Add interface to use symmetric eigendecomposition with different lapack method

NEWS.md

@@ -88,6 +88,9 @@ Standard library changes
 #### LinearAlgebra
 
 * `rank` can now take a `QRPivoted` matrix to allow rank estimation via QR factorization ([#54283]).
+* Added keyword argument `alg` to `eigen`, `eigen!`, `eigvals` and `eigvals!` for self-adjoint
+  matrix types (i.e., the type union `RealHermSymComplexHerm`) that allows one to switch
+  between different eigendecomposition algorithms ([#49355]).
 
 #### Logging
 

stdlib/LinearAlgebra/src/LinearAlgebra.jl

@@ -190,6 +190,7 @@ end
 abstract type Algorithm end
 struct DivideAndConquer <: Algorithm end
 struct QRIteration <: Algorithm end
+struct RobustRepresentations <: Algorithm end
 
 # Pivoting strategies for matrix factorization algorithms.
 abstract type PivotingStrategy end

stdlib/LinearAlgebra/src/lapack.jl

@@ -5762,11 +5762,6 @@ syevr!(jobz::AbstractChar, range::AbstractChar, uplo::AbstractChar, A::AbstractM
 Finds the eigenvalues (`jobz = N`) or eigenvalues and eigenvectors
 (`jobz = V`) of a symmetric matrix `A`. If `uplo = U`, the upper triangle
 of `A` is used. If `uplo = L`, the lower triangle of `A` is used.
-
-Use the divide-and-conquer method, instead of the QR iteration used by
-`syev!` or multiple relatively robust representations used by `syevr!`.
-See James W. Demmel et al, SIAM J. Sci. Comput. 30, 3, 1508 (2008) for
-a comparison of the accuracy and performatce of different methods.
 """
 syevd!(jobz::AbstractChar, uplo::AbstractChar, A::AbstractMatrix)
 

stdlib/LinearAlgebra/src/svd.jl

@@ -149,8 +149,9 @@ 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()`.
+`alg` specifies which algorithm and LAPACK method to use for SVD:
+- `alg = DivideAndConquer()` (default): Calls `LAPACK.gesdd!`.
+- `alg = QRIteration()`: Calls `LAPACK.gesvd!` (typically slower but more accurate) .
 
 !!! compat "Julia 1.3"
     The `alg` keyword argument requires Julia 1.3 or later.

stdlib/LinearAlgebra/src/symmetriceigen.jl

@@ -5,13 +5,19 @@
 eigencopy_oftype(A::Hermitian, S) = Hermitian(copytrito!(similar(parent(A), S, size(A)), A.data, A.uplo), sym_uplo(A.uplo))
 eigencopy_oftype(A::Symmetric, S) = Symmetric(copytrito!(similar(parent(A), S, size(A)), A.data, A.uplo), sym_uplo(A.uplo))
 
-# Eigensolvers for symmetric and Hermitian matrices
-eigen!(A::RealHermSymComplexHerm{<:BlasReal,<:StridedMatrix}; sortby::Union{Function,Nothing}=nothing) =
-    Eigen(sorteig!(LAPACK.syevr!('V', 'A', A.uplo, A.data, 0.0, 0.0, 0, 0, -1.0)..., sortby)...)
+default_eigen_alg(A) = DivideAndConquer()
 
-function eigen(A::RealHermSymComplexHerm; sortby::Union{Function,Nothing}=nothing)
-    S = eigtype(eltype(A))
-    eigen!(eigencopy_oftype(A, S), sortby=sortby)
+# Eigensolvers for symmetric and Hermitian matrices
+function eigen!(A::RealHermSymComplexHerm{<:BlasReal,<:StridedMatrix}, alg::Algorithm = default_eigen_alg(A); sortby::Union{Function,Nothing}=nothing)
+    if alg === DivideAndConquer()
+        Eigen(sorteig!(LAPACK.syevd!('V', A.uplo, A.data)..., sortby)...)
+    elseif alg === QRIteration()
+        Eigen(sorteig!(LAPACK.syev!('V', A.uplo, A.data)..., sortby)...)
+    elseif alg === RobustRepresentations()
+        Eigen(sorteig!(LAPACK.syevr!('V', 'A', A.uplo, A.data, 0.0, 0.0, 0, 0, -1.0)..., sortby)...)
+    else
+        throw(ArgumentError("Unsupported value for `alg` keyword."))
+    end
 end
 function eigen(A::RealHermSymComplexHerm{Float16}; sortby::Union{Function,Nothing}=nothing)
     S = eigtype(eltype(A))
@@ -21,6 +27,36 @@ function eigen(A::RealHermSymComplexHerm{Float16}; sortby::Union{Function,Nothin
     return Eigen(values, vectors)
 end
 
+"""
+    eigen(A::Union{Hermitian, Symmetric}, alg::Algorithm = default_eigen_alg(A)) -> Eigen
+
+Compute the eigenvalue decomposition of `A`, returning an [`Eigen`](@ref) factorization object `F`
+which contains the eigenvalues in `F.values` and the eigenvectors in the columns of the
+matrix `F.vectors`. (The `k`th eigenvector can be obtained from the slice `F.vectors[:, k]`.)
+
+Iterating the decomposition produces the components `F.values` and `F.vectors`.
+
+`alg` specifies which algorithm and LAPACK method to use for eigenvalue decomposition:
+- `alg = DivideAndConquer()` (default): Calls `LAPACK.syevd!`.
+- `alg = QRIteration()`: Calls `LAPACK.syev!`.
+- `alg = RobustRepresentations()`: Multiple relatively robust representations method, Calls `LAPACK.syevr!`.
+
+See James W. Demmel et al, SIAM J. Sci. Comput. 30, 3, 1508 (2008) for
+a comparison of the accuracy and performance of different algorithms.
+
+The default `alg` used may change in the future.
+
+!!! compat "Julia 1.12"
+    The `alg` keyword argument requires Julia 1.12 or later.
+
+The following functions are available for `Eigen` objects: [`inv`](@ref), [`det`](@ref), and [`isposdef`](@ref).
+"""
+function eigen(A::RealHermSymComplexHerm, alg::Algorithm = default_eigen_alg(A); sortby::Union{Function,Nothing}=nothing)
+    S = eigtype(eltype(A))
+    eigen!(eigencopy_oftype(A, S), alg; sortby)
+end
+
+
 eigen!(A::RealHermSymComplexHerm{<:BlasReal,<:StridedMatrix}, irange::UnitRange) =
     Eigen(LAPACK.syevr!('V', 'I', A.uplo, A.data, 0.0, 0.0, irange.start, irange.stop, -1.0)...)
 
@@ -71,17 +107,42 @@ function eigen(A::RealHermSymComplexHerm, vl::Real, vh::Real)
     eigen!(eigencopy_oftype(A, S), vl, vh)
 end
 
-function eigvals!(A::RealHermSymComplexHerm{<:BlasReal,<:StridedMatrix}; sortby::Union{Function,Nothing}=nothing)
-    vals = LAPACK.syevr!('N', 'A', A.uplo, A.data, 0.0, 0.0, 0, 0, -1.0)[1]
+
+function eigvals!(A::RealHermSymComplexHerm{<:BlasReal,<:StridedMatrix}, alg::Algorithm = default_eigen_alg(A); sortby::Union{Function,Nothing}=nothing)
+    vals::Vector{real(eltype(A))} = if alg === DivideAndConquer()
+        LAPACK.syevd!('N', A.uplo, A.data)
+    elseif alg === QRIteration()
+        LAPACK.syev!('N', A.uplo, A.data)
+    elseif alg === RobustRepresentations()
+        LAPACK.syevr!('N', 'A', A.uplo, A.data, 0.0, 0.0, 0, 0, -1.0)[1]
+    else
+        throw(ArgumentError("Unsupported value for `alg` keyword."))
+    end
     !isnothing(sortby) && sort!(vals, by=sortby)
     return vals
 end
 
-function eigvals(A::RealHermSymComplexHerm; sortby::Union{Function,Nothing}=nothing)
+"""
+    eigvals(A::Union{Hermitian, Symmetric}, alg::Algorithm = default_eigen_alg(A))) -> values
+
+Return the eigenvalues of `A`.
+
+`alg` specifies which algorithm and LAPACK method to use for eigenvalue decomposition:
+- `alg = DivideAndConquer()` (default): Calls `LAPACK.syevd!`.
+- `alg = QRIteration()`: Calls `LAPACK.syev!`.
+- `alg = RobustRepresentations()`: Multiple relatively robust representations method, Calls `LAPACK.syevr!`.
+
+See James W. Demmel et al, SIAM J. Sci. Comput. 30, 3, 1508 (2008) for
+a comparison of the accuracy and performance of different methods.
+
+The default `alg` used may change in the future.
+"""
+function eigvals(A::RealHermSymComplexHerm, alg::Algorithm = default_eigen_alg(A); sortby::Union{Function,Nothing}=nothing)
     S = eigtype(eltype(A))
-    eigvals!(eigencopy_oftype(A, S), sortby=sortby)
+    eigvals!(eigencopy_oftype(A, S), alg; sortby)
 end
 
+
 """
     eigvals!(A::Union{SymTridiagonal, Hermitian, Symmetric}, irange::UnitRange) -> values
 

stdlib/LinearAlgebra/test/symmetric.jl

@@ -720,6 +720,22 @@ end
     end
 end
 
+@testset "eigendecomposition Algorithms" begin
+    using LinearAlgebra: DivideAndConquer, QRIteration, RobustRepresentations
+    for T in (Float64, ComplexF64, Float32, ComplexF32)
+        n = 4
+        A = T <: Real ? Symmetric(randn(T, n, n)) : Hermitian(randn(T, n, n))
+        d, v = eigen(A)
+        for alg in (DivideAndConquer(), QRIteration(), RobustRepresentations())
+            @test (@inferred eigvals(A, alg)) ≈ d
+            d2, v2 = @inferred eigen(A, alg)
+            @test d2 ≈ d
+            @test A * v2 ≈ v2 * Diagonal(d2)
+        end
+    end
+end
+
+const BASE_TEST_PATH = joinpath(Sys.BINDIR, "..", "share", "julia", "test")
 isdefined(Main, :ImmutableArrays) || @eval Main include(joinpath($(BASE_TEST_PATH), "testhelpers", "ImmutableArrays.jl"))
 using .Main.ImmutableArrays