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Improve Cholesky docstrings #39964

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Improve Cholesky docstrings
nalimilan Mar 9, 2021
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Update stdlib/LinearAlgebra/src/cholesky.jl
nalimilan Mar 10, 2021
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nalimilan Mar 11, 2021
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53 changes: 49 additions & 4 deletions stdlib/LinearAlgebra/src/cholesky.jl
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Expand Up @@ -35,7 +35,10 @@ positive definite matrix `A`. This is the return type of [`cholesky`](@ref),
the corresponding matrix factorization function.

The triangular Cholesky factor can be obtained from the factorization `F::Cholesky`
via `F.L` and `F.U`.
via `F.L` and `F.U`, where `A ≈ F.U' * F.U ≈ F.L * F.L'`.
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@nalimilan nalimilan Mar 9, 2021 •
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Examples use == but I imagine only ≈ holds generally? Or maybe this may not even be exactly the case?


The following functions are available for `Cholesky` objects: [`size`](@ref), [`\\`](@ref),
[`inv`](@ref), [`det`](@ref), [`logdet`](@ref) and [`isposdef`](@ref).

Iterating the decomposition produces the components `L` and `U`.

Expand Down Expand Up @@ -107,7 +110,11 @@ positive semi-definite matrix `A`. This is the return type of [`cholesky(_, Val(
the corresponding matrix factorization function.

The triangular Cholesky factor can be obtained from the factorization `F::CholeskyPivoted`
via `F.L` and `F.U`.
via `F.L` and `F.U`, and the permutation via `F.p`, where `A[F.p, F.p] ≈ F.U' * F.U ≈ F.L * F.L'`,
or alternatively `A ≈ F.U[:, F.p]' * F.U[:, F.p] ≈ F.L[F.p, :] * F.L[F.p, :]'`.

The following functions are available for `CholeskyPivoted` objects:
[`size`](@ref), [`\\`](@ref), [`inv`](@ref), [`det`](@ref), and [`rank`](@ref).

Iterating the decomposition produces the components `L` and `U`.

Expand Down Expand Up @@ -332,7 +339,10 @@ end
Compute the Cholesky factorization of a dense symmetric positive definite matrix `A`
and return a [`Cholesky`](@ref) factorization. The matrix `A` can either be a [`Symmetric`](@ref) or [`Hermitian`](@ref)
[`StridedMatrix`](@ref) or a *perfectly* symmetric or Hermitian `StridedMatrix`.
The triangular Cholesky factor can be obtained from the factorization `F` with: `F.L` and `F.U`.

The triangular Cholesky factor can be obtained from the factorization `F` via `F.L` and `F.U`,
where `A ≈ F.U' * F.U ≈ F.L * F.L'`.

The following functions are available for `Cholesky` objects: [`size`](@ref), [`\\`](@ref),
[`inv`](@ref), [`det`](@ref), [`logdet`](@ref) and [`isposdef`](@ref).

Expand Down Expand Up @@ -386,9 +396,14 @@ cholesky(A::Union{StridedMatrix,RealHermSymComplexHerm{<:Real,<:StridedMatrix}},
Compute the pivoted Cholesky factorization of a dense symmetric positive semi-definite matrix `A`
and return a [`CholeskyPivoted`](@ref) factorization. The matrix `A` can either be a [`Symmetric`](@ref)
or [`Hermitian`](@ref) [`StridedMatrix`](@ref) or a *perfectly* symmetric or Hermitian `StridedMatrix`.
The triangular Cholesky factor can be obtained from the factorization `F` with: `F.L` and `F.U`.

The triangular Cholesky factor can be obtained from the factorization `F` via `F.L` and `F.U`,
and the permutation via `F.p`, where `A[F.p, F.p] ≈ F.U' * F.U ≈ F.L * F.L'`, or alternatively
`A ≈ F.U[:, F.p]' * F.U[:, F.p] ≈ F.L[F.p, :] * F.L[F.p, :]'`.

The following functions are available for `CholeskyPivoted` objects:
[`size`](@ref), [`\\`](@ref), [`inv`](@ref), [`det`](@ref), and [`rank`](@ref).

The argument `tol` determines the tolerance for determining the rank.
For negative values, the tolerance is the machine precision.

Expand All @@ -398,6 +413,36 @@ wrap it in `Hermitian(A)` before passing it to `cholesky` in order to treat it a
When `check = true`, an error is thrown if the decomposition fails.
When `check = false`, responsibility for checking the decomposition's
validity (via [`issuccess`](@ref)) lies with the user.

# Examples
```jldoctest
julia> A = [4. 12. -16.; 12. 37. -43.; -16. -43. 98.]
3×3 Matrix{Float64}:
4.0 12.0 -16.0
12.0 37.0 -43.0
-16.0 -43.0 98.0

julia> C = cholesky(A, Val(true))
CholeskyPivoted{Float64, Matrix{Float64}}
U factor with rank 3:
3×3 UpperTriangular{Float64, Matrix{Float64}}:
9.89949 -4.34366 -1.61624
⋅ 4.25825 1.1694
⋅ ⋅ 0.142334
permutation:
3-element Vector{Int64}:
3
2
1

julia> C.U[:, C.p]' * C.U[:, C.p] ≈ A
true

julia> l, u = C; # destructuring via iteration

julia> l == C.L && u == C.U
true
```
"""
cholesky(A::Union{StridedMatrix,RealHermSymComplexHerm{<:Real,<:StridedMatrix}},
::Val{true}; tol = 0.0, check::Bool = true) =
Expand Down