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Adding rtol and atol for pinv and nullspace #29998

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Dec 11, 2018
Merged

Adding rtol and atol for pinv and nullspace #29998

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759afc8
Add code for rtol and atol
Nov 10, 2018
62c9adf
add tests
Nov 11, 2018
278321e
Merge branch 'master' of https://github.com/JuliaLang/julia into pinv…
Nov 20, 2018
82afdf2
resolve comment
Nov 21, 2018
46d3cd6
fix typo
Nov 22, 2018
c3d8fc6
fix tests
Nov 23, 2018
20346b5
Merge branch 'master' of https://github.com/JuliaLang/julia into pinv…
Dec 7, 2018
3e28425
add news.md item
Dec 7, 2018
6dbd52a
Not deprecated yet.
simonbyrne Dec 7, 2018
d90190d
Tweak docs slightly
simonbyrne Dec 7, 2018
6418489
Merge branch 'master' into pinvNull
sam0410 Dec 10, 2018
367c63b
typo from diff
simonbyrne Dec 11, 2018
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Tweak docs slightly
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@simonbyrne
simonbyrne authored Dec 7, 2018
commit d90190d3d44a252150ab43e393d92d582a6cf7e6
25 changes: 14 additions & 11 deletions stdlib/LinearAlgebra/src/dense.jl
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Expand Up @@ -1237,20 +1237,21 @@ factorize(A::Transpose) = transpose(factorize(parent(A)))
## Moore-Penrose pseudoinverse

"""
pinv(M; atol::Real, rtol::Real)
pinv(M; atol::Real=0, rtol::Real=atol>0 ? 0 : n*ϵ)
pinv(M, rtol::Real) = pinv(M; rtol=rtol) # to be deprecated in Julia 2.0

Computes the Moore-Penrose pseudoinverse.

For matrices `M` with floating point elements, it is convenient to compute
the pseudoinverse by inverting only singular values greater than
`max(atol, rtol * maximum(svdvals(M)))`.
`max(atol, rtol*σ₁)` where `σ₁` is the largest singular value of `M`.

The optimal choice of `rtol` varies both with the value of `M` and the intended application
of the pseudoinverse. The default value of `rtol` is
`eps(real(float(one(eltype(M)))))*minimum(size(M))`, which is essentially machine epsilon
for the real part of a matrix element multiplied by the larger matrix dimension. For
inverting dense ill-conditioned matrices in a least-squares sense,
The optimal choice of absolute (`atol`) and relative tolerance (`rtol`) varies
both with the value of `M` and the intended application of the pseudoinverse.
The default relative tolerance is `n*ϵ`, where `n` is the size of the smallest
dimension of `M`, and `ϵ` is the [`eps`](@ref) of the element type of `M`.

For inverting dense ill-conditioned matrices in a least-squares sense,
`rtol = sqrt(eps(real(float(one(eltype(M))))))` is recommended.

For more information, see [^issue8859], [^B96], [^S84], [^KY88].
Expand Down Expand Up @@ -1320,14 +1321,16 @@ end
## Basis for null space

"""
nullspace(M; atol::Real, rtol::Real)
nullspace(M; atol::Real=0, rtol::Rea=atol>0 ? 0 : n*ϵ)
nullspace(M, rtol::Real) = nullspace(M; rtol=rtol) # to be deprecated in Julia 2.0

Computes a basis for the nullspace of `M` by including the singular
vectors of A whose singular have magnitude are greater than `max(atol, rtol*σ₁)`,
where `σ₁` is `A`'s largest singular value. By default, the value of
`rtol` is the smallest dimension of `A` multiplied by the [`eps`](@ref)
of the [`eltype`](@ref) of `A`.
where `σ₁` is `M`'s largest singularvalue.

By default, the relative tolerance `rtol` is `n*ϵ`, where `n`
is the size of the smallest dimension of `M`, and `ϵ` is the [`eps`](@ref) of
the element type of `M`.

# Examples
```jldoctest
Expand Down