stable pinv least-squares via rtol,atol in svd, ldiv!

src/dense.jl

@@ -1731,10 +1731,20 @@ 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(oneunit(eltype(M))))))` is recommended.
+For solving dense, ill-conditioned equations in a least-square sense, it
+is better to *not* explicitly form the pseudoinverse matrix, since this
+can lead to numerical instability at low tolerances.  The default `M \\ b`
+algorithm instead uses pivoted QR factorization ([`qr`](@ref)).  To use an
+SVD-based algorithm, it is better to employ the SVD directly via `svd(M; rtol, atol) \\ b`
+or `ldiv!(svd(M), b; rtol, atol)`.
 
-For more information, see [^issue8859], [^B96], [^S84], [^KY88].
+One can also pass `M = svd(A)` as the argument to `pinv` in order to re-use
+an existing [`SVD`](@ref) factorization.
+
+!!! compat "Julia 1.13"
+    Passing an `SVD` object to `pinv` requires Julia 1.13 or later.
+
+For more information, see [^pr1387], [^B96], [^S84], [^KY88].
 
 # Examples
 ```jldoctest
@@ -1754,15 +1764,15 @@ julia> M * N
  4.44089e-16   1.0
 ```
 
-[^issue8859]: Issue 8859, "Fix least squares", [https://github.com/JuliaLang/julia/pull/8859](https://github.com/JuliaLang/julia/pull/8859)
+[^pr1387]: PR 1387, "stable pinv least-squares", [LinearAlgebra.jl#1387](https://github.com/JuliaLang/LinearAlgebra.jl/pull/1387)
 
 [^B96]: Åke Björck, "Numerical Methods for Least Squares Problems",  SIAM Press, Philadelphia, 1996, "Other Titles in Applied Mathematics", Vol. 51. [doi:10.1137/1.9781611971484](http://epubs.siam.org/doi/book/10.1137/1.9781611971484)
 
 [^S84]: G. W. Stewart, "Rank Degeneracy", SIAM Journal on Scientific and Statistical Computing, 5(2), 1984, 403-413. [doi:10.1137/0905030](http://epubs.siam.org/doi/abs/10.1137/0905030)
 
 [^KY88]: Konstantinos Konstantinides and Kung Yao, "Statistical analysis of effective singular values in matrix rank determination", IEEE Transactions on Acoustics, Speech and Signal Processing, 36(5), 1988, 757-763. [doi:10.1109/29.1585](https://doi.org/10.1109/29.1585)
 """
-function pinv(A::AbstractMatrix{T}; atol::Real = 0.0, rtol::Real = (eps(real(float(oneunit(T))))*min(size(A)...))*iszero(atol)) where T
+function pinv(A::AbstractMatrix{T}; atol::Real=0, rtol::Real = (eps(real(float(oneunit(T))))*min(size(A)...))*iszero(atol)) where T
     m, n = size(A)
     Tout = typeof(zero(T)/sqrt(oneunit(T) + oneunit(T)))
     if m == 0 || n == 0
@@ -1830,7 +1840,7 @@ julia> nullspace(M, atol=0.95)
  1.0
 ```
 """
-function nullspace(A::AbstractVecOrMat; atol::Real = 0.0, rtol::Real = (min(size(A, 1), size(A, 2))*eps(real(float(oneunit(eltype(A))))))*iszero(atol))
+function nullspace(A::AbstractVecOrMat; atol::Real=0, rtol::Real = (min(size(A, 1), size(A, 2))*eps(real(float(oneunit(eltype(A))))))*iszero(atol))
     m, n = size(A, 1), size(A, 2)
     (m == 0 || n == 0) && return Matrix{eigtype(eltype(A))}(I, n, n)
     SVD = svd(A; full=true)

src/svd.jl

@@ -92,21 +92,22 @@
 
 
 """
-    svd!(A; full::Bool = false, alg::Algorithm = default_svd_alg(A)) -> SVD
+    svd!(A; full::Bool = false, alg::Algorithm = default_svd_alg(A), atol::Real=0, rtol::Real=0) -> SVD
 
 `svd!` is the same as [`svd`](@ref), but saves space by
 overwriting the input `A`, instead of creating a copy. See documentation of [`svd`](@ref) for details.
 """
-function svd!(A::StridedMatrix{T}; full::Bool = false, alg::Algorithm = default_svd_alg(A)) where {T<:BlasFloat}
+function svd!(A::StridedMatrix{T}; full::Bool = false, alg::Algorithm = default_svd_alg(A), atol::Real=0, rtol::Real=0) where {T<:BlasFloat}
     m, n = size(A)
     if m == 0 || n == 0
         u, s, vt = (Matrix{T}(I, m, full ? m : n), real(zeros(T,0)), Matrix{T}(I, n, n))
     else
         u, s, vt = _svd!(A, full, alg)
     end
-    SVD(u, s, vt)
+    s[_count_svdvals(s, atol, rtol)+1:end] .= 0
+    return SVD(u, s, vt)
 end
-function svd!(A::StridedVector{T}; full::Bool = false, alg::Algorithm = default_svd_alg(A)) where {T<:BlasFloat}
+function svd!(A::StridedVector{T}; full::Bool = false, alg::Algorithm = default_svd_alg(A), atol::Real=0, rtol::Real=0) where {T<:BlasFloat}
     m = length(A)
     normA = norm(A)
     if iszero(normA)
@@ -116,6 +117,7 @@
         return SVD(reshape(A, (m, 1)), [normA], ones(T, 1, 1))
     else
         u, s, vt = _svd!(reshape(A, (m, 1)), full, alg)
+        s[_count_svdvals(s, atol, rtol)+1:end] .= 0
         return SVD(u, s, vt)
     end
 end
@@ -129,10 +131,15 @@
     u, s, vt = LAPACK.gesvd!(c, c, A)
 end
 
-
+# count positive singular values S ≥ given tolerances, S assumed sorted
+function _count_svdvals(S, atol::Real, rtol::Real)
+    isempty(S) && return 0
+    tol = max(rtol * S[1], atol)
+    return iszero(S[1]) ? 0 : searchsortedlast(S, tol, rev=true)
+end
 
 """
-    svd(A; full::Bool = false, alg::Algorithm = default_svd_alg(A)) -> SVD
+    svd(A; full::Bool = false, alg::Algorithm = default_svd_alg(A), atol::Real=0, rtol::Real=0) -> SVD
 
 Compute the singular value decomposition (SVD) of `A` and return an `SVD` object.
 
@@ -151,11 +158,18 @@
 
 `alg` specifies which algorithm and LAPACK method to use for SVD:
 - `alg = LinearAlgebra.DivideAndConquer()` (default): Calls `LAPACK.gesdd!`.
-- `alg = LinearAlgebra.QRIteration()`: Calls `LAPACK.gesvd!` (typically slower but more accurate) .
+- `alg = LinearAlgebra.QRIteration()`: Calls `LAPACK.gesvd!` (typically slower but more accurate).
+
+The `atol` and `rtol` parameters specify optional tolerances to truncate the SVD,
+dropping (setting to zero) singular values less than `max(atol, rtol*σ₁)` where
+`σ₁` is the largest singular value of `A`.
 
 !!! compat "Julia 1.3"
     The `alg` keyword argument requires Julia 1.3 or later.
 
+!!! compat "Julia 1.13"
+    The `atol` and `rtol` arguments require Julia 1.13 or later.
+
 # Examples
 ```jldoctest
 julia> A = rand(4,3);
@@ -176,25 +190,25 @@
 true
 ```
 """
-function svd(A::AbstractVecOrMat{T}; full::Bool = false, alg::Algorithm = default_svd_alg(A)) where {T}
-    svd!(eigencopy_oftype(A, eigtype(T)), full = full, alg = alg)
+function svd(A::AbstractVecOrMat{T}; full::Bool = false, alg::Algorithm = default_svd_alg(A), atol::Real=0, rtol::Real=0) where {T}
+    svd!(eigencopy_oftype(A, eigtype(T)); full, alg, atol, rtol)
 end
-function svd(A::AbstractVecOrMat{T}; full::Bool = false, alg::Algorithm = default_svd_alg(A)) where {T <: Union{Float16,Complex{Float16}}}
-    A = svd!(eigencopy_oftype(A, eigtype(T)), full = full, alg = alg)
+function svd(A::AbstractVecOrMat{T}; full::Bool = false, alg::Algorithm = default_svd_alg(A), atol::Real=0, rtol::Real=0) where {T <: Union{Float16,Complex{Float16}}}
+    A = svd!(eigencopy_oftype(A, eigtype(T)); full, alg, atol, rtol)
     return SVD{T}(A)
 end
-function svd(x::Number; full::Bool = false, alg::Algorithm = default_svd_alg(x))
-    SVD(x == 0 ? fill(one(x), 1, 1) : fill(x/abs(x), 1, 1), [abs(x)], fill(one(x), 1, 1))
+function svd(x::Number; full::Bool = false, alg::Algorithm = default_svd_alg(x), atol::Real=0, rtol::Real=0)
+    SVD(iszero(x) || abs(x) < atol ? fill(one(x), 1, 1) : fill(x/abs(x), 1, 1), [abs(x)], fill(one(x), 1, 1))
 end
-function svd(x::Integer; full::Bool = false, alg::Algorithm = default_svd_alg(x))
-    svd(float(x), full = full, alg = alg)
+function svd(x::Integer; full::Bool = false, alg::Algorithm = default_svd_alg(x), atol::Real=0, rtol::Real=0)
+    svd(float(x); full, alg, atol, rtol)
 end
-function svd(A::Adjoint; full::Bool = false, alg::Algorithm = default_svd_alg(A))
-    s = svd(A.parent, full = full, alg = alg)
+function svd(A::Adjoint; full::Bool = false, alg::Algorithm = default_svd_alg(A), atol::Real=0, rtol::Real=0)
+    s = svd(A.parent; full, alg, atol, rtol)
     return SVD(s.Vt', s.S, s.U')
 end
-function svd(A::Transpose; full::Bool = false, alg::Algorithm = default_svd_alg(A))
-    s = svd(A.parent, full = full, alg = alg)
+function svd(A::Transpose; full::Bool = false, alg::Algorithm = default_svd_alg(A), atol::Real=0, rtol::Real=0)
+    s = svd(A.parent; full, alg, atol, rtol)
     return SVD(transpose(s.Vt), s.S, transpose(s.U))
 end
 
@@ -248,7 +262,7 @@
 """
     rank(S::SVD{<:Any, T}; atol::Real=0, rtol::Real=min(n,m)*ϵ) where {T}
 
-Compute the numerical rank of the SVD object `S` by counting how many singular values are greater 
+Compute the numerical rank of the SVD object `S` by counting how many singular values are greater
 than `max(atol, rtol*σ₁)` where `σ₁` is the largest calculated singular value.
 This is equivalent to the default [`rank(::AbstractMatrix)`](@ref) method except that it re-uses an existing SVD factorization.
 `atol` and `rtol` are the absolute and relative tolerances, respectively.
@@ -258,25 +272,51 @@
 !!! compat "Julia 1.12"
     The `rank(::SVD)` method requires at least Julia 1.12.
 """
-function rank(S::SVD; atol::Real = 0.0, rtol::Real = (min(size(S)...)*eps(real(float(eltype(S))))))
+function rank(S::SVD; atol::Real=0, rtol::Real = (min(size(S)...)*eps(real(float(eltype(S))))))
     tol = max(atol, rtol*S.S[1])
     count(>(tol), S.S)
 end
 
 ### SVD least squares ###
-function ldiv!(A::SVD{T}, B::AbstractVecOrMat) where T
-    m, n = size(A)
-    k = searchsortedlast(A.S, eps(real(T))*A.S[1], rev=true)
-    mul!(view(B, 1:n, :), view(A.Vt, 1:k, :)', view(A.S, 1:k) .\ (view(A.U, :, 1:k)' * _cut_B(B, 1:m)))
+"""
+    ldiv!(F::SVD, B; atol::Real=0, rtol::Real=atol>0 ? 0 : n*ϵ)
+
+Given the SVD `F` of an ``m \\times n`` matrix, multiply the first ``m`` rows of `B` in-place
+by the Moore-Penrose pseudoinverse, storing the result in the first ``n`` rows of `B`, returning `B`.
+This is equivalent to a least-squares solution (for ``m \\ge n``) or a minimum-norm solution (for ``m \\le n``).
+
+Similar to the [`pinv`](@ref) function, the solution can be regularized by truncating the SVD,
+dropping any singular values less than `max(atol, rtol*σ₁)` where `σ₁` is the largest singular value.
+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`.
+
+!!! compat "Julia 1.13"
+    The `atol` and `rtol` arguments require Julia 1.13 or later.
+"""
+function ldiv!(F::SVD{T}, B::AbstractVecOrMat; atol::Real=0, rtol::Real = (eps(real(float(oneunit(T))))*min(size(F)...))*iszero(atol)) where T
+    m, n = size(F)
+    k = _count_svdvals(F.S, atol, rtol)
+    if k == 0
+        B[1:n, :] .= 0
+    else
+        temp = view(F.U, :, 1:k)' * _cut_B(B, 1:m)
+        ldiv!(Diagonal(view(F.S, 1:k)), temp)
+        mul!(view(B, 1:n, :), view(F.Vt, 1:k, :)', temp)
+    end
     return B
 end
 
-function inv(F::SVD{T}) where T
+function pinv(F::SVD{T}; atol::Real=0, rtol::Real = (eps(real(float(oneunit(T))))*min(size(F)...))*iszero(atol)) where T
+    k = _count_svdvals(F.S, atol, rtol)
+    @views (F.S[1:k] .\ F.Vt[1:k, :])' * F.U[:,1:k]'
+end
+
+function inv(F::SVD)
+    # TODO: checksquare(F)
     @inbounds for i in eachindex(F.S)
         iszero(F.S[i]) && throw(SingularException(i))
     end
-    k = searchsortedlast(F.S, eps(real(T))*F.S[1], rev=true)
-    @views (F.S[1:k] .\ F.Vt[1:k, :])' * F.U[:,1:k]'
+    pinv(F; rtol=eps(real(eltype(F))))
 end
 
 size(A::SVD, dim::Integer) = dim == 1 ? size(A.U, dim) : size(A.Vt, dim)

test/svd.jl

@@ -53,6 +53,7 @@ using LinearAlgebra: BlasComplex, BlasFloat, BlasReal, QRPivoted
     @test sf2.U*Diagonal(sf2.S)*sf2.Vt' ≊ m2
 
     @test ldiv!([0., 0.], svd(Matrix(I, 2, 2)), [1., 1.]) ≊ [1., 1.]
+    # @test_throws DimensionMismatch inv(svd(Matrix(I, 3, 2)))
     @test inv(svd(Matrix(I, 2, 2))) ≈ I
     @test inv(svd([1 2; 3 4])) ≈ [-2.0 1.0; 1.5 -0.5]
     @test inv(svd([1 0 1; 0 1 0])) ≈ [0.5 0.0; 0.0 1.0; 0.5 0.0]
@@ -235,7 +236,20 @@ end
     @test Uc * diagm(0=>Sc) * transpose(V) ≈ complex.(A) rtol=1e-3
 end
 
-@testset "Issue 40944. ldiV!(SVD) should update rhs" begin
+@testset "SVD pinv and truncation" begin
+    m, n = 10,5
+    A = randn(m,n) * [1/(i+j-1) for i = 1:n, j=1:n] # badly conditioned Hilbert matrix
+    @test pinv(A) ≈ pinv(svd(A))                                  rtol=1e-13
+    pinv_3 = pinv(A, rtol=1e-3)
+    @test pinv_3 ≈ pinv(svd(A), rtol=1e-3)                        rtol=1e-13
+    @test pinv_3 ≈ pinv(svd(A, rtol=1e-3))                        rtol=1e-13
+    b = float([1:m;]) # arbitrary rhs
+    @test pinv_3 * b ≈ svd(A, rtol=1e-3) \ b                      rtol=1e-13
+    @test pinv_3 * b ≈ ldiv!(svd(A), copy(b), rtol=1e-3)[1:n]     rtol=1e-13
+    @test pinv(A, atol=100) == pinv(svd(A), atol=100) == pinv(svd(A, atol=100)) == zeros(5,10)
+end
+
+@testset "Issue 40944. ldiv!(SVD) should update rhs" begin
     F = svd(randn(2, 2))
     b = randn(2)
     x = ldiv!(F, b)