Introduction

FillArrays allows one to lazily represent arrays filled with a single entry, as well as identity matrices. This package exports the following types: Eye, Fill, Ones, Zeros, Trues and Falses. Among these, the FillArrays.AbstractFill types represent lazy versions of dense arrays where all elements have the same value. Eye, on the other hand, represents a Diagonal matrix with ones along the principal diagonal. All these types accept sizes or axes as arguments, so one may create arrays of arbitrary sizes and dimensions. A rectangular Eye matrix may be constructed analogously, by passing the size of the matrix to Eye.

Quick Start

Create a 2x2 zero matrix

julia> z = Zeros(2,2)
2×2 Zeros{Float64}

julia> Array(z)
2×2 Matrix{Float64}:
 0.0  0.0
 0.0  0.0

We may specify the element type as

julia> z = Zeros{Int}(2,2)
2×2 Zeros{Int64}

julia> Array(z)
2×2 Matrix{Int64}:
 0  0
 0  0

We may create arrays with any number of dimensions. A Vector of ones may be created as

julia> a = Ones(4)
4-element Ones{Float64}

julia> Array(a)
4-element Vector{Float64}:
 1.0
 1.0
 1.0
 1.0

Similarly, a 2x3x2 array, where every element is equal to 10, may be created as

julia> f = Fill(10, 2,3,2)
2×3×2 Fill{Int64}, with entries equal to 10

julia> Array(f)
2×3×2 Array{Int64, 3}:
[:, :, 1] =
 10  10  10
 10  10  10

[:, :, 2] =
 10  10  10
 10  10  10

The elements of a Fill array don't need to be restricted to numbers, and these may be any Julia object. For example, we may construct an array of strings using

julia> f = Fill("hello", 2,5)
2×5 Fill{String}, with entries equal to "hello"

julia> Array(f)
2×5 Matrix{String}:
 "hello"  "hello"  "hello"  "hello"  "hello"
 "hello"  "hello"  "hello"  "hello"  "hello"

Conversion to a sparse form

These Fill array types may be converted to sparse arrays as well, which might be useful in certain cases

julia> using SparseArrays

julia> z = Zeros{Int}(2,2)
2×2 Zeros{Int64}

julia> sparse(z)
2×2 SparseMatrixCSC{Int64, Int64} with 0 stored entries:
 ⋅  ⋅
 ⋅  ⋅

Note, however, that most Fill arrays are not sparse, despite being lazily evaluated.

These types have methods that perform many operations efficiently, including elementary algebra operations like multiplication and addition, as well as linear algebra methods like norm, adjoint, transpose and vec.

Custom axes

The various Fill equivalents all support offset or custom axes, where instead of the size, one may pass a Tuple of axes. So, for example, one may use a SOneTo axis from StaticArrays.jl to construct a statically sized Fill.

julia> using StaticArrays

julia> f = Fill(2, (SOneTo(4), SOneTo(5)))
4×5 Fill{Int64, 2, Tuple{SOneTo{4}, SOneTo{5}}} with indices SOneTo(4)×SOneTo(5), with entries equal to 2

The size of such an array would be known at compile time, permitting compiler optimizations.

We may construct infinite fill arrays by passing infinite-sized axes, see InfiniteArrays.jl.

Other lazy types

A lazy representation of an identity matrix may be constructured using Eye. For example, a 4x4 identity matrix with Float32 elements may be constructed as

julia> id = Eye{Float32}(4)
4×4 Eye{Float32}

julia> Array(id)
4×4 Matrix{Float32}:
 1.0  0.0  0.0  0.0
 0.0  1.0  0.0  0.0
 0.0  0.0  1.0  0.0
 0.0  0.0  0.0  1.0

julia> sparse(id)
4×4 SparseMatrixCSC{Float32, Int64} with 4 stored entries:
 1.0   ⋅    ⋅    ⋅
  ⋅   1.0   ⋅    ⋅
  ⋅    ⋅   1.0   ⋅
  ⋅    ⋅    ⋅   1.0

julia> idrect = Eye(2,5) # rectangular matrix
2×5 Eye{Float64}

julia> sparse(idrect)
2×5 SparseMatrixCSC{Float64, Int64} with 2 stored entries:
 1.0   ⋅    ⋅    ⋅    ⋅
  ⋅   1.0   ⋅    ⋅    ⋅

Note that an Eye actually returns a Diagonal matrix, where the diagonal is a Ones vector.

Warning about map and broadcasting

Broadcasting operations, and map and mapreduce, are also done efficiently, by evaluating the function being applied only once:

julia> map(sqrt, Fill(4, 2,5))  # one evaluation, not 10, to save time
2×5 Fill{Float64}, with entries equal to 2.0

julia> println.(Fill(pi, 10))
π
10-element Fill{Nothing}, with entries equal to nothing

Notice that this will only match the behaviour of a dense matrix from fill if the function is pure. And that this shortcut is taken before any other fused broadcast:

julia> map(_ -> rand(), Fill("pi", 2,5))  # not a pure function!
2×5 Fill{Float64}, with entries equal to 0.32597672886359486

julia> map(_ -> rand(), fill("4", 2,5))  # 10 evaluations, different answer!
2×5 Matrix{Float64}:
 0.549051  0.894245  0.394255  0.795547  0.748415
 0.218587  0.353112  0.953125  0.49425   0.578232

julia> ones(1,5) .+ (_ -> rand()).(Fill("vec", 2))  # Fill broadcast is done first
2×5 Matrix{Float64}:
 1.72794  1.72794  1.72794  1.72794  1.72794
 1.72794  1.72794  1.72794  1.72794  1.72794

julia> ones(1,5) .+ (_ -> rand()).(fill("vec", 2))  # fused, 10 evaluations
2×5 Matrix{Float64}:
 1.00745  1.43924  1.95674  1.99667  1.11008
 1.19938  1.68253  1.64786  1.74919  1.49138

API

FillArrays.FillArrays — Module

FillArrays module to lazily represent matrices with a single value