FillArrays.jl
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Julia package for lazily representing matrices filled with a single entry
FillArrays.jl
Julia package to lazily represent matrices filled with a single entry,
as well as identity matrices. This package exports the following types:
Eye
, Fill
, Ones
, Zeros
, Trues
and Falses
.
The primary purpose of this package is to present a unified way of constructing
matrices. For example, to construct a 5-by-5 CLArray
of all zeros, one would use
julia> CLArray(Zeros(5,5))
Because Zeros
is lazy, this can be accomplished on the GPU with no memory transfer.
Similarly, to construct a 5-by-5 BandedMatrix
of all zeros with bandwidths (1,2)
, one would use
julia> BandedMatrix(Zeros(5,5), (1, 2))
Usage
Here are the matrix types:
julia> Zeros(5, 6)
5×6 Zeros{Float64}
julia> Zeros{Int}(2, 3)
2×3 Zeros{Int64}
julia> Ones{Int}(5)
5-element Ones{Int64}
julia> Eye{Int}(5)
5×5 Diagonal{Int64,Ones{Int64,1,Tuple{Base.OneTo{Int64}}}}:
1 ⋅ ⋅ ⋅ ⋅
⋅ 1 ⋅ ⋅ ⋅
⋅ ⋅ 1 ⋅ ⋅
⋅ ⋅ ⋅ 1 ⋅
⋅ ⋅ ⋅ ⋅ 1
julia> Fill(7.0f0, 3, 2)
3×2 Fill{Float32}: entries equal to 7.0
julia> Trues(2, 3)
2×3 Ones{Bool}
julia> Falses(2)
2-element Zeros{Bool}
They support conversion to other matrix types like Array
, SparseVector
, SparseMatrix
, and Diagonal
:
julia> Matrix(Zeros(5, 5))
5×5 Array{Float64,2}:
0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0
julia> SparseMatrixCSC(Zeros(5, 5))
5×5 SparseMatrixCSC{Float64,Int64} with 0 stored entries
julia> Array(Fill(7, (2,3)))
2×3 Array{Int64,2}:
7 7 7
7 7 7
There is also support for offset index ranges,
and the type includes the axes
:
julia> Ones((-3:2, 1:2))
6×2 Ones{Float64,2,Tuple{UnitRange{Int64},UnitRange{Int64}}} with indices -3:2×1:2
julia> Fill(7, ((0:2), (-1:0)))
3×2 Fill{Int64,2,Tuple{UnitRange{Int64},UnitRange{Int64}}} with indices 0:2×-1:0: entries equal to 7
julia> typeof(Zeros(5,6))
Zeros{Float64,2,Tuple{Base.OneTo{Int64},Base.OneTo{Int64}}}
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
.
Warning!
Broadcasting operations and map
, 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}: entries equal to 2.0
julia> println.(Fill(pi, 10))
π
10-element Fill{Nothing}: 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}: entries equal to 0.7201617100284206
julia> map(_ -> rand(), fill("4", 2,5)) # 10 evaluations, different answer!
2×5 Matrix{Float64}:
0.43675 0.270809 0.56536 0.0948089 0.24655
0.959363 0.79598 0.238662 0.401909 0.317716
julia> ones(1,5) .+ (_ -> rand()).(Fill("vec", 2)) # Fill broadcast is done first
2×5 Matrix{Float64}:
1.51796 1.51796 1.51796 1.51796 1.51796
1.51796 1.51796 1.51796 1.51796 1.51796
julia> ones(1,5) .+ (_ -> rand()).(fill("vec", 2)) # fused, 10 evaluations
2×5 Matrix{Float64}:
1.51337 1.17578 1.19815 1.43035 1.2987
1.30253 1.21909 1.61755 1.02645 1.77681