Refactor changebasis_I

[hyrodium]

Before this PR

julia> using BasicBSpline, BenchmarkTools

julia> p = 3
3

julia> P = BSplineSpace{p}(KnotVector(1:10))
BSplineSpace{3, Int64, KnotVector{Int64}}(KnotVector([1, 2, 3, 4, 5, 6, 7, 8, 9, 10]))

julia> P′ = BSplineSpace{p}(KnotVector([1,2,3,4,4.5,5,6,7,8.1,9.1,10.5]))
BSplineSpace{3, Float64, KnotVector{Float64}}(KnotVector([1.0, 2.0, 3.0, 4.0, 4.5, 5.0, 6.0, 7.0, 8.1, 9.1, 10.5]))

julia> changebasis(P,P′)
6×7 SparseArrays.SparseMatrixCSC{Float64, Int32} with 13 stored entries:
 1.0  0.166667   ⋅    ⋅         ⋅     ⋅           ⋅ 
  ⋅   0.833333  0.5   ⋅         ⋅     ⋅           ⋅ 
  ⋅    ⋅        0.5  0.833333   ⋅     ⋅           ⋅ 
  ⋅    ⋅         ⋅   0.166667  1.0  -0.0333333   0.0183333
  ⋅    ⋅         ⋅    ⋅         ⋅    1.03333    -0.103333
  ⋅    ⋅         ⋅    ⋅         ⋅     ⋅          1.085

julia> @benchmark changebasis(P,P′)
BenchmarkTools.Trial: 10000 samples with 1 evaluation per sample.
 Range (min … max):  17.111 μs …  17.679 ms  ┊ GC (min … max): 0.00% … 99.49%
 Time  (median):     19.486 μs               ┊ GC (median):    0.00%
 Time  (mean ± σ):   22.863 μs ± 176.635 μs  ┊ GC (mean ± σ):  7.69% ±  0.99%

      ▄█▄                                                       
  ▁▁▄████▆▅▃▃▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▁▂▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁ ▂
  17.1 μs         Histogram: frequency by time         39.6 μs <

 Memory estimate: 31.19 KiB, allocs estimate: 604.

After this PR

julia> using BasicBSpline, BenchmarkTools

julia> p = 3
3

julia> P = BSplineSpace{p}(KnotVector(1:10))
BSplineSpace{3, Int64, KnotVector{Int64}}(KnotVector([1, 2, 3, 4, 5, 6, 7, 8, 9, 10]))

julia> P′ = BSplineSpace{p}(KnotVector([1,2,3,4,4.5,5,6,7,8.1,9.1,10.5]))
BSplineSpace{3, Float64, KnotVector{Float64}}(KnotVector([1.0, 2.0, 3.0, 4.0, 4.5, 5.0, 6.0, 7.0, 8.1, 9.1, 10.5]))

julia> changebasis(P,P′)
6×7 SparseArrays.SparseMatrixCSC{Float64, Int32} with 13 stored entries:
 1.0  0.166667   ⋅    ⋅         ⋅     ⋅           ⋅ 
  ⋅   0.833333  0.5   ⋅         ⋅     ⋅           ⋅ 
  ⋅    ⋅        0.5  0.833333   ⋅     ⋅           ⋅ 
  ⋅    ⋅         ⋅   0.166667  1.0  -0.0333333   0.0183333
  ⋅    ⋅         ⋅    ⋅         ⋅    1.03333    -0.103333
  ⋅    ⋅         ⋅    ⋅         ⋅     ⋅          1.085

julia> @benchmark changebasis(P,P′)
BenchmarkTools.Trial: 10000 samples with 7 evaluations per sample.
 Range (min … max):  4.190 μs …  2.114 ms  ┊ GC (min … max): 0.00% … 99.34%
 Time  (median):     4.500 μs              ┊ GC (median):    0.00%
 Time  (mean ± σ):   5.036 μs ± 25.724 μs  ┊ GC (mean ± σ):  7.07% ±  1.40%

    ▃▂▆▆█▇██▃▆▂▄ ▂▂ ▁ ▁  ▁ ▁ ▂▁▁▂▁▂ ▂▁▁▁ ▂ ▁                 ▂
  ▆▇███████████████▇█▆█▅██▇█▇████████████████▇▇▆▆▇▆▅▆▇▆▅▅▆▅▅ █
  4.19 μs      Histogram: log(frequency) by time     6.43 μs <

 Memory estimate: 5.94 KiB, allocs estimate: 112.

x-ref: #323 (comment)

[codecov]

Codecov Report

✅ All modified and coverable lines are covered by tests.
✅ Project coverage is 94.58%. Comparing base (e173f9b) to head (4b7a2c1).
⚠️ Report is 2 commits behind head on main.

Additional details and impacted files

@@            Coverage Diff             @@
##             main     #399      +/-   ##
==========================================
- Coverage   95.11%   94.58%   -0.53%     
==========================================
  Files          14       14              
  Lines        2066     2086      +20     
==========================================
+ Hits         1965     1973       +8     
- Misses        101      113      +12     

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[hyrodium]

Note that this PR introduces 0.0 value element. (may be fixed in the future PR)

Before this PR

julia> P1 = BSplineSpace{2}(KnotVector([-3, -3, -2, 1, 2, 2, 2]))
BSplineSpace{2, Int64, KnotVector{Int64}}(KnotVector([-3, -3, -2, 1, 2, 2, 2]))

julia> P2 = BSplineSpace{4}(KnotVector([-5, -4, -2, -2, -2, 1, 1, 1, 2, 2, 2, 2, 2]))
BSplineSpace{4, Int64, KnotVector{Int64}}(KnotVector([-5, -4, -2, -2, -2, 1, 1, 1, 2, 2, 2, 2, 2]))

julia> changebasis_I(P1, P2)
4×8 SparseArrays.SparseMatrixCSC{Float64, Int32} with 16 stored entries:
 1.0  0.375  0.125   ⋅      ⋅      ⋅          ⋅    ⋅ 
  ⋅   0.625  0.75   0.625  0.125  0.0416667   ⋅    ⋅ 
  ⋅    ⋅     0.125  0.375  0.875  0.791667   0.5   ⋅ 
  ⋅    ⋅      ⋅      ⋅      ⋅     0.166667   0.5  1.0

After this PR

julia> P1 = BSplineSpace{2}(KnotVector([-3, -3, -2, 1, 2, 2, 2]))
BSplineSpace{2, Int64, KnotVector{Int64}}(KnotVector([-3, -3, -2, 1, 2, 2, 2]))

julia> P2 = BSplineSpace{4}(KnotVector([-5, -4, -2, -2, -2, 1, 1, 1, 2, 2, 2, 2, 2]))
BSplineSpace{4, Int64, KnotVector{Int64}}(KnotVector([-5, -4, -2, -2, -2, 1, 1, 1, 2, 2, 2, 2, 2]))

julia> changebasis_I(P1, P2)
4×8 SparseArrays.SparseMatrixCSC{Float64, Int32} with 17 stored entries:
 1.0  0.375  0.125   ⋅      ⋅      ⋅          ⋅    ⋅ 
 0.0  0.625  0.75   0.625  0.125  0.0416667   ⋅    ⋅ 
  ⋅    ⋅     0.125  0.375  0.875  0.791667   0.5   ⋅ 
  ⋅    ⋅      ⋅      ⋅      ⋅     0.166667   0.5  1.0