Functions of matrices and linear maps

function exponentiate(A, t::Number, x; kwargs...)
function exponentiate(A, t::Number, x, algorithm)

Compute $y = exp(t*A) x$, where A is a general linear map, i.e. a AbstractMatrix or just a general function or callable object and x is of any Julia type with vector like behavior.

Arguments:

The linear map A can be an AbstractMatrix (dense or sparse) or a general function or callable object that implements the action of the linear map on a vector. If A is an AbstractMatrix, x is expected to be an AbstractVector, otherwise x can be of any type that behaves as a vector and supports the required methods (see KrylovKit docs).

The time parameter t can be real or complex, and it is better to choose t e.g. imaginary and A hermitian, then to absorb the imaginary unit in an antihermitian A. For the former, the Lanczos scheme is used to built a Krylov subspace, in which an approximation to the exponential action of the linear map is obtained. The argument x can be of any type and should be in the domain of A.

Return values:

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

• y: the result of the computation, i.e. y = exp(t*A)*x
• info: an object of type [ConvergenceInfo], which has the following fields
  • info.converged::Int: 0 or 1 if the solution y was approximated up to the requested tolerance tol.
  • info.residual::Nothing: value nothing, there is no concept of a residual in this case
  • info.normres::Real: an estimate (upper bound) of the error between the approximate and exact solution
  • 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.