diff --git a/dev/.documenter-siteinfo.json b/dev/.documenter-siteinfo.json index 8618767..e87e979 100644 --- a/dev/.documenter-siteinfo.json +++ b/dev/.documenter-siteinfo.json @@ -1 +1 @@ -{"documenter":{"julia_version":"1.10.2","generation_timestamp":"2024-04-18T06:11:55","documenter_version":"1.3.0"}} \ No newline at end of file +{"documenter":{"julia_version":"1.10.2","generation_timestamp":"2024-04-18T18:20:16","documenter_version":"1.3.0"}} \ No newline at end of file diff --git a/dev/index.html b/dev/index.html index c31f0c5..71dcb2f 100644 --- a/dev/index.html +++ b/dev/index.html @@ -1,5 +1,5 @@ -Home · BlockBandedMatrices.jl
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BlockBandedMatrices.jl Documentation

Creating block-banded and banded-block-banded matrices

BlockBandedMatrices.BlockBandedMatrix — Method
BlockBandedMatrix(A::Union{AbstractMatrix,UniformScaling},
+Home · BlockBandedMatrices.jl
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BlockBandedMatrices.jl Documentation
Creating block-banded and banded-block-banded matrices
BlockBandedMatrices.BlockBandedMatrix — Method
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
 
@@ -29,7 +29,7 @@
  ─────┼────────────┼───────────────
   ⋅   │   ⋅    ⋅   │  1.0  1.0  1.0
   ⋅   │   ⋅    ⋅   │  1.0  1.0  1.0
-  ⋅   │   ⋅    ⋅   │  1.0  1.0  1.0
source
BlockBandedMatrices.BlockBandedMatrix — Method
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
+  ⋅   │   ⋅    ⋅   │  1.0  1.0  1.0
source
BlockBandedMatrices.BlockBandedMatrix — Method
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);
 
@@ -42,8 +42,8 @@
  ─────┼────────────┼───────────────
   ⋅   │  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
source
BlockBandedMatrices.BlockBandedMatrix — Method
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.
source
BlockBandedMatrices.BlockBandedMatrix — Method
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.
source
BlockBandedMatrices.BandedBlockBandedMatrix — Type
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))
@@ -74,7 +74,7 @@
  ────────────┼─────────────────┼────────────
  ⋅  ⋅  ⋅  ⋅  │  1  1  ⋅  ⋅  ⋅  │  1  1  ⋅  ⋅
  ⋅  ⋅  ⋅  ⋅  │  1  1  1  ⋅  ⋅  │  1  1  1  ⋅
- ⋅  ⋅  ⋅  ⋅  │  ⋅  1  1  1  ⋅  │  ⋅  1  1  1
source
BlockBandedMatrices.BandedBlockBandedMatrix — Method
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).
source
BlockBandedMatrices.BandedBlockBandedMatrix — Method
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).
source
BlockBandedMatrices.BlockSkylineMatrix — Type
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]))
@@ -105,7 +105,7 @@
  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
source
Accessing block-banded and banded-block-banded matrices
BlockBandedMatrices.subblockbandwidths — Function
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).
source
BlockBandedMatrices.subblockbandwidth — Function
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.
source
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 │  ⋅    ⋅
+ 1.0  │   ⋅    ⋅   │  1.0  1.0  1.0
source

Accessing block-banded and banded-block-banded matrices

BlockBandedMatrices.subblockbandwidths — Function
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).

source
BlockBandedMatrices.subblockbandwidth — Function
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.

source

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
@@ -123,4 +123,4 @@
  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

+ 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 — Type
AbstractBlockBandedLayout

isa a MemoryLayout that indicates that the array implements the block-banded interface.

source
BlockBandedMatrices.AbstractBandedBlockBandedLayout — Type
AbstractBandedBlockBandedLayout

isa a MemoryLayout that indicates that the array implements the banded-block-banded interface.

source