# LinearMaps.jl

A Julia package for defining and working with linear maps, also known as linear transformations or linear operators acting on vectors. The only requirement for a LinearMap is that it can act on a vector (by multiplication) efficiently.

## Installation

LinearMaps.jl is a registered package and can be installed via

pkg> add LinearMaps

in package mode, which can be entered by typing ] in the Julia REPL.

## Examples

Let

A = LinearMap(rand(10, 10))
B = LinearMap(cumsum, reverse∘cumsum∘reverse, 10)

be a matrix- and function-based linear map, respectively. Then the following code just works, indistinguishably from the case when A and B are both AbstractMatrix-typed objects.

3.0A + 2B
A + I
A*B'
[A B; B A]
kron(A, B)

The LinearMap type and corresponding methods combine well with the following packages:

using LinearMaps
import Arpack, IterativeSolvers, KrylovKit, TSVD, ArnoldiMethod

# Example 1, 1-dimensional Laplacian with periodic boundary conditions
function leftdiff!(y::AbstractVector, x::AbstractVector) # left difference assuming periodic boundary conditions
N = length(x)
axes(y) == axes(x) || throw(DimensionMismatch())
@inbounds for i in eachindex(x, y)
y[i] = x[i] - x[mod1(i-1, N)]
end
return y
end

function mrightdiff!(y::AbstractVector, x::AbstractVector) # minus right difference
N = length(x)
axes(y) == axes(x) || throw(DimensionMismatch())
@inbounds for i in eachindex(x, y)
y[i] = x[i] - x[mod1(i+1, N)]
end
return y
end

D = LinearMap(leftdiff!, mrightdiff!, 100; ismutating=true) # by default has eltype(D) = Float64

Arpack.eigs(D'D; nev=3, which=:SR) # note that D'D is recognized as symmetric => real eigenfact
Arpack.svds(D; nsv=3)

ArnoldiMethod.partialschur(D'D; nev=3, which=ArnoldiMethod.SR())

KrylovKit.eigsolve(D'D, 100, 3, :SR)

Σ, L = IterativeSolvers.svdl(D; nsv=3)

TSVD.tsvd(D, 3)

# Example 2, 3 smallest eigenvalues of 1-dimensional Laplacian
A = LinearMap(100; issymmetric=true, ismutating=true) do C, B
C = -2B + B
for i in 2:length(B)-1
C[i] = B[i-1] - 2B[i] + B[i+1]
end
C[end] = B[end-1] - 2B[end]
return C
end

Arpack.eigs(-A; nev=3, which=:SR)

ArnoldiMethod.partialschur(-A; nev=3, which=ArnoldiMethod.SR())

KrylovKit.eigsolve(-A, size(A, 1), 3, :SR)

# Example 3, 2-dimensional Laplacian
Δ = kronsum(A, A)

Arpack.eigs(Δ; nev=3, which=:LR)
ArnoldiMethod.partialeigen(ArnoldiMethod.partialschur(Δ; nev=3, which=ArnoldiMethod.LR()))
KrylovKit.eigsolve(Δ, size(Δ, 1), 3, :LR)

In the last line above we leverage the fact that objects of type L <: LinearMap are callable.

### Inverse map with conjugate gradient

The InverseMap type can be used to lazily represent the inverse of an operator. When this map acts on a vector the linear system is solved. This can be used to solve a system of the form $Sx = (C A^{-1} B) x = b$ without explicitly computing $A^{-1}$ (see for example solving a linear system using the Schur complement).

using LinearMaps, IterativeSolvers

A = [2.0 1.5 0.0
1.5 3.0 0.0
0.0 0.0 4.0]
B = [2.0 0.0
0.0 1.0
0.0 0.0]
C = B'
b = [2.0, 3.0]

# Use IterativeSolvers.cg! to solve the system with 0 as the initial guess
linsolve = (x, A, b) -> IterativeSolvers.cg!(fill!(x, 0), A, b)

# Construct the linear map S
S = C * InverseMap(A; solver=linsolve) * B

# Solve the system
IterativeSolvers.cg(S, b)

In every CG iteration the linear map S will act on a vector v. Since S is a composed linear map, S * v is roughly equivalent to

# Apply first linear map B to v
tmp1 = B * v
# Apply second linear map: solve linear system with vector tmp1 as RHS
tmp2 = A \ tmp1
# Apply third linear map C to tmp2
result = C * tmp2

i.e. inside the CG solver for solving Sx = b we use CG to solve another inner linear system.

## Philosophy

Several iterative linear algebra methods such as linear solvers or eigensolvers only require an efficient evaluation of the matrix-vector product, where the concept of a matrix can be formalized / generalized to a linear map (or linear operator in the special case of a square matrix).

The LinearMaps package provides the following functionality:

1. A LinearMap type that shares with the AbstractMatrix type that it responds to the functions size, eltype, isreal, issymmetric, ishermitian and isposdef, transpose and adjoint and multiplication with a vector using both * or the in-place version mul!. Linear algebra functions that use duck-typing for their arguments can handle LinearMap objects similar to AbstractMatrix objects, provided that they can be written using the above methods. Unlike AbstractMatrix types, LinearMap objects cannot be indexed, neither using getindex or setindex!.

2. A single function LinearMap that acts as a general purpose constructor (though it is only an abstract type) and allows to construct linear map objects from functions, or to wrap objects of type AbstractMatrix or LinearMap. The latter functionality is useful to (re)define the properties (isreal, issymmetric, ishermitian, isposdef) of the existing matrix or linear map.

3. A framework for combining objects of type LinearMap and of type AbstractMatrix using linear combinations, transposition, composition, concatenation and Kronecker product/sums, where the linear map resulting from these operations is never explicitly evaluated but only its matrix-vector product is defined (i.e. lazy evaluation). The matrix-vector product is written to minimize memory allocation by using a minimal number of temporary vectors. There is full support for the in-place version mul!, which should be preferred for higher efficiency in critical algorithms. In addition, it tries to recognize the properties of combinations of linear maps. In particular, compositions such as A'*A for arbitrary A or even A'*B*C*B'*A with arbitrary A and B and positive definite C are recognized as being positive definite and hermitian. In case a certain property of the resulting LinearMap object is not correctly inferred, the LinearMap method can be called to redefine the properties.