在 Julia 中定义任意维度的多维数组

Question

语境

这个问题与这个有关。

在 Julia 中，我想制作一个 5 x 5 的二维数组，其中 (i, j) 元素具有[i,j]如下所示：

5×5 Matrix{Vector{Int64}}:
 [1, 1]  [1, 2]  [1, 3]  [1, 4]  [1, 5]
 [2, 1]  [2, 2]  [2, 3]  [2, 4]  [2, 5]
 [3, 1]  [3, 2]  [3, 3]  [3, 4]  [3, 5]
 [4, 1]  [4, 2]  [4, 3]  [4, 4]  [4, 5]
 [5, 1]  [5, 2]  [5, 3]  [5, 4]  [5, 5]

我使用数组理解尝试了这个：

N = 5
L_2 = [[x1,x2] for x1 = 1:N, x2 = 1:N]

我想做的事

我想将这个定义推广到任意维度D 。

L_1 = [[x1] for x1 = 1:N] # 1-dimensional
L_2 = [[x1,x2] for x1 = 1:N, x2 = 1:N] # 2-dimensional
L_3 = [[x1,x2,x3] for x1 = 1:N, x2 = 1:N,x3 = 1:N] # 3-dimensional
...

#L_D = ??? # D-dimensional

我该如何定义？

不使用数组理解也没关系。

任何信息，将不胜感激。

Answer 1

您可以概括我在其他答案中发布的vcat方法，如下所示：

julia> lattice(N, D) = vcat.((reshape(1:N, ntuple(j -> j == i ? N : 1, D)) for i in 1:D)...)
lattice (generic function with 1 method)

julia> lattice(2, 1)
2-element Vector{Vector{Int64}}:
 [1]
 [2]

julia> lattice(2, 2)
2×2 Matrix{Vector{Int64}}:
 [1, 1]  [1, 2]
 [2, 1]  [2, 2]

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

[:, :, 2] =
 [1, 1, 2]  [1, 2, 2]
 [2, 1, 2]  [2, 2, 2]

julia> lattice(2, 4)
2×2×2×2 Array{Vector{Int64}, 4}:
[:, :, 1, 1] =
 [1, 1, 1, 1]  [1, 2, 1, 1]
 [2, 1, 1, 1]  [2, 2, 1, 1]

[:, :, 2, 1] =
 [1, 1, 2, 1]  [1, 2, 2, 1]
 [2, 1, 2, 1]  [2, 2, 2, 1]

[:, :, 1, 2] =
 [1, 1, 1, 2]  [1, 2, 1, 2]
 [2, 1, 1, 2]  [2, 2, 1, 2]

[:, :, 2, 2] =
 [1, 1, 2, 2]  [1, 2, 2, 2]
 [2, 1, 2, 2]  [2, 2, 2, 2]

julia> lattice(2, 5)
2×2×2×2×2 Array{Vector{Int64}, 5}:
[:, :, 1, 1, 1] =
 [1, 1, 1, 1, 1]  [1, 2, 1, 1, 1]
 [2, 1, 1, 1, 1]  [2, 2, 1, 1, 1]

[:, :, 2, 1, 1] =
 [1, 1, 2, 1, 1]  [1, 2, 2, 1, 1]
 [2, 1, 2, 1, 1]  [2, 2, 2, 1, 1]

[:, :, 1, 2, 1] =
 [1, 1, 1, 2, 1]  [1, 2, 1, 2, 1]
 [2, 1, 1, 2, 1]  [2, 2, 1, 2, 1]

[:, :, 2, 2, 1] =
 [1, 1, 2, 2, 1]  [1, 2, 2, 2, 1]
 [2, 1, 2, 2, 1]  [2, 2, 2, 2, 1]

[:, :, 1, 1, 2] =
 [1, 1, 1, 1, 2]  [1, 2, 1, 1, 2]
 [2, 1, 1, 1, 2]  [2, 2, 1, 1, 2]

[:, :, 2, 1, 2] =
 [1, 1, 2, 1, 2]  [1, 2, 2, 1, 2]
 [2, 1, 2, 1, 2]  [2, 2, 2, 1, 2]

[:, :, 1, 2, 2] =
 [1, 1, 1, 2, 2]  [1, 2, 1, 2, 2]
 [2, 1, 1, 2, 2]  [2, 2, 1, 2, 2]

[:, :, 2, 2, 2] =
 [1, 1, 2, 2, 2]  [1, 2, 2, 2, 2]
 [2, 1, 2, 2, 2]  [2, 2, 2, 2, 2]julia> lattice(N, D) = vcat.([reshape(1:N, ntuple(j -> j == i ? N : 1, D)) for i in 1:D]...)
lattice (generic function with 1 method)

julia> lattice(2, 1)
2-element Vector{Vector{Int64}}:
 [1]
 [2]

julia> lattice(2, 2)
2×2 Matrix{Vector{Int64}}:
 [1, 1]  [1, 2]
 [2, 1]  [2, 2]

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

[:, :, 2] =
 [1, 1, 2]  [1, 2, 2]
 [2, 1, 2]  [2, 2, 2]

julia> lattice(2, 4)
2×2×2×2 Array{Vector{Int64}, 4}:
[:, :, 1, 1] =
 [1, 1, 1, 1]  [1, 2, 1, 1]
 [2, 1, 1, 1]  [2, 2, 1, 1]

[:, :, 2, 1] =
 [1, 1, 2, 1]  [1, 2, 2, 1]
 [2, 1, 2, 1]  [2, 2, 2, 1]

[:, :, 1, 2] =
 [1, 1, 1, 2]  [1, 2, 1, 2]
 [2, 1, 1, 2]  [2, 2, 1, 2]

[:, :, 2, 2] =
 [1, 1, 2, 2]  [1, 2, 2, 2]
 [2, 1, 2, 2]  [2, 2, 2, 2]

julia> lattice(2, 5)
2×2×2×2×2 Array{Vector{Int64}, 5}:
[:, :, 1, 1, 1] =
 [1, 1, 1, 1, 1]  [1, 2, 1, 1, 1]
 [2, 1, 1, 1, 1]  [2, 2, 1, 1, 1]

[:, :, 2, 1, 1] =
 [1, 1, 2, 1, 1]  [1, 2, 2, 1, 1]
 [2, 1, 2, 1, 1]  [2, 2, 2, 1, 1]

[:, :, 1, 2, 1] =
 [1, 1, 1, 2, 1]  [1, 2, 1, 2, 1]
 [2, 1, 1, 2, 1]  [2, 2, 1, 2, 1]

[:, :, 2, 2, 1] =
 [1, 1, 2, 2, 1]  [1, 2, 2, 2, 1]
 [2, 1, 2, 2, 1]  [2, 2, 2, 2, 1]

[:, :, 1, 1, 2] =
 [1, 1, 1, 1, 2]  [1, 2, 1, 1, 2]
 [2, 1, 1, 1, 2]  [2, 2, 1, 1, 2]

[:, :, 2, 1, 2] =
 [1, 1, 2, 1, 2]  [1, 2, 2, 1, 2]
 [2, 1, 2, 1, 2]  [2, 2, 2, 1, 2]

[:, :, 1, 2, 2] =
 [1, 1, 1, 2, 2]  [1, 2, 1, 2, 2]
 [2, 1, 1, 2, 2]  [2, 2, 1, 2, 2]

[:, :, 2, 2, 2] =
 [1, 1, 2, 2, 2]  [1, 2, 2, 2, 2]
 [2, 1, 2, 2, 2]  [2, 2, 2, 2, 2]

Answer 2

您似乎不需要 CartesianIndices ，但为了记录，CartesianIndices 可以采用任何 Int 元组（更准确地说Dims aka NTuple{N,Int} where N ）来表示数组的大小。 CartesianIndices((5,5))用于 5x5， CartesianIndices((2,8,3))用于 2x8x3 等。您可以快速创建 NxNxNx... 表示具有 D 维度的大小的元组NtotheD(N,D) = ntuple(i -> N, D) 。