Julia's Extended Linear Algebra Interface

The following ArrayInterface functions extend Julia's Base LinearAlgebra interface to improve the ability to do generic linear algebra.

Generic Matrix Constructors

These functions allow for the construction of matrices from array information in a generic way. It handles cases like how if x is a vector on the GPU, its associated matrix type should also be GPU-based, and thus appropriately placed with zeros/undef values.

ArrayInterface.zeromatrix — Function

zeromatrix(u::AbstractVector)

Creates the zero'd matrix version of u. Note that this is unique because similar(u,length(u),length(u)) returns a mutable type, so it is not type-matching, while fill(zero(eltype(u)),length(u),length(u)) doesn't match the array type, i.e., you'll get a CPU array from a GPU array. The generic fallback is u .* u' .* false, which works on a surprising number of types, but can be broken with weird (recursive) broadcast overloads. For higher-order tensors, this returns the matrix linear operator type which acts on the vec of the array.

ArrayInterface.undefmatrix — Function

undefmatrix(u::AbstractVector)

Creates the matrix version of u with possibly undefined values. Note that this is unique because similar(u,length(u),length(u)) returns a mutable type, so it is not type-matching, while fill(zero(eltype(u)),length(u),length(u)) doesn't match the array type, i.e., you'll get a CPU array from a GPU array. The generic fallback is u .* u', which works on a surprising number of types, but can be broken with weird (recursive) broadcast overloads. For higher-order tensors, this returns the matrix linear operator type which acts on the vec of the array.

Generic Matrix Functions

These query allow for easily learning properties of a general matrix.

ArrayInterface.issingular — Function

issingular(A::AbstractMatrix) -> Bool

Determine whether a given abstract matrix is singular.

Factorization Instance Functions

These functions allow for generating the instance of a factorization's result without running the full factorization. This thus allows for building types to hold the factorization without having to perform expensive extra computations.

ArrayInterface.bunchkaufman_instance — Function

bunchkaufman_instance(A, pivot = LinearAlgebra.RowMaximum()) -> bunchkaufman_factorization_instance

Returns an instance of the Bunch-Kaufman factorization object with the correct type cheaply.

bunchkaufman_instance(a::Number) -> a

Returns the number.

bunchkaufman_instance(a::Any) -> cholesky(a, check=false)

Returns the number.

ArrayInterface.cholesky_instance — Function

cholesky_instance(A, pivot = LinearAlgebra.NoPivot()) -> cholesky_factorization_instance

Returns an instance of the Cholesky factorization object with the correct type cheaply.

cholesky_instance(a::Number, pivot = LinearAlgebra.NoPivot()) -> a

Returns the number.

cholesky_instance(a::Any, pivot = LinearAlgebra.NoPivot()) -> cholesky(a, check=false)

Slow fallback which gets the instance via factorization. Should get specialized for new matrix types.

ArrayInterface.ldlt_instance — Function

ldlt_instance(A) -> ldlt_factorization_instance

Returns an instance of the LDLT factorization object with the correct type cheaply.

ldlt_instance(a::Number) -> a

Returns the number.

ldlt_instance(a::Any) -> ldlt(a, check=false)

Slow fallback which gets the instance via factorization. Should get specialized for new matrix types.

ArrayInterface.lu_instance — Function

lu_instance(A) -> lu_factorization_instance

Returns an instance of the LU factorization object with the correct type cheaply.

lu_instance(a::Number) -> a

Returns the number.

lu_instance(a::Any) -> lu(a, check = false)

Slow fallback which gets the instance via factorization. Should get specialized for new matrix types.

ArrayInterface.qr_instance — Function

qr_instance(A, pivot = NoPivot()) -> qr_factorization_instance

Returns an instance of the QR factorization object with the correct type cheaply.

qr_instance(a::Number) -> a

Returns the number.

qr_instance(a::Any) -> qr(a)

Slow fallback which gets the instance via factorization. Should get specialized for new matrix types.

ArrayInterface.svd_instance — Function

svd_instance(A) -> qr_factorization_instance

Returns an instance of the SVD factorization object with the correct type cheaply.

svd_instance(a::Number) -> a

Returns the number.

svd_instance(a::Any) -> svd(a)

Slow fallback which gets the instance via factorization. Should get specialized for new matrix types.

Addtional Linear Algebra Interface Tools

If dealing with general linear algebra, consider:

LinearSolve.jl: An extended linear solving library with support for generic arrays.