# Krylov.jl documentation

This package provides implementations of certain of the most useful Krylov method for a variety of problems:

1 - Square or rectangular full-rank systems

$$$Ax = b$$$

should be solved when b lies in the range space of A. This situation occurs when

• A is square and nonsingular,
• A is tall and has full column rank and b lies in the range of A.

2 - Linear least-squares problems

$$$\min \|b - Ax\|$$$

should be solved when b is not in the range of A (inconsistent systems), regardless of the shape and rank of A. This situation mainly occurs when

• A is square and singular,
• A is tall and thin.

Underdetermined sytems are less common but also occur.

If there are infinitely many such x (because A is column rank-deficient), one with minimum norm is identified

$$$\min \|x\| \quad \text{subject to} \quad x \in \argmin \|b - Ax\|.$$$

3 - Linear least-norm problems

$$$\min \|x\| \quad \text{subject to} \quad Ax = b$$$

sould be solved when A is column rank-deficient but b is in the range of A (consistent systems), regardless of the shape of A. This situation mainly occurs when

• A is square and singular,
• A is short and wide.

Overdetermined sytems are less common but also occur.

4 - Adjoint systems

$$$Ax = b \quad \text{and} \quad A^H y = c$$$

where A can have any shape.

5 - Saddle-point and symmetric quasi-definite (SQD) systems

$$$\begin{bmatrix} M & \phantom{-}A \\ A^H & -N \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \left(\begin{bmatrix} b \\ 0 \end{bmatrix},\begin{bmatrix} 0 \\ c \end{bmatrix},\begin{bmatrix} b \\ c \end{bmatrix}\right)$$$

where A can have any shape.

6 - Generalized saddle-point and unsymmetric partitioned systems

$$$\begin{bmatrix} M & A \\ B & N \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} b \\ c \end{bmatrix}$$$

where A can have any shape and B has the shape of Aᴴ. A, B, b and c must be all nonzero.

Krylov solvers are particularly appropriate in situations where such problems must be solved but a factorization is not possible, either because:

• A is not available explicitly,
• A would be dense or would consume an excessive amount of memory if it were materialized,
• factors would consume an excessive amount of memory.

Iterative methods are recommended in either of the following situations:

• the problem is sufficiently large that a factorization is not feasible or would be slow,
• 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 better complexity than if it were materialized as a matrix. Certain fast operators would materialize as dense matrices.

## Features

All solvers in Krylov.jl have in-place version, are compatible with GPU and work in any floating-point data type.

## How to Install

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

julia> ]
pkg> test Krylov