Krylov.jl: A Julia basket of hand-picked Krylov methods
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:
- 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,
- 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,
- 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> import Pkg
- 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
- 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.