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KrylovKit.jl

A Julia package collecting a number of Krylov-based algorithms for linear problems, singular value and eigenvalue problems and the application of functions of linear maps or operators to vectors.

Overview

KrylovKit.jl accepts general functions or callable objects as linear maps, and general Julia objects with vector like behavior (see below) as vectors.

The high level interface of KrylovKit is provided by the following functions:

Package features and alternatives

This section could also be titled "Why did I create KrylovKit.jl"?

There are already a fair number of packages with Krylov-based or other iterative methods, such as

KrylovKit.jl distinguishes itself from the previous packages in the following ways

  1. KrylovKit accepts general functions to represent the linear map or operator that defines the problem, without having to wrap them in a LinearMap or LinearOperator type. Of course, subtypes of AbstractMatrix are also supported. If the linear map (always the first argument) is a subtype of AbstractMatrix, matrix vector multiplication is used, otherwise is applied as a function call.

  2. KrylovKit does not assume that the vectors involved in the problem are actual subtypes of AbstractVector. Any Julia object that behaves as a vector is supported, so in particular higher-dimensional arrays or any custom user type that supports the following functions (with v and w two instances of this type and α a scalar (Number)):

    • Base.eltype(v): the scalar type (i.e. <:Number) of the data in v
    • Base.similar(v, [T::Type<:Number]): a way to construct additional similar vectors, possibly with a different scalar type T.
    • Base.copyto!(w, v): copy the contents of v to a preallocated vector w
    • Base.fill!(w, α): fill all the scalar entries of w with value α; this is only used in combination with α = 0 to create a zero vector. Note that Base.zero(v) does not work for this purpose if we want to change the scalar eltype. We can also not use rmul!(v, 0) (see below), since NaN*0 yields NaN.
    • LinearAlgebra.mul!(w, v, α): out of place scalar multiplication; multiply vector v with scalar α and store the result in w
    • LinearAlgebra.rmul!(v, α): in-place scalar multiplication of v with α.
    • LinearAlgebra.axpy!(α, v, w): store in w the result of α*v + w
    • LinearAlgebra.axpby!(α, v, β, w): store in w the result of α*v + β*w
    • LinearAlgebra.dot(v,w): compute the inner product of two vectors
    • LinearAlgebra.norm(v): compute the 2-norm of a vector

    Furthermore, KrylovKit provides two types satisfying the above requirements that might facilitate certain applications:

    • RecursiveVec can be used for grouping a set of vectors into a single vector like

    structure (can be used recursively). The reason that e.g. Vector{<:Vector} cannot be used for this is that it returns the wrong eltype and methods like similar(v, T) and fill!(v, α) don't work correctly.

    • InnerProductVec can be used to redefine the inner product (i.e. dot) and corresponding

    norm (norm) of an already existing vector like object. The latter should help with implementing certain type of preconditioners and solving generalized eigenvalue problems with a positive definite matrix in the right hand side.

  3. To the best of my knowledge, KrylovKit.jl is the only package that provides a native Julia implementation of a Krylov method for eigenvalues of general matrices (in particular the Krylov-Schur algorithm). As such, is the only pure Julia alternative to Arpack.jl.

Current functionality

The following algorithms are currently implemented

Future functionality?

Here follows a wish list / to-do list for the future. Any help is welcomed and appreciated.