alg keyword for svd and svd! (#31057)

* alg keyword for LinearAlgebra.svd

* SVDAlgorithms -> Algorithms

* default_svd_alg

* refined docstring

Co-Authored-By: Andreas Noack <[email protected]>

* rename to QRIteration; _svd! dispatch

* compat annotation

Author: carstenbauer

NEWS.md

@@ -74,6 +74,7 @@ Standard library changes
 * The BLAS submodule no longer exports `dot`, which conflicts with that in LinearAlgebra ([#31838]).
 * `diagm` and `spdiagm` now accept optional `m,n` initial arguments to specify a size ([#31654]).
 * `Hessenberg` factorizations `H` now support efficient shifted solves `(H+µI) \ b` and determinants, and use a specialized tridiagonal factorization for Hermitian matrices. There is also a new `UpperHessenberg` matrix type ([#31853]).
+* Added keyword argument `alg` to `svd` and `svd!` that allows one to switch between different SVD algorithms ([#31057]).
 * Five-argument `mul!(C, A, B, α, β)` now implements inplace multiplication fused with addition _C = A B α + C β_ ([#23919]).
 
 #### SparseArrays

stdlib/LinearAlgebra/src/LinearAlgebra.jl

@@ -156,6 +156,12 @@ else
     const BlasInt = Int32
 end
 
+
+abstract type Algorithm end
+struct DivideAndConquer <: Algorithm end
+struct QRIteration <: Algorithm end
+
+
 # Check that stride of matrix/vector is 1
 # Writing like this to avoid splatting penalty when called with multiple arguments,
 # see PR 16416

stdlib/LinearAlgebra/src/bidiag.jl

@@ -200,8 +200,8 @@ function svd!(M::Bidiagonal{<:BlasReal}; full::Bool = false)
     d, e, U, Vt, Q, iQ = LAPACK.bdsdc!(M.uplo, 'I', M.dv, M.ev)
     SVD(U, d, Vt)
 end
-function svd(M::Bidiagonal; full::Bool = false)
-    svd!(copy(M), full = full)
+function svd(M::Bidiagonal; kw...)
+    svd!(copy(M), kw...)
 end
 
 ####################

stdlib/LinearAlgebra/src/svd.jl

@@ -54,14 +54,19 @@ function SVD{T}(U::AbstractArray, S::AbstractVector{Tr}, Vt::AbstractArray) wher
         convert(AbstractArray{T}, Vt))
 end
 
+
 # iteration for destructuring into components
 Base.iterate(S::SVD) = (S.U, Val(:S))
 Base.iterate(S::SVD, ::Val{:S}) = (S.S, Val(:V))
 Base.iterate(S::SVD, ::Val{:V}) = (S.V, Val(:done))
 Base.iterate(S::SVD, ::Val{:done}) = nothing
 
+
+default_svd_alg(A) = DivideAndConquer()
+
+
 """
-    svd!(A; full::Bool = false) -> SVD
+    svd!(A; full::Bool = false, alg::Algorithm = default_svd_alg(A)) -> SVD
 
 `svd!` is the same as [`svd`](@ref), but saves space by
 overwriting the input `A`, instead of creating a copy.
@@ -92,18 +97,28 @@ julia> A
   0.0       0.0  -2.0  0.0  0.0
 ```
 """
-function svd!(A::StridedMatrix{T}; full::Bool = false) where T<:BlasFloat
+function svd!(A::StridedMatrix{T}; full::Bool = false, alg::Algorithm = default_svd_alg(A)) 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 = LAPACK.gesdd!(full ? 'A' : 'S', A)
+        u,s,vt = _svd!(A,full,alg)
     end
     SVD(u,s,vt)
 end
 
+
+_svd!(A::StridedMatrix{T}, full::Bool, alg::Algorithm) where T<:BlasFloat = throw(ArgumentError("Unsupported value for `alg` keyword."))
+_svd!(A::StridedMatrix{T}, full::Bool, alg::DivideAndConquer) where T<:BlasFloat = LAPACK.gesdd!(full ? 'A' : 'S', A)
+function _svd!(A::StridedMatrix{T}, full::Bool, alg::QRIteration) where T<:BlasFloat
+    c = full ? 'A' : 'S'
+    u,s,vt = LAPACK.gesvd!(c, c, A)
+end
+
+
+
 """
-    svd(A; full::Bool = false) -> SVD
+    svd(A; full::Bool = false, alg::Algorithm = default_svd_alg(A)) -> SVD
 
 Compute the singular value decomposition (SVD) of `A` and return an `SVD` object.
 
@@ -120,6 +135,12 @@ and `V` is `N \\times N`, while in the thin factorization `U` is `M
 \\times K` and `V` is `N \\times K`, where `K = \\min(M,N)` is the
 number of singular values.
 
+If `alg = DivideAndConquer()` a divide-and-conquer algorithm is used to calculate the SVD.
+Another (typically slower but more accurate) option is `alg = QRIteration()`.
+
+!!! compat "Julia 1.3"
+    The `alg` keyword argument requires Julia 1.3 or later.
+
 # Examples
 ```jldoctest
 julia> A = [1. 0. 0. 0. 2.; 0. 0. 3. 0. 0.; 0. 0. 0. 0. 0.; 0. 2. 0. 0. 0.]
@@ -144,21 +165,21 @@ julia> u == F.U && s == F.S && v == F.V
 true
 ```
 """
-function svd(A::StridedVecOrMat{T}; full::Bool = false) where T
-    svd!(copy_oftype(A, eigtype(T)), full = full)
+function svd(A::StridedVecOrMat{T}; full::Bool = false, alg::Algorithm = default_svd_alg(A)) where T
+    svd!(copy_oftype(A, eigtype(T)), full = full, alg = alg)
 end
-function svd(x::Number; full::Bool = false)
+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))
 end
-function svd(x::Integer; full::Bool = false)
-    svd(float(x), full = full)
+function svd(x::Integer; full::Bool = false, alg::Algorithm = default_svd_alg(x))
+    svd(float(x), full = full, alg = alg)
 end
-function svd(A::Adjoint; full::Bool = false)
-    s = svd(A.parent, full = full)
+function svd(A::Adjoint; full::Bool = false, alg::Algorithm = default_svd_alg(A))
+    s = svd(A.parent, full = full, alg = alg)
     return SVD(s.Vt', s.S, s.U')
 end
-function svd(A::Transpose; full::Bool = false)
-    s = svd(A.parent, full = full)
+function svd(A::Transpose; full::Bool = false, alg::Algorithm = default_svd_alg(A))
+    s = svd(A.parent, full = full, alg = alg)
     return SVD(transpose(s.Vt), s.S, transpose(s.U))
 end
 

stdlib/LinearAlgebra/src/triangular.jl

@@ -2532,7 +2532,7 @@ eigen(A::AbstractTriangular) = Eigen(eigvals(A), eigvecs(A))
 # Generic singular systems
 for func in (:svd, :svd!, :svdvals)
     @eval begin
-        ($func)(A::AbstractTriangular) = ($func)(copyto!(similar(parent(A)), A))
+        ($func)(A::AbstractTriangular; kwargs...) = ($func)(copyto!(similar(parent(A)), A); kwargs...)
     end
 end
 

stdlib/LinearAlgebra/test/svd.jl

@@ -143,4 +143,28 @@ aimg  = randn(n,n)/2
     end
 end
 
+
+
+@testset "SVD Algorithms" begin
+    ≊(x,y) = isapprox(x,y,rtol=1e-15)
+
+    x = [0.1 0.2; 0.3 0.4]
+
+    for alg in [LinearAlgebra.QRIteration(), LinearAlgebra.DivideAndConquer()]
+        sx1 = svd(x, alg = alg)
+        @test sx1.U * Diagonal(sx1.S) * sx1.Vt ≊ x
+        @test sx1.V * sx1.Vt ≊ I
+        @test sx1.U * sx1.U' ≊ I
+        @test all(sx1.S .≥ 0)
+
+        sx2 = svd!(copy(x), alg = alg)
+        @test sx2.U * Diagonal(sx2.S) * sx2.Vt ≊ x
+        @test sx2.V * sx2.Vt ≊ I
+        @test sx2.U * sx2.U' ≊ I
+        @test all(sx2.S .≥ 0)
+    end
+end
+
+
+
 end # module TestSVD