, independent of the underlying arithmetic.
• info: an object of type [ConvergenceInfo], which has the following fields
  • info.converged::Int: indicates how many eigenvalues and Schur vectors were actually converged to the specified tolerance (see below under keyword arguments)
  • info.residuals::Vector: a list of the same length as vals containing the actual residuals julia info.residuals[i] = f(vecs[i]) - sum(vecs[j]*T[j,i] for j = 1:i+1) where T[i+1,i] is definitely zero in case of complex arithmetic and possibly zero in case of real arithmetic
  • info.normres::Vector{<:Real}: list of the same length as vals containing the norm of the residual for every Schur vector, i.e. info.normes[i] = norm(info.residual[i])
  • info.numops::Int: number of times the linear map was applied, i.e. number of times f was called, or a vector was multiplied with A
  • info.numiter::Int: number of times the Krylov subspace was restarted (see below)

Algorithm

The actual algorithm is an implementation of the Krylov-Schur algorithm, where the Arnoldi algorithm is used to generate the Krylov subspace. During the algorith, the Krylov subspace is dynamically grown and shrunk, i.e. the restarts are so-called thick restarts where a part of the current Krylov subspace is kept.