# BlockBandedMatrices.jl Documentation

## Creating block-banded and banded-block-banded matrices

BlockBandedMatrices.BlockBandedMatrixType
BlockBandedMatrix

A BlockBandedMatrix is a subtype of BlockMatrix of BlockArrays.jl whose layout of non-zero blocks is banded.

BlockBandedMatrices.BlockBandedMatrixMethod
BlockBandedMatrix(A::Union{AbstractMatrix,UniformScaling},
rows::AbstractVector{Int}, cols::AbstractVector{Int},
(l,u)::NTuple{2,Int})

Return a sum(rows) × sum(cols) BlockBandedMatrix, with rows by cols blocks, with (l,u) as the block-bandwidth. The structural non-zero entries are equal to the corresponding indices of A.

Examples

julia> using LinearAlgebra, FillArrays

julia> l,u = 0,1; # block bandwidths

julia> nrowblk, ncolblk = 3, 3; # number of row/column blocks

julia> rows = 1:nrowblk; cols = 1:ncolblk; # block sizes

julia> BlockBandedMatrix(I, rows, cols, (l,u))
3×3-blocked 6×6 BlockBandedMatrix{Bool}:
1  │  0  0  │  ⋅  ⋅  ⋅
───┼────────┼─────────
⋅  │  1  0  │  0  0  0
⋅  │  0  1  │  0  0  0
───┼────────┼─────────
⋅  │  ⋅  ⋅  │  1  0  0
⋅  │  ⋅  ⋅  │  0  1  0
⋅  │  ⋅  ⋅  │  0  0  1

julia> BlockBandedMatrix(Ones(sum(rows),sum(cols)), rows, cols, (l,u))
3×3-blocked 6×6 BlockBandedMatrix{Float64}:
1.0  │  1.0  1.0  │   ⋅    ⋅    ⋅
─────┼────────────┼───────────────
⋅   │  1.0  1.0  │  1.0  1.0  1.0
⋅   │  1.0  1.0  │  1.0  1.0  1.0
─────┼────────────┼───────────────
⋅   │   ⋅    ⋅   │  1.0  1.0  1.0
⋅   │   ⋅    ⋅   │  1.0  1.0  1.0
⋅   │   ⋅    ⋅   │  1.0  1.0  1.0
BlockBandedMatrices.BlockBandedMatrixMethod
BlockBandedMatrix(A::AbstractMatrix, (l,u)::NTuple{2,Int})

Return a BlockBandedMatrix with block-bandwidths (l,u), where the structural non-zero blocks correspond to those of A.

Examples

julia> using BlockArrays

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

julia> BlockBandedMatrix(B, (1,1))
3×3-blocked 6×6 BlockBandedMatrix{Float64}:
1.0  │  1.0  1.0  │   ⋅    ⋅    ⋅
─────┼────────────┼───────────────
1.0  │  1.0  1.0  │  1.0  1.0  1.0
1.0  │  1.0  1.0  │  1.0  1.0  1.0
─────┼────────────┼───────────────
⋅   │  1.0  1.0  │  1.0  1.0  1.0
⋅   │  1.0  1.0  │  1.0  1.0  1.0
⋅   │  1.0  1.0  │  1.0  1.0  1.0
BlockBandedMatrices.BlockBandedMatrixMethod
BlockBandedMatrix{T}(undef, rows::AbstractVector{Int}, cols::AbstractVector{Int},
(l,u)::NTuple{2,Int})

Return an unitialized sum(rows) × sum(cols) BlockBandedMatrix having eltype T, with rows by cols blocks and (l,u) as the block-bandwidth.

BlockBandedMatrices.BandedBlockBandedMatrixType
BandedBlockBandedMatrix(M::Union{UniformScaling,AbstractMatrix},
rows, cols, (l, u), (λ, μ))

Return a sum(rows) × sum(cols) banded-block-banded matrix A, with block-bandwidths (l,u) and where A[Block(K,J)] is a BandedMatrix of size rows[K]×cols[J] with bandwidths (λ,μ). The structural non-zero elements of the returned matrix corresponds to those of M.

Examples

julia> using LinearAlgebra, FillArrays

julia> BandedBlockBandedMatrix(I, [3,4,3], [3,4,3], (1,1), (1,1))
3×3-blocked 10×10 BandedBlockBandedMatrix{Bool} with block-bandwidths (1, 1) and sub-block-bandwidths block-bandwidths (1, 1):
1  0  ⋅  │  0  0  ⋅  ⋅  │  ⋅  ⋅  ⋅
0  1  0  │  0  0  0  ⋅  │  ⋅  ⋅  ⋅
⋅  0  1  │  ⋅  0  0  0  │  ⋅  ⋅  ⋅
─────────┼──────────────┼─────────
0  0  ⋅  │  1  0  ⋅  ⋅  │  0  0  ⋅
0  0  0  │  0  1  0  ⋅  │  0  0  0
⋅  0  0  │  ⋅  0  1  0  │  ⋅  0  0
⋅  ⋅  0  │  ⋅  ⋅  0  1  │  ⋅  ⋅  0
─────────┼──────────────┼─────────
⋅  ⋅  ⋅  │  0  0  ⋅  ⋅  │  1  0  ⋅
⋅  ⋅  ⋅  │  0  0  0  ⋅  │  0  1  0
⋅  ⋅  ⋅  │  ⋅  0  0  0  │  ⋅  0  1

julia> BandedBlockBandedMatrix(Ones{Int}(10,13), [3,4,3], [4,5,4], (1,1), (1,1))
3×3-blocked 10×13 BandedBlockBandedMatrix{Int64} with block-bandwidths (1, 1) and sub-block-bandwidths block-bandwidths (1, 1):
1  1  ⋅  ⋅  │  1  1  ⋅  ⋅  ⋅  │  ⋅  ⋅  ⋅  ⋅
1  1  1  ⋅  │  1  1  1  ⋅  ⋅  │  ⋅  ⋅  ⋅  ⋅
⋅  1  1  1  │  ⋅  1  1  1  ⋅  │  ⋅  ⋅  ⋅  ⋅
────────────┼─────────────────┼────────────
1  1  ⋅  ⋅  │  1  1  ⋅  ⋅  ⋅  │  1  1  ⋅  ⋅
1  1  1  ⋅  │  1  1  1  ⋅  ⋅  │  1  1  1  ⋅
⋅  1  1  1  │  ⋅  1  1  1  ⋅  │  ⋅  1  1  1
⋅  ⋅  1  1  │  ⋅  ⋅  1  1  1  │  ⋅  ⋅  1  1
────────────┼─────────────────┼────────────
⋅  ⋅  ⋅  ⋅  │  1  1  ⋅  ⋅  ⋅  │  1  1  ⋅  ⋅
⋅  ⋅  ⋅  ⋅  │  1  1  1  ⋅  ⋅  │  1  1  1  ⋅
⋅  ⋅  ⋅  ⋅  │  ⋅  1  1  1  ⋅  │  ⋅  1  1  1
BlockBandedMatrices.BandedBlockBandedMatrixMethod
BandedBlockBandedMatrix{T}(undef, rows, cols, (l, u), (λ, μ))

Return an unitialized BandedBlockBandedMatrix having eltype T, with block-bandwidths (l,u) and where A[Block(K,J)] is a BandedMatrix{T} of size rows[K]×cols[J] with bandwidths (λ,μ).

BlockBandedMatrices.BlockSkylineMatrixType
BlockSkylineMatrix{T,LL,UU}(M::Union{UndefInitializer,UniformScaling,AbstractMatrix},
rows, cols, (l::LL, u::UU))

returns a sum(rows)×sum(cols) block-banded matrix A having elements of type T, with block-bandwidths (l,u), and where A[Block(K,J)] is a Matrix{T} of size rows[K]×cols[J].

(l,u) may be integers for constant bandwidths, or integer vectors of length length(cols) for ragged bands. In the latter case, l and u represent the number of sub and super-block-bands in each column.

Examples

julia> using LinearAlgebra, FillArrays

julia> BlockSkylineMatrix(I, [2,2,2,4], [1,2,3], ([2,0,1],[0,1,1]))
4×3-blocked 10×6 BlockSkylineMatrix{Bool, Vector{Bool}, BlockBandedMatrices.BlockSkylineSizes{Tuple{BlockArrays.BlockedOneTo{Int64, Vector{Int64}}, BlockArrays.BlockedOneTo{Int64, Vector{Int64}}}, Vector{Int64}, Vector{Int64}, BandedMatrices.BandedMatrix{Int64, Matrix{Int64}, Base.OneTo{Int64}}, Vector{Int64}}}:
1  │  0  0  │  ⋅  ⋅  ⋅
0  │  1  0  │  ⋅  ⋅  ⋅
───┼────────┼─────────
0  │  0  1  │  0  0  0
0  │  0  0  │  1  0  0
───┼────────┼─────────
0  │  ⋅  ⋅  │  0  1  0
0  │  ⋅  ⋅  │  0  0  1
───┼────────┼─────────
⋅  │  ⋅  ⋅  │  0  0  0
⋅  │  ⋅  ⋅  │  0  0  0
⋅  │  ⋅  ⋅  │  0  0  0
⋅  │  ⋅  ⋅  │  0  0  0

julia> BlockSkylineMatrix(Ones(9,6), [2,3,4], [1,2,3], ([2,0,0],[0,1,1]))
3×3-blocked 9×6 BlockSkylineMatrix{Float64, Vector{Float64}, BlockBandedMatrices.BlockSkylineSizes{Tuple{BlockArrays.BlockedOneTo{Int64, Vector{Int64}}, BlockArrays.BlockedOneTo{Int64, Vector{Int64}}}, Vector{Int64}, Vector{Int64}, BandedMatrices.BandedMatrix{Int64, Matrix{Int64}, Base.OneTo{Int64}}, Vector{Int64}}}:
1.0  │  1.0  1.0  │   ⋅    ⋅    ⋅
1.0  │  1.0  1.0  │   ⋅    ⋅    ⋅
─────┼────────────┼───────────────
1.0  │  1.0  1.0  │  1.0  1.0  1.0
1.0  │  1.0  1.0  │  1.0  1.0  1.0
1.0  │  1.0  1.0  │  1.0  1.0  1.0
─────┼────────────┼───────────────
1.0  │   ⋅    ⋅   │  1.0  1.0  1.0
1.0  │   ⋅    ⋅   │  1.0  1.0  1.0
1.0  │   ⋅    ⋅   │  1.0  1.0  1.0
1.0  │   ⋅    ⋅   │  1.0  1.0  1.0

## Accessing block-banded and banded-block-banded matrices

BlockBandedMatrices.subblockbandwidthsFunction
subblockbandwidths(A)

returns the sub-block bandwidths of A, where A is a banded-block-banded matrix. In other words, A[Block(K,J)] will return a BandedMatrix with bandwidths given by subblockbandwidths(A).

BlockBandedMatrices.subblockbandwidthFunction
subblockbandwidth(A, i)

returns the sub-block lower (i == 1) or upper (i == 2) bandwidth of A, where A is a banded-block-banded matrix. In other words, A[Block(K,J)] will return a BandedMatrix with the returned lower/upper bandwidth.

## Implementation

A BlockBandedMatrix stores the entries in a single vector, ordered by columns. For example, if A is a BlockBandedMatrix with block-bandwidths (A.l,A.u) == (1,0) and the block sizes fill(2, N) where N = 3 is the number of row and column blocks, then A has zero structure

[ a_11 a_12 │  ⋅    ⋅
a_21 a_22 │  ⋅    ⋅
──────────┼──────────
a_31 a_32 │ a_33 a_34
a_41 a_42 │ a_43 a_44
──────────┼──────────
⋅    ⋅   │ a_53 a_54
⋅    ⋅   │ a_63 a_64 ]

and is stored in memory via A.data as a single vector by columns, containing:

[a_11,a_21,a_31,a_41,a_12,a_22,a_32,a_42,a_33,a_43,a_53,a_63,a_34,a_44,a_54,a_64]

The reasoning behind this storage scheme as that each block still satisfies the strided matrix interface, but we can also use BLAS and LAPACK to, for example, upper-triangularize a block column all at once.

A BandedBlockBandedMatrix stores the entries as a BlockedMatrix, with the number of row blocks equal to A.l + A.u + 1, and the row block sizes are all A.μ + A.λ + 1. The column block sizes of the storage is the same as the the column block sizes of the BandedBlockBandedMatrix. This is a block-wise version of the storage of BandedMatrix.

For example, if A is a BandedBlockBandedMatrix with block-bandwidths (A.l,A.u) == (1,0) and subblock-bandwidths (A.λ, A.μ) == (1,0), and the block sizes fill(2, N) where N = 3 is the number of row and column blocks, then A has zero structure

[ a_11  ⋅   │  ⋅    ⋅
a_21 a_22 │  ⋅    ⋅
──────────┼──────────
a_31  ⋅   │ a_33  ⋅
a_41 a_42 │ a_43 a_44
──────────┼──────────
⋅    ⋅   │ a_53  ⋅
⋅    ⋅   │ a_63 a_64 ]

and is stored in memory via A.data as a BlockedMatrix, which has block sizes 2 x 2, containing entries:

[a_11 a_22 │ a_33 a_44
a_21  ×   │ a_43  ×
──────────┼──────────
a_31 a_42 │ a_53 a_64
a_41  ×   │ a_63  ×   ]

where × is an entry in memory that is not used.

The reasoning behind this storage scheme as that each block still satisfies the banded matrix interface.