# 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`

, `Falses`

, and `OneElement`

.

The primary purpose of this package is to present a unified way of constructing
matrices.
For example, 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> Zeros(Int, 2, 3) # can also specify the type as an argument
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}
julia> OneElement(3.0, (2,1), (5,6))
5×6 OneElement{Float64, 2, Tuple{Int64, Int64}, Tuple{Base.OneTo{Int64}, Base.OneTo{Int64}}}:
⋅ ⋅ ⋅ ⋅ ⋅ ⋅
3.0 ⋅ ⋅ ⋅ ⋅ ⋅
⋅ ⋅ ⋅ ⋅ ⋅ ⋅
⋅ ⋅ ⋅ ⋅ ⋅ ⋅
⋅ ⋅ ⋅ ⋅ ⋅ ⋅
```

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
```