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.

Documentation	Build Status	Digital Object Idenitifier	License

Release notes for the latest version

v0.9

KrylovKit v0.9 adds two new sets of functionality:

• The function lssolve can be used to solve linear least squares problems, i.e. problems of the form x = argmin(norm(A*x - b))
  for a given linear map A and vector b. Currently, only one algorithm is implemented, namely the LSMR algorithm
  of Fong and Saunders.
• There are now new functions reallinsolve and realeigsolve (and reallssolve since v0.9.4), which are useful when
  using vectors with complex arithmetic, but where the linear map (implemented as a function f) acts as a real linear map,
  meaning that it only satisfies f(α*x) = α*f(x) when α is a real number. This occurs for example when computing the
  Jacobian of a complex function that is not holomorphic, e.g. in the context of automatic differentation. This is implemented
  by simply wrapping the vector as RealVec, which is a specific InnerProductVec type where the redefined inner product
  forgets about the imaginary part of the original inner function, thereby effectively treating the vector as living in a
  real vector space. Furthermore, in this setting, only real linear combinations of vectors are allowed, so that for the
  case of eigsolve, only real eigenvalues and eigenvectors are computed. An error will be thrown if the requested list
  of eigenvalues contains complex eigenvalues.

In addition, the following is technically a breaking change:

• The verbosity system, the different verbosity levels and the output formatting have been redesigned (both in the primal methods
  and the rrules). The default verbosity level is now 1, which means that warnings will be printed by default, but all other output
  (info messages) are suppressed. Before, the default verbosity was such that all output (including warnings) were suppressed.

Overview

KrylovKit.jl accepts general functions or callable objects as linear maps, and general Julia
objects with vector like behavior (as defined in the docs) as vectors.

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

• linsolve: solve linear systems
• lssolve: solve least squares problems
• eigsolve: find a few eigenvalues and corresponding eigenvectors
• geneigsolve: find a few generalized eigenvalues and corresponding vectors
• svdsolve: find a few singular values and corresponding left and right singular vectors
• exponentiate: apply the exponential of a linear map to a vector
• expintegrator: exponential integrator
  for a linear non-homogeneous ODE, computes a linear combination of the ϕⱼ functions which generalize ϕ₀(z) = exp(z).

Installation

KrylovKit.jl can be installed with the Julia package manager.
From the Julia REPL, type ] to enter the Pkg REPL mode and run:

pkg> add KrylovKit

Or, equivalently, via the Pkg API:

julia> import Pkg; Pkg.add("KrylovKit.jl")

Documentation

• STABLE - documentation of the most recently tagged version.
• DEVEL - documentation of the in-development version.

Project Status

The package is tested against Julia 1.6, the long-term stable release (1.10), the current stable release as well
as nightly builds of the Julia master branch on Linux, macOS, and Windows 64-bit architecture and with 1 and 4 threads.

Questions and Contributions

Contributions are very welcome, as are feature requests and suggestions. Please open an issue if you encounter any problems.