Fix precision of 2x2 eigenvectors (#1191)

* Fix precision of 2x2 eigenvectors

* Use same code for upper and lower diag (2x2 eigen)

* Add tests and fix some issues

* Fix test on julia 1.10

* Remove todo comments

* Bump patch version

Author: KnutAM

Project.toml

@@ -1,6 +1,6 @@
 name = "StaticArrays"
 uuid = "90137ffa-7385-5640-81b9-e52037218182"
-version = "1.6.2"
+version = "1.6.3"
 
 [deps]
 LinearAlgebra = "37e2e46d-f89d-539d-b4ee-838fcccc9c8e"

src/eigen.jl

@@ -159,48 +159,26 @@ end
 @inline function _eig(::Size{(2,2)}, A::LinearAlgebra.RealHermSymComplexHerm{T}, permute, scale) where {T <: Real}
     a = A.data
     TA = eltype(A)
-    @inbounds if A.uplo == 'U'
-        if !iszero(a[3]) # A is not diagonal
-            t_half = real(a[1] + a[4]) / 2
-            tmp = norm(SVector((a[1] - a[4])/2, a[3]'))
-            vals = SVector(t_half - tmp, t_half + tmp)
-
-            v11 = vals[1] - a[4]
-            n1 = sqrt(v11' * v11 + a[3]' * a[3])
-            v11 = v11 / n1
-            v12 = a[3]' / n1
-
-            v21 = vals[2] - a[4]
-            n2 = sqrt(v21' * v21 + a[3]' * a[3])
-            v21 = v21 / n2
-            v22 = a[3]' / n2
-
-            vecs = @SMatrix [ v11  v21 ;
-                              v12  v22 ]
-
-            return Eigen(vals, vecs)
-        end
-    else # A.uplo == 'L'
-        if !iszero(a[2]) # A is not diagonal
-            t_half = real(a[1] + a[4]) / 2
-            tmp = norm(SVector((a[1] - a[4])/2, a[2]))
-            vals = SVector(t_half - tmp, t_half + tmp)
-
-            v11 = vals[1] - a[4]
-            n1 = sqrt(v11' * v11 + a[2]' * a[2])
-            v11 = v11 / n1
-            v12 = a[2] / n1
-
-            v21 = vals[2] - a[4]
-            n2 = sqrt(v21' * v21 + a[2]' * a[2])
-            v21 = v21 / n2
-            v22 = a[2] / n2
-
-            vecs = @SMatrix [ v11  v21 ;
-                              v12  v22 ]
-
-            return Eigen(vals,vecs)
-        end
+    @inbounds a21 = A.uplo == 'U' ? a[3]' : a[2]
+    @inbounds if !iszero(a21) # A is not diagonal
+        t_half = real(a[1] + a[4]) / 2
+        diag_avg_diff = (a[1] - a[4])/2
+        tmp = norm(SVector(diag_avg_diff, a21))
+        vals = SVector(t_half - tmp, t_half + tmp)
+        v11 = -tmp + diag_avg_diff
+        n1 = sqrt(v11' * v11 + a21 * a21')
+        v11 = v11 / n1
+        v12 = a21 / n1
+
+        v21 = tmp + diag_avg_diff
+        n2 = sqrt(v21' * v21 + a21 * a21')
+        v21 = v21 / n2
+        v22 = a21 / n2
+
+        vecs = @SMatrix [ v11  v21 ;
+                            v12  v22 ]
+
+        return Eigen(vals, vecs)
     end
 
     # A must be diagonal if we reached this point; treatment of uplo 'L' and 'U' is then identical

test/eigen.jl

@@ -83,6 +83,25 @@ using StaticArrays, Test, LinearAlgebra
             @test eigvecs(E) * SDiagonal(eigvals(E)) * eigvecs(E)' ≈ A
         end
 
+        # issue #956
+        v1, v2 = let A = SMatrix{2,2}((1.0, 1e-100, 1e-100, 1.0))
+            vecs = eigvecs(eigen(Hermitian(A)))
+            vecs[:,1], vecs[:,2]
+        end
+        @test norm(v1) ≈ 1
+        @test norm(v2) ≈ 1
+        @test acos(v1 ⋅ v2)*2 ≈ π
+
+        # Test that correct value is taken for upper or lower sym 
+        for T in (Float64, Complex{Float64})
+            af = SMatrix{2,2}(rand(T, 4))
+            a = af + af' # Make hermitian
+            au = Hermitian(SMatrix{2,2}((a[1,1], zero(T), a[1,2], a[2,2])), 'U')
+            al = Hermitian(SMatrix{2,2}((a[1,1], a[2,1], zero(T), a[2,2])), 'L')
+            @test eigvals(eigen(a)) ≈ eigvals(eigen(au)) ≈ eigvals(eigen(al))
+            @test eigvals(eigen(a)) ≈ eigvals(eigen(au)) ≈ eigvals(eigen(al))
+        end
+
         m1_a = randn(2,2)
         m1_a = m1_a*m1_a'
         m1 = SMatrix{2,2}(m1_a)

test/initializers.jl

@@ -50,7 +50,11 @@ if VERSION >= v"1.7.0"
     @test SA[1 2;1 2 ;;; 1 2;1 2] === SArray{Tuple{2,2,2}}(Tuple([1 2;1 2 ;;; 1 2;1 2]))
     @test_inlined SA_test_hvncat1(3)
     @test_inlined SA_test_hvncat2(2)
-    @test_throws ArgumentError SA[1;2;;3]
+    if VERSION < v"1.10-DEV"
+        @test_throws ArgumentError SA[1;2;;3]
+    else
+        @test_throws DimensionMismatch SA[1;2;;3]
+    end
 end
 
 # https://github.com/JuliaArrays/StaticArrays.jl/pull/685