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andreasnoack merged 5 commits into from
Apr 3, 2025
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fixed example in readme #21

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6e2ff84
fixed example in readme
ajinkya-k Apr 2, 2025
ee320a8
fixed hermitian example
ajinkya-k Apr 2, 2025
a95f319
using floats everywhere
ajinkya-k Apr 3, 2025
f94b0d2
removed extra spaces
ajinkya-k Apr 3, 2025
b7c1ab7
removed float wrapper
ajinkya-k Apr 3, 2025
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126 changes: 63 additions & 63 deletions README.md
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Original file line number Diff line number Diff line change
Expand Up @@ -13,49 +13,49 @@ For example, starting with a 6 by 6 matrix whose elements are numbered 1 to 36 i
```julia
julia> using LinearAlgebra, RectangularFullPacked

julia> A = reshape(1:36, (6, 6))
6×6 reshape(::UnitRange{Int64}, 6, 6) with eltype Int64:
1 7 13 19 25 31
2 8 14 20 26 32
3 9 15 21 27 33
4 10 16 22 28 34
5 11 17 23 29 35
6 12 18 24 30 36
julia> A = reshape(1.:36., (6, 6))
6×6 reshape(::StepRangeLen{Float64, Base.TwicePrecision{Float64}, Base.TwicePrecision{Float64}, Int64}, 6, 6) with eltype Float64:
1.0 7.0 13.0 19.0 25.0 31.0
2.0 8.0 14.0 20.0 26.0 32.0
3.0 9.0 15.0 21.0 27.0 33.0
4.0 10.0 16.0 22.0 28.0 34.0
5.0 11.0 17.0 23.0 29.0 35.0
6.0 12.0 18.0 24.0 30.0 36.0
```
the lower triangular matrix `AL` is constructed by replacing the elements above the diagonal with zero.
```julia
julia> AL = tril!(collect(A))
6×6 Matrix{Int64}:
1 0 0 0 0 0
2 8 0 0 0 0
3 9 15 0 0 0
4 10 16 22 0 0
5 11 17 23 29 0
6 12 18 24 30 36
```
`AL` requires the same amount of storage as does `A` even though there are only 21 potential non-zeros in `AL`.
6×6 Matrix{Float64}:
1.0 0.0 0.0 0.0 0.0 0.0
2.0 8.0 0.0 0.0 0.0 0.0
3.0 9.0 15.0 0.0 0.0 0.0
4.0 10.0 16.0 22.0 0.0 0.0
5.0 11.0 17.0 23.0 29.0 0.0
6.0 12.0 18.0 24.0 30.0 36.0
```
`AL` requires the same amount of storage as does `A` even though there are only 21 potential non-zeros in `AL`.
The RFP version of the lower triangular matrix
```julia
julia> ArfpL = Int.(TriangularRFP(float.(A), :L))
6×6 Matrix{Int64}:
1 0 0 0 0 0
2 8 0 0 0 0
3 9 15 0 0 0
4 10 16 22 0 0
5 11 17 23 29 0
6 12 18 24 30 36
```julia
julia> ArfpL = TriangularRFP(A, :L)
6×6 TriangularRFP{Float64}:
1.0 0.0 0.0 0.0 0.0 0.0
2.0 8.0 0.0 0.0 0.0 0.0
3.0 9.0 15.0 0.0 0.0 0.0
4.0 10.0 16.0 22.0 0.0 0.0
5.0 11.0 17.0 23.0 29.0 0.0
6.0 12.0 18.0 24.0 30.0 36.0
```
provides the same displayed form but the underlying, "parent" array is 7 by 3
```julia
julia> ALparent = Int.(ArfpL.data)
7×3 Matrix{Int64}:
22 23 24
1 29 30
2 8 36
3 9 15
4 10 16
5 11 17
6 12 18
provides the same displayed form but the underlying, "parent" array is 7 by 3
```julia
julia> ALparent = ArfpL.data
7×3 Matrix{Float64}:
22.0 23.0 24.0
1.0 29.0 30.0
2.0 8.0 36.0
3.0 9.0 15.0
4.0 10.0 16.0
5.0 11.0 17.0
6.0 12.0 18.0
```

The three blocks of `AL` are the lower triangle of `AL[1:3, 1:3]`, stored as the lower triangle of `ALparent[2:4, :]`; the square block `AL[4:6, 1:3]` stored in `ALparent[5:7, :]`; and the lower triangle of `AL[4:6, 4:6]` stored transposed in `ALparent[1:3, :]`.
Expand All @@ -64,50 +64,50 @@ For odd values of n, the parent is of size `(n, div(n + 1, 2))` and the non-zero

For example,
```julia
julia> AL = tril!(collect(reshape(1:25, 5, 5)))
5×5 Matrix{Int64}:
1 0 0 0 0
2 7 0 0 0
3 8 13 0 0
4 9 14 19 0
5 10 15 20 25
julia> AL = tril!(collect(reshape(1.:25., 5, 5)))
5×5 Matrix{Float64}:
1.0 0.0 0.0 0.0 0.0
2.0 7.0 0.0 0.0 0.0
3.0 8.0 13.0 0.0 0.0
4.0 9.0 14.0 19.0 0.0
5.0 10.0 15.0 20.0 25.0

julia> ArfpL = Int.(TriangularRFP(float(AL), :L).data)
5×3 Matrix{Int64}:
1 19 20
2 7 25
3 8 13
4 9 14
5 10 15
julia> ArfpL = TriangularRFP(AL, :L).data
5×3 Matrix{Float64}:
1.0 19.0 20.0
2.0 7.0 25.0
3.0 8.0 13.0
4.0 9.0 14.0
5.0 10.0 15.0
```

RFP storage is especially useful for large positive definite Hermitian matrices because the Cholesky factor can be evaluated nearly as quickly (by applying Level-3 BLAS to the blocks) as in full storage mode but requiring about half the storage.
RFP storage is especially useful for large positive definite Hermitian matrices because the Cholesky factor can be evaluated nearly as quickly (by applying Level-3 BLAS to the blocks) as in full storage mode but requiring about half the storage.

A trivial example is
```julia
julia> A = [2 1 2; 1 2 0; 1 0 2]
3×3 Matrix{Int64}:
2 1 2
1 2 0
1 0 2
A trivial example is
```julia
julia> A = [2. 1 2; 1 2 0; 1 0 2]
3×3 Matrix{Float64}:
2.0 1.0 2.0
1.0 2.0 0.0
1.0 0.0 2.0

julia> cholesky(Hermitian(A, :L))
julia> cholesky(Hermitian(A, :L))
Cholesky{Float64, Matrix{Float64}}
L factor:
3×3 LowerTriangular{Float64, Matrix{Float64}}:
1.41421 ⋅ ⋅
0.707107 1.22474 ⋅
0.707107 -0.408248 1.1547

julia> ArfpL = HermitianRFP(TriangularRFP(float.(A), :L))
julia> ArfpL = Hermitian(TriangularRFP(A, :L))
3×3 HermitianRFP{Float64}:
2.0 1.0 1.0
1.0 2.0 0.0
1.0 0.0 2.0

julia> cholesky!(ArfpL).data
julia> cholesky!(ArfpL).data
3×2 Matrix{Float64}:
1.41421 1.1547
0.707107 1.22474
0.707107 -0.408248
```
```
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