BlockBandedMatrices.jl Documentation
Creating block-banded and banded-block-banded matrices
BlockBandedMatrices.BlockBandedMatrix — TypeBlockBandedMatrixA BlockBandedMatrix is a subtype of BlockMatrix of BlockArrays.jl whose layout of non-zero blocks is banded.
BlockBandedMatrices.BlockBandedMatrix — MethodBlockBandedMatrix(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 1.0BlockBandedMatrices.BlockBandedMatrix — MethodBlockBandedMatrix(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 1.0BlockBandedMatrices.BlockBandedMatrix — MethodBlockBandedMatrix{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.BandedBlockBandedMatrix — TypeBandedBlockBandedMatrix(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 1BlockBandedMatrices.BandedBlockBandedMatrix — MethodBandedBlockBandedMatrix{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.BlockSkylineMatrix — TypeBlockSkylineMatrix{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.BlockedUnitRange{Vector{Int64}}, BlockArrays.BlockedUnitRange{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.BlockedUnitRange{Vector{Int64}}, BlockArrays.BlockedUnitRange{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 1.0Accessing block-banded and banded-block-banded matrices
BlockBandedMatrices.isblockbanded — Functionisblockbanded(A)returns true if a matrix implements the block banded interface.
BlockBandedMatrices.blockbandwidths — Functionblockbandwidths(A)Returns a tuple containing the upper and lower blockbandwidth of A.
BlockBandedMatrices.blockbandwidth — Functionblockbandwidth(A,i)Returns the lower blockbandwidth (i==1) or the upper blockbandwidth (i==2).
BlockBandedMatrices.blockbandrange — Functionblockbandrange(A)Returns the range -blockbandwidth(A,1):blockbandwidth(A,2).
BlockBandedMatrices.subblockbandwidths — Functionsubblockbandwidths(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.subblockbandwidth — Functionsubblockbandwidth(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 PseudoBlockMatrix, 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 PseudoBlockMatrix, 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.
Layout
BlockBandedMatrices.AbstractBlockBandedLayout — TypeAbstractBlockBandedLayoutisa a MemoryLayout that indicates that the array implements the block-banded interface.
BlockBandedMatrices.AbstractBandedBlockBandedLayout — TypeAbstractBandedBlockBandedLayoutisa a MemoryLayout that indicates that the array implements the banded-block-banded interface.