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BlockArrays

Creating BlockArrays from an array

An AbstractArray can be repacked into a BlockArray with BlockArray(array, block_sizes...). The block sizes are each an AbstractVector{Int} which determines the size of the blocks in that dimension (so the sum of block_sizes in every dimension must match the size of array in that dimension).

julia> BlockArray(Array(reshape(1:16, 4, 4)), [2,2], [1,1,2])
2×3-blocked 4×4 BlockMatrix{Int64}:
 1  │  5  │   9  13
 2  │  6  │  10  14
 ───┼─────┼────────
 3  │  7  │  11  15
 4  │  8  │  12  16

julia> S = spzeros(4,5); S[1,2] = S[4,3] = 1;

julia> block_array_sparse = BlockArray(S, [1,3], [2,3])
2×2-blocked 4×5 BlockMatrix{Float64, Matrix{SparseMatrixCSC{Float64, Int64}}, Tuple{BlockedOneTo{Int64, Vector{Int64}}, BlockedOneTo{Int64, Vector{Int64}}}}:
  ⋅   1.0  │   ⋅    ⋅    ⋅ 
 ──────────┼───────────────
  ⋅    ⋅   │   ⋅    ⋅    ⋅ 
  ⋅    ⋅   │   ⋅    ⋅    ⋅ 
  ⋅    ⋅   │  1.0   ⋅    ⋅

Creating uninitialized BlockArrays

A block array can be created with uninitialized values (but initialized blocks) using the BlockArray{T}(undef, block_sizes) function. The block_sizes are each an AbstractVector{Int} which determines the size of the blocks in that dimension. We here create a block matrix of Float32s:

julia> BlockArray{Float32}(undef, [1,2,1], [1,1,1])
3×3-blocked 4×3 BlockMatrix{Float32}:
 -2.15145e-35  │   1.4013e-45   │  -1.77199e-35
 ──────────────┼────────────────┼──────────────
  1.4013e-45   │  -1.77199e-35  │  -1.72473e-34
  1.4013e-45   │   4.57202e-41  │   4.57202e-41
 ──────────────┼────────────────┼──────────────
  0.0          │  -1.36568e-33  │  -1.72473e-34

We can also any other user defined array type that supports similar.

Creating BlockArrays with uninitialized blocks.

A BlockArray can be created with the blocks left uninitialized using the BlockArray(undef_blocks[, block_type], block_sizes...) function. We here create a [1,2]×[3,2] block matrix of Float32s:

julia> BlockArray{Float32}(undef_blocks, [1,2], [3,2])
2×2-blocked 3×5 BlockMatrix{Float32}:
 #undef  #undef  #undef  │  #undef  #undef
 ────────────────────────┼────────────────
 #undef  #undef  #undef  │  #undef  #undef
 #undef  #undef  #undef  │  #undef  #undef

The block_type should be an array type. It specifies the internal block type, which defaults to an Array of the according dimension. We can also use a SparseVector or any other user defined array type:

julia> BlockArray(undef_blocks, SparseVector{Float64, Int}, [1,2])
2-blocked 3-element BlockVector{Float64, Vector{SparseVector{Float64, Int64}}, Tuple{BlockedOneTo{Int64, Vector{Int64}}}}:
 #undef
 ──────
 #undef
 #undef
Warning

Note that accessing an undefined block will throw an "access to undefined reference"-error! If you create an array with undefined blocks, you have to initialize it block-wise); whole-array functions like fill! will not work:

julia> fill!(BlockArray{Float32}(undef_blocks, [1,2], [3,2]), 0)
ERROR: UndefRefError: access to undefined reference

Setting and getting blocks and values

A block can be set by block_array[Block(i...)] = v. The indexing may equivalently be carried out as block_array[Block.(i)...].

julia> block_array = BlockArray{Float64}(undef_blocks, [1,2], [2,2])
2×2-blocked 3×4 BlockMatrix{Float64}:
 #undef  #undef  │  #undef  #undef
 ────────────────┼────────────────
 #undef  #undef  │  #undef  #undef
 #undef  #undef  │  #undef  #undef

julia> block_array[Block(2,1)] = reshape([1:4;], 2, 2);

julia> block_array[Block(1),Block(1)] = [1 2];

julia> block_array
2×2-blocked 3×4 BlockMatrix{Float64}:
 1.0  2.0  │  #undef  #undef
 ──────────┼────────────────
 1.0  3.0  │  #undef  #undef
 2.0  4.0  │  #undef  #undef

Note that this will "take ownership" of the passed in array, that is, no copy is made.

A block can be retrieved with view(block_array, Block(i...)), or if a copy is desired, block_array[Block(i...)]:

julia> view(block_array, Block(1, 1))
1×2 Matrix{Float64}:
 1.0  2.0

julia> block_array[Block(1, 1)] # makes a copy
1×2 Matrix{Float64}:
 1.0  2.0

julia> block_array[Block(1), Block(1)]  # equivalent to above
1×2 Matrix{Float64}:
 1.0  2.0

For setting and getting a single scalar element, the usual setindex! and getindex are available.

julia> block_array[1, 2]
2.0

Views of blocks

To view and modify blocks of BlockArray use the view syntax.

julia> A = BlockArray(ones(6), 1:3);

julia> view(A, Block(2))
2-element Vector{Float64}:
 1.0
 1.0

julia> view(A, Block(2)) .= [3,4]; A[Block(2)]
2-element Vector{Float64}:
 3.0
 4.0

julia> view(A, Block.(1:2))
3-element view(::BlockVector{Float64, Vector{Vector{Float64}}, Tuple{BlockedOneTo{Int64, ArrayLayouts.RangeCumsum{Int64, UnitRange{Int64}}}}}, BlockSlice(BlockRange(1:2),1:1:3)) with eltype Float64 with indices BlockedOneTo([1, 3]):
 1.0
 3.0
 4.0

Converting between BlockArray and normal arrays

An array can be repacked into a BlockArray with BlockArray(array, block_sizes...):

julia> S = spzeros(4,5); S[1,2] = S[4,3] = 1;

julia> block_array_sparse = BlockArray(S, [1,3], [2,3])
2×2-blocked 4×5 BlockMatrix{Float64, Matrix{SparseMatrixCSC{Float64, Int64}}, Tuple{BlockedOneTo{Int64, Vector{Int64}}, BlockedOneTo{Int64, Vector{Int64}}}}:
  ⋅   1.0  │   ⋅    ⋅    ⋅ 
 ──────────┼───────────────
  ⋅    ⋅   │   ⋅    ⋅    ⋅ 
  ⋅    ⋅   │   ⋅    ⋅    ⋅ 
  ⋅    ⋅   │  1.0   ⋅    ⋅

To get back the underlying sparse array, use sparse:

julia> sparse(block_array_sparse)
4×5 SparseMatrixCSC{Float64, Int64} with 2 stored entries:
  ⋅   1.0   ⋅    ⋅    ⋅ 
  ⋅    ⋅    ⋅    ⋅    ⋅ 
  ⋅    ⋅    ⋅    ⋅    ⋅ 
  ⋅    ⋅   1.0   ⋅    ⋅

To get a dense array, use Array:

julia> Array(block_array_sparse)
4×5 Matrix{Float64}:
 0.0  1.0  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  1.0  0.0  0.0