A Julia Basket of Hand-Picked Krylov Methods


Krylov.jl: A Julia basket of hand-picked Krylov methods

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This package implements iterative methods for the solution of linear systems of equations

Ax = b,

and linear least-squares problems

minimize ‖b - Ax‖.

It is appropriate, in particular, in situations where such a problem must be solved but a factorization is not possible, either because:

  • the operator is not available explicitly,
  • the operator is dense, or
  • factors would consume an excessive amount of memory and/or disk space.

Iterative methods are particularly appropriate in either of the following situations:

  • the problem is sufficiently large that a factorization is not feasible or would be slower,
  • an effective preconditioner is known in cases where the problem has unfavorable spectral structure,
  • the operator can be represented efficiently as a sparse matrix,
  • the operator is fast, i.e., can be applied with far better complexity than if it were materialized as a matrix. Often, fast operators would materialize as dense matrices.

Objective: solve Ax ≈ b

Given a linear operator A and a right-hand side b, solve Ax ≈ b, which means:

  1. when A has full column rank and b lies in the range space of A, find the unique x such that Ax = b; this situation occurs when
    • A is square and nonsingular, or
    • A is tall and has full column rank and b lies in the range of A,
  2. when A is column-rank deficient but b is in the range of A, find x with minimum norm such that Ax = b; this situation occurs when b is in the range of A and
    • A is square but singular, or
    • A is short and wide,
  3. when b is not in the range of A, regardless of the shape and rank of A, find x that minimizes the residual ‖b - Ax‖. If there are infinitely many such x (because A is rank deficient), identify the one with minimum norm.

How to Install

Krylov can be installed and tested through the Julia package manager:

julia> Pkg.add("Krylov")
julia> Pkg.test("Krylov")

Long-Term Goals

  • provide implementations of certain of the most useful Krylov method for linear systems with special emphasis on methods for linear least-squares problems and saddle-point linear system (including symmetric quasi-definite systems)
  • provide state-of-the-art implementations alongside simple implementations of equivalent methods in exact artithmetic (e.g., LSQR vs. CGLS, MINRES vs. CR, LSMR vs. CRLS, etc.)
  • provide simple, consistent calling signatures and avoid over-typing
  • ensure those implementations are fast and stable.

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