Finite sections of JacobiTriangle Jacobi matrices are ApplyArrays instead of BandedBlockBandedMatrices

[MikaelSlevinsky]

I don't think this used to be the case.

julia> jacobimatrix(Val(1), RectPolynomial(ChebyshevT(), ChebyshevT()))[Block.(1:5), Block.(1:5)]
5×5-blocked 15×15 BandedBlockBandedMatrix{Float64} with block-bandwidths (1, 1) and sub-block-bandwidths block-bandwidths (0, 0) with data 3×5-blocked 3×15 BlockedMatrix{Float64}:
 0.0  │  0.5   ⋅   │   ⋅    ⋅    ⋅   │   ⋅    ⋅    ⋅    ⋅   │   ⋅    ⋅    ⋅    ⋅    ⋅ 
 ─────┼────────────┼─────────────────┼──────────────────────┼─────────────────────────
 1.0  │  0.0   ⋅   │  0.5   ⋅    ⋅   │   ⋅    ⋅    ⋅    ⋅   │   ⋅    ⋅    ⋅    ⋅    ⋅ 
  ⋅   │   ⋅   0.0  │   ⋅   0.5   ⋅   │   ⋅    ⋅    ⋅    ⋅   │   ⋅    ⋅    ⋅    ⋅    ⋅ 
 ─────┼────────────┼─────────────────┼──────────────────────┼─────────────────────────
  ⋅   │  0.5   ⋅   │  0.0   ⋅    ⋅   │  0.5   ⋅    ⋅    ⋅   │   ⋅    ⋅    ⋅    ⋅    ⋅ 
  ⋅   │   ⋅   1.0  │   ⋅   0.0   ⋅   │   ⋅   0.5   ⋅    ⋅   │   ⋅    ⋅    ⋅    ⋅    ⋅ 
  ⋅   │   ⋅    ⋅   │   ⋅    ⋅   0.0  │   ⋅    ⋅   0.5   ⋅   │   ⋅    ⋅    ⋅    ⋅    ⋅ 
 ─────┼────────────┼─────────────────┼──────────────────────┼─────────────────────────
  ⋅   │   ⋅    ⋅   │  0.5   ⋅    ⋅   │  0.0   ⋅    ⋅    ⋅   │  0.5   ⋅    ⋅    ⋅    ⋅ 
  ⋅   │   ⋅    ⋅   │   ⋅   0.5   ⋅   │   ⋅   0.0   ⋅    ⋅   │   ⋅   0.5   ⋅    ⋅    ⋅ 
  ⋅   │   ⋅    ⋅   │   ⋅    ⋅   1.0  │   ⋅    ⋅   0.0   ⋅   │   ⋅    ⋅   0.5   ⋅    ⋅ 
  ⋅   │   ⋅    ⋅   │   ⋅    ⋅    ⋅   │   ⋅    ⋅    ⋅   0.0  │   ⋅    ⋅    ⋅   0.5   ⋅ 
 ─────┼────────────┼─────────────────┼──────────────────────┼─────────────────────────
  ⋅   │   ⋅    ⋅   │   ⋅    ⋅    ⋅   │  0.5   ⋅    ⋅    ⋅   │  0.0   ⋅    ⋅    ⋅    ⋅ 
  ⋅   │   ⋅    ⋅   │   ⋅    ⋅    ⋅   │   ⋅   0.5   ⋅    ⋅   │   ⋅   0.0   ⋅    ⋅    ⋅ 
  ⋅   │   ⋅    ⋅   │   ⋅    ⋅    ⋅   │   ⋅    ⋅   0.5   ⋅   │   ⋅    ⋅   0.0   ⋅    ⋅ 
  ⋅   │   ⋅    ⋅   │   ⋅    ⋅    ⋅   │   ⋅    ⋅    ⋅   1.0  │   ⋅    ⋅    ⋅   0.0   ⋅ 
  ⋅   │   ⋅    ⋅   │   ⋅    ⋅    ⋅   │   ⋅    ⋅    ⋅    ⋅   │   ⋅    ⋅    ⋅    ⋅   0.0

julia> typeof(ans)
BlockBandedMatrices.BandedBlockBandedMatrix{Float64, BlockedMatrix{Float64, Matrix{Float64}, Tuple{BlockedOneTo{Int64, StepRangeLen{Int64, Int64, Int64, Int64}}, BlockedOneTo{Int64, Vector{Int64}}}}, BlockedOneTo{Int64, Vector{Int64}}}

julia> jacobimatrix(Val(1), JacobiTriangle())[Block.(1:5), Block.(1:5)]
(5×5-blocked 15×15 BandedBlockBandedMatrix{Float64} with block-bandwidths (1, 0) and sub-block-bandwidths block-bandwidths (0, 0) with data 2×5-blocked 2×15 BlockVcat{Float64}) * (5×5-blocked 15×15 BandedBlockBandedMatrix{Float64} with block-bandwidths (0, 1) and sub-block-bandwidths block-bandwidths (0, 0) with data 2×5-blocked 2×15 BlockVcat{Float64}) with indices BlockedOneTo([1, 3, 6, 10, 15])×BlockedOneTo([1, 3, 6, 10, 15]):
 0.333333  │  0.166667   ⋅   │   ⋅         ⋅         ⋅        │   ⋅         ⋅         ⋅         ⋅        │   ⋅         ⋅         ⋅         ⋅          ⋅       
 ──────────┼─────────────────┼────────────────────────────────┼──────────────────────────────────────────┼────────────────────────────────────────────────────
 0.333333  │  0.466667   ⋅   │  0.2        ⋅         ⋅        │   ⋅         ⋅         ⋅         ⋅        │   ⋅         ⋅         ⋅         ⋅          ⋅       
  ⋅        │   ⋅        0.2  │   ⋅        0.133333   ⋅        │   ⋅         ⋅         ⋅         ⋅        │   ⋅         ⋅         ⋅         ⋅          ⋅       
 ──────────┼─────────────────┼────────────────────────────────┼──────────────────────────────────────────┼────────────────────────────────────────────────────
  ⋅        │  0.3        ⋅   │  0.485714   ⋅         ⋅        │  0.214286   ⋅         ⋅         ⋅        │   ⋅         ⋅         ⋅         ⋅          ⋅       
  ⋅        │   ⋅        0.2  │   ⋅        0.371429   ⋅        │   ⋅        0.178571   ⋅         ⋅        │   ⋅         ⋅         ⋅         ⋅          ⋅       
  ⋅        │   ⋅         ⋅   │   ⋅         ⋅        0.142857  │   ⋅         ⋅        0.107143   ⋅        │   ⋅         ⋅         ⋅         ⋅          ⋅       
 ──────────┼─────────────────┼────────────────────────────────┼──────────────────────────────────────────┼────────────────────────────────────────────────────
  ⋅        │   ⋅         ⋅   │  0.285714   ⋅         ⋅        │  0.492063   ⋅         ⋅         ⋅        │  0.222222   ⋅         ⋅         ⋅          ⋅       
  ⋅        │   ⋅         ⋅   │   ⋅        0.238095   ⋅        │   ⋅        0.428571   ⋅         ⋅        │   ⋅        0.2        ⋅         ⋅          ⋅       
  ⋅        │   ⋅         ⋅   │   ⋅         ⋅        0.142857  │   ⋅         ⋅        0.301587   ⋅        │   ⋅         ⋅        0.155556   ⋅          ⋅       
  ⋅        │   ⋅         ⋅   │   ⋅         ⋅         ⋅        │   ⋅         ⋅         ⋅        0.111111  │   ⋅         ⋅         ⋅        0.0888889   ⋅       
 ──────────┼─────────────────┼────────────────────────────────┼──────────────────────────────────────────┼────────────────────────────────────────────────────
  ⋅        │   ⋅         ⋅   │   ⋅         ⋅         ⋅        │  0.277778   ⋅         ⋅         ⋅        │  0.494949   ⋅         ⋅         ⋅          ⋅       
  ⋅        │   ⋅         ⋅   │   ⋅         ⋅         ⋅        │   ⋅        0.25       ⋅         ⋅        │   ⋅        0.454545   ⋅         ⋅          ⋅       
  ⋅        │   ⋅         ⋅   │   ⋅         ⋅         ⋅        │   ⋅         ⋅        0.194444   ⋅        │   ⋅         ⋅        0.373737   ⋅          ⋅       
  ⋅        │   ⋅         ⋅   │   ⋅         ⋅         ⋅        │   ⋅         ⋅         ⋅        0.111111  │   ⋅         ⋅         ⋅        0.252525    ⋅       
  ⋅        │   ⋅         ⋅   │   ⋅         ⋅         ⋅        │   ⋅         ⋅         ⋅         ⋅        │   ⋅         ⋅         ⋅         ⋅         0.0909091

julia> typeof(ans)
LazyArrays.ApplyArray{Float64, 2, typeof(*), Tuple{BlockBandedMatrices.BandedBlockBandedMatrix{Float64, LazyBandedMatrices.BlockVcat{Float64, 2, Tuple{Adjoint{Float64, BlockVector{Float64, Vector{Vector{Float64}}, Tuple{BlockedOneTo{Int64, Vector{Int64}}}}}, Adjoint{Float64, BlockVector{Float64, Vector{Vector{Float64}}, Tuple{BlockedOneTo{Int64, Vector{Int64}}}}}}, Tuple{BlockedOneTo{Int64, StaticArraysCore.SVector{2, Int64}}, BlockedOneTo{Int64, Vector{Int64}}}}, BlockedOneTo{Int64, Vector{Int64}}}, BlockBandedMatrices.BandedBlockBandedMatrix{Float64, LazyBandedMatrices.BlockVcat{Float64, 2, Tuple{Adjoint{Float64, BlockVector{Float64, Vector{Vector{Float64}}, Tuple{BlockedOneTo{Int64, Vector{Int64}}}}}, Adjoint{Float64, BlockVector{Float64, Vector{Vector{Float64}}, Tuple{BlockedOneTo{Int64, Vector{Int64}}}}}}, Tuple{BlockedOneTo{Int64, StaticArraysCore.SVector{2, Int64}}, BlockedOneTo{Int64, Vector{Int64}}}}, BlockedOneTo{Int64, Vector{Int64}}}}}

[dlfivefifty]

Does this matter in practice? You can always convert explicitly:

julia> @time BandedBlockBandedMatrix(jacobimatrix(Val(1), JacobiTriangle())[Block.(1:5), Block.(1:5)])
  0.000032 seconds (425 allocations: 23.297 KiB)
5×5-blocked 15×15 BandedBlockBandedMatrix{Float64} with block-bandwidths (1, 1) and sub-block-bandwidths block-bandwidths (0, 0) with data 3×5-blocked 3×15 BlockedMatrix{Float64}:
 0.333333  │  0.166667   ⋅   │   ⋅         ⋅         ⋅        │   ⋅         ⋅         ⋅         ⋅        │   ⋅         ⋅         ⋅         ⋅          ⋅       
 ──────────┼─────────────────┼────────────────────────────────┼──────────────────────────────────────────┼────────────────────────────────────────────────────
 0.333333  │  0.466667   ⋅   │  0.2        ⋅         ⋅        │   ⋅         ⋅         ⋅         ⋅        │   ⋅         ⋅         ⋅         ⋅          ⋅       
  ⋅        │   ⋅        0.2  │   ⋅        0.133333   ⋅        │   ⋅         ⋅         ⋅         ⋅        │   ⋅         ⋅         ⋅         ⋅          ⋅       
 ──────────┼─────────────────┼────────────────────────────────┼──────────────────────────────────────────┼────────────────────────────────────────────────────
  ⋅        │  0.3        ⋅   │  0.485714   ⋅         ⋅        │  0.214286   ⋅         ⋅         ⋅        │   ⋅         ⋅         ⋅         ⋅          ⋅       
  ⋅        │   ⋅        0.2  │   ⋅        0.371429   ⋅        │   ⋅        0.178571   ⋅         ⋅        │   ⋅         ⋅         ⋅         ⋅          ⋅       
  ⋅        │   ⋅         ⋅   │   ⋅         ⋅        0.142857  │   ⋅         ⋅        0.107143   ⋅        │   ⋅         ⋅         ⋅         ⋅          ⋅       
 ──────────┼─────────────────┼────────────────────────────────┼──────────────────────────────────────────┼────────────────────────────────────────────────────
  ⋅        │   ⋅         ⋅   │  0.285714   ⋅         ⋅        │  0.492063   ⋅         ⋅         ⋅        │  0.222222   ⋅         ⋅         ⋅          ⋅       
  ⋅        │   ⋅         ⋅   │   ⋅        0.238095   ⋅        │   ⋅        0.428571   ⋅         ⋅        │   ⋅        0.2        ⋅         ⋅          ⋅       
  ⋅        │   ⋅         ⋅   │   ⋅         ⋅        0.142857  │   ⋅         ⋅        0.301587   ⋅        │   ⋅         ⋅        0.155556   ⋅          ⋅       
  ⋅        │   ⋅         ⋅   │   ⋅         ⋅         ⋅        │   ⋅         ⋅         ⋅        0.111111  │   ⋅         ⋅         ⋅        0.0888889   ⋅       
 ──────────┼─────────────────┼────────────────────────────────┼──────────────────────────────────────────┼────────────────────────────────────────────────────
  ⋅        │   ⋅         ⋅   │   ⋅         ⋅         ⋅        │  0.277778   ⋅         ⋅         ⋅        │  0.494949   ⋅         ⋅         ⋅          ⋅       
  ⋅        │   ⋅         ⋅   │   ⋅         ⋅         ⋅        │   ⋅        0.25       ⋅         ⋅        │   ⋅        0.454545   ⋅         ⋅          ⋅       
  ⋅        │   ⋅         ⋅   │   ⋅         ⋅         ⋅        │   ⋅         ⋅        0.194444   ⋅        │   ⋅         ⋅        0.373737   ⋅          ⋅       
  ⋅        │   ⋅         ⋅   │   ⋅         ⋅         ⋅        │   ⋅         ⋅         ⋅        0.111111  │   ⋅         ⋅         ⋅        0.252525    ⋅       
  ⋅        │   ⋅         ⋅   │   ⋅         ⋅         ⋅        │   ⋅         ⋅         ⋅         ⋅        │   ⋅         ⋅         ⋅         ⋅         0.0909091

[dlfivefifty]

There is a fundamental question that there's ambiguity about when something should be left lazy versus materialised. The change probably came about to support something infinite dimensional, and as no tests failed it was merged.

Without a firm guideline on what the right choice is I'm inclined to not worry about this, unless there's a very argument why it matters.