RFC: treat small negative λ as 0 for sqrt(::Hermitian)

NEWS.md

@@ -83,6 +83,7 @@ Standard library changes
 * The BLAS submodule now supports the level-2 BLAS subroutine `hpmv!` ([#34211]).
 * `normalize` now supports multidimensional arrays ([#34239])
 * `lq` factorizations can now be used to compute the minimum-norm solution to under-determined systems ([#34350]).
+* `sqrt(::Hermitian)` now treats slightly negative eigenvalues as zero for nearly semidefinite matrices, and accepts a new `rtol` keyword argument for this tolerance ([#35057]).
 * The BLAS submodule now supports the level-2 BLAS subroutine `spmv!` ([#34320]).
 
 #### Markdown

stdlib/LinearAlgebra/src/dense.jl

@@ -706,8 +706,15 @@ If `A` has no negative real eigenvalues, compute the principal matrix square roo
 that is the unique matrix ``X`` with eigenvalues having positive real part such that
 ``X^2 = A``. Otherwise, a nonprincipal square root is returned.
 
-If `A` is symmetric or Hermitian, its eigendecomposition ([`eigen`](@ref)) is
-used to compute the square root. Otherwise, the square root is determined by means of the
+If `A` is real-symmetric or Hermitian, its eigendecomposition ([`eigen`](@ref)) is
+used to compute the square root.   For such matrices, eigenvalues λ that
+appear to be slightly negative due to roundoff errors are treated as if they were zero
+More precisely, matrices with all eigenvalues `≥ -rtol*(max |λ|)` are treated as semidefinite
+(yielding a Hermitian square root), with negative eigenvalues taken to be zero.
+`rtol` is a keyword argument to `sqrt` (in the Hermitian/real-symmetric case only) that
+defaults to machine precision scaled by `size(A,1)`.
+
+Otherwise, the square root is determined by means of the
 Björck-Hammarling method [^BH83], which computes the complex Schur form ([`schur`](@ref))
 and then the complex square root of the triangular factor.
 

stdlib/LinearAlgebra/src/symmetric.jl

@@ -993,22 +993,27 @@ end
 
 
 for func in (:log, :sqrt)
+    # sqrt has rtol arg to handle matrices that are semidefinite up to roundoff errors
+    rtolarg = func === :sqrt ? Any[Expr(:kw, :(rtol::Real), :(eps(real(float(one(T))))*size(A,1)))] : Any[]
+    rtolval = func === :sqrt ? :(-maximum(abs, F.values) * rtol) : 0
     @eval begin
-        function ($func)(A::HermOrSym{<:Real})
+        function ($func)(A::HermOrSym{T}; $(rtolarg...)) where {T<:Real}
             F = eigen(A)
-            if all(λ -> λ ≥ 0, F.values)
-                retmat = (F.vectors * Diagonal(($func).(F.values))) * F.vectors'
+            λ₀ = $rtolval # treat λ ≥ λ₀ as "zero" eigenvalues up to roundoff
+            if all(λ -> λ ≥ λ₀, F.values)
+                retmat = (F.vectors * Diagonal(($func).(max.(0, F.values)))) * F.vectors'
             else
                 retmat = (F.vectors * Diagonal(($func).(complex.(F.values)))) * F.vectors'
             end
             return Symmetric(retmat)
         end
 
-        function ($func)(A::Hermitian{<:Complex})
+        function ($func)(A::Hermitian{T}; $(rtolarg...)) where {T<:Complex}
             n = checksquare(A)
             F = eigen(A)
-            if all(λ -> λ ≥ 0, F.values)
-                retmat = (F.vectors * Diagonal(($func).(F.values))) * F.vectors'
+            λ₀ = $rtolval # treat λ ≥ λ₀ as "zero" eigenvalues up to roundoff
+            if all(λ -> λ ≥ λ₀, F.values)
+                retmat = (F.vectors * Diagonal(($func).(max.(0, F.values)))) * F.vectors'
                 for i = 1:n
                     retmat[i,i] = real(retmat[i,i])
                 end

stdlib/LinearAlgebra/test/symmetric.jl

@@ -663,4 +663,19 @@ end
     @test LinearAlgebra.hermitian_type(Int) == Int
 end
 
+@testset "sqrt(nearly semidefinite)" begin
+    let A = [0.9999999999999998 4.649058915617843e-16 -1.3149405273715513e-16 9.9959579317056e-17; -8.326672684688674e-16 1.0000000000000004 2.9280733590254494e-16 -2.9993900031619594e-16; 9.43689570931383e-16 -1.339206523454095e-15 1.0000000000000007 -8.550505126287743e-16; -6.245004513516506e-16 -2.0122792321330962e-16 1.183061278035052e-16 1.0000000000000002],
+        B = [0.09648289218436859 0.023497875751503007 0.0 0.0; 0.023497875751503007 0.045787575150300804 0.0 0.0; 0.0 0.0 0.0 0.0; 0.0 0.0 0.0 0.0],
+        C = Symmetric(A*B*A'), # semidefinite up to roundoff
+        Csqrt = sqrt(C)
+        @test Csqrt isa Symmetric{Float64}
+        @test Csqrt*Csqrt ≈ C rtol=1e-14
+    end
+    let D = Symmetric(Matrix(Diagonal([1 0; 0 -1e-14])))
+        @test sqrt(D) ≈ [1 0; 0 1e-7im] rtol=1e-14
+        @test sqrt(D, rtol=1e-13) ≈ [1 0; 0 0] rtol=1e-14
+        @test sqrt(D, rtol=1e-13)^2 ≈ D rtol=1e-13
+    end
+end
+
 end # module TestSymmetric