# OffsetArrays.jl

OffsetArrays provides Julia users with arrays that have arbitrary indices, similar to those found in some other programming languages like Fortran. Below is the basic usage found in the README, followed by a couple of short examples illustrating circumstances in which OffsetArrays can be useful. For a lengthier discussion, see this blog post.

## Usage

You can construct such arrays as follows:

`OA = OffsetArray(A, axis1, axis2, ...)`

where you want `OA`

to have axes `(axis1, axis2, ...)`

and be indexed by values that fall within these axis ranges.

`julia> using OffsetArrays`

`julia> A = Float64.(reshape(1:15, 3, 5))`

`3×5 Matrix{Float64}: 1.0 4.0 7.0 10.0 13.0 2.0 5.0 8.0 11.0 14.0 3.0 6.0 9.0 12.0 15.0`

`julia> OA = OffsetArray(A, -1:1, 0:4) # OA will have axes (-1:1, 0:4)`

`3×5 OffsetArray(::Matrix{Float64}, -1:1, 0:4) with eltype Float64 with indices -1:1×0:4: 1.0 4.0 7.0 10.0 13.0 2.0 5.0 8.0 11.0 14.0 3.0 6.0 9.0 12.0 15.0`

`julia> OA = OffsetArray(A, CartesianIndex(-1, 0):CartesianIndex(1, 4))`

`3×5 OffsetArray(::Matrix{Float64}, -1:1, 0:4) with eltype Float64 with indices -1:1×0:4: 1.0 4.0 7.0 10.0 13.0 2.0 5.0 8.0 11.0 14.0 3.0 6.0 9.0 12.0 15.0`

`julia> OA[-1,0], OA[1,4]`

`(1.0, 15.0)`

You could also pass integers as offsets, where `0`

means no offsets are applied:

`julia> OA = OffsetArray(A, -2, -1)`

`3×5 OffsetArray(::Matrix{Float64}, -1:1, 0:4) with eltype Float64 with indices -1:1×0:4: 1.0 4.0 7.0 10.0 13.0 2.0 5.0 8.0 11.0 14.0 3.0 6.0 9.0 12.0 15.0`

When you create a new `OffsetArray`

on the top of another `OffsetArray`

, the offsets are accumulated:

`julia> OOA = OffsetArray(OA, 2, 1)`

`3×5 OffsetArray(::Matrix{Float64}, 1:3, 1:5) with eltype Float64 with indices 1:3×1:5: 1.0 4.0 7.0 10.0 13.0 2.0 5.0 8.0 11.0 14.0 3.0 6.0 9.0 12.0 15.0`

For the special cases that you want to compensate the offset back to the ordinary 1-based array, you can use `OffsetArrays.no_offset_view(A)`

. Furthermore, you could use `Base.require_one_based_indexing`

if you want to ensure the array does not have offsets.

`julia> OffsetArrays.no_offset_view(OA)`

`3×5 Matrix{Float64}: 1.0 4.0 7.0 10.0 13.0 2.0 5.0 8.0 11.0 14.0 3.0 6.0 9.0 12.0 15.0`

`julia> Base.require_one_based_indexing(ans)`

`true`

`julia> Base.require_one_based_indexing(OA)`

`ERROR: ArgumentError: offset arrays are not supported but got an array with index other than 1`

`OffsetArrays.Origin`

can be convenient if you want to directly specify the origin of the output OffsetArray, it will automatically compute the corresponding offsets. For example:

`julia> OffsetArray(A, OffsetArrays.Origin(-1, -1))`

`3×5 OffsetArray(::Matrix{Float64}, -1:1, -1:3) with eltype Float64 with indices -1:1×-1:3: 1.0 4.0 7.0 10.0 13.0 2.0 5.0 8.0 11.0 14.0 3.0 6.0 9.0 12.0 15.0`

`julia> OffsetArray(OA, OffsetArrays.Origin(-1, -1))`

`3×5 OffsetArray(::Matrix{Float64}, -1:1, -1:3) with eltype Float64 with indices -1:1×-1:3: 1.0 4.0 7.0 10.0 13.0 2.0 5.0 8.0 11.0 14.0 3.0 6.0 9.0 12.0 15.0`

An equivalent — but possibly more convenient — way to specify the origin of an array is

`julia> OffsetArrays.Origin(-1, -1)(A)`

`3×5 OffsetArray(::Matrix{Float64}, -1:1, -1:3) with eltype Float64 with indices -1:1×-1:3: 1.0 4.0 7.0 10.0 13.0 2.0 5.0 8.0 11.0 14.0 3.0 6.0 9.0 12.0 15.0`

Sometimes, it will be convenient to shift the center coordinate of the given array to `(0, 0, ...)`

, `OffsetArrays.centered`

is a helper for this very purpose:

`julia> Ao = OffsetArrays.centered(A)`

`3×5 OffsetArray(::Matrix{Float64}, -1:1, -2:2) with eltype Float64 with indices -1:1×-2:2: 1.0 4.0 7.0 10.0 13.0 2.0 5.0 8.0 11.0 14.0 3.0 6.0 9.0 12.0 15.0`

`julia> Ao[0, 0] == 8.0`

`true`

and `OffsetArrays.center`

tells you the center coordinate of given array:

`julia> c = OffsetArrays.center(A)`

`(2, 3)`

`julia> A[c...] == 8.0`

`true`

## Example: Relativistic Notation

Suppose we have a position vector `r = [:x, :y, :z]`

which is naturally one-based, ie. `r[1] == :x`

, `r[2] == :y`

, `r[3] == :z`

and we also want to construct a relativistic position vector which includes time as the 0th component. This can be done with OffsetArrays like

```
julia> r = [:x, :y, :z];
julia> x = OffsetVector([:t, r...], 0:3)
4-element OffsetArray(::Vector{Symbol}, 0:3) with eltype Symbol with indices 0:3:
:t
:x
:y
:z
julia> x[0]
:t
julia> x[1:3]
3-element Vector{Symbol}:
:x
:y
:z
```

## Example: Polynomials

Suppose one wants to represent the Laurent polynomial

\[6/x + 5 - 2*x + 3*x^2 + x^3\]

The coefficients of this polynomial are a naturally `-1`

based list, since the `n`

th element of the list (counting from `-1`

) `6, 5, -2, 3, 1`

is the coefficient corresponding to the `n`

th power of `x`

. This Laurent polynomial can be evaluated at say `x = 2`

as follows.

```
julia> coeffs = OffsetVector([6, 5, -2, 3, 1], -1:3)
5-element OffsetArray(::Vector{Int64}, -1:3) with eltype Int64 with indices -1:3:
6
5
-2
3
1
julia> polynomial(x, coeffs) = sum(coeffs[n]*x^n for n in eachindex(coeffs))
polynomial (generic function with 1 method)
julia> polynomial(2.0, coeffs)
24.0
```

Notice our use of the `eachindex`

function which does not assume that the given array starts at `1`

.