Loosen strictness on hermitian checks

ext/MatrixAlgebraKitAMDGPUExt/MatrixAlgebraKitAMDGPUExt.jl

@@ -4,7 +4,7 @@ using MatrixAlgebraKit
 using MatrixAlgebraKit: @algdef, Algorithm, check_input
 using MatrixAlgebraKit: one!, zero!, uppertriangular!, lowertriangular!
 using MatrixAlgebraKit: diagview, sign_safe
-using MatrixAlgebraKit: LQViaTransposedQR, TruncationByValue
+using MatrixAlgebraKit: LQViaTransposedQR, TruncationByValue, AbstractAlgorithm
 using MatrixAlgebraKit: default_qr_algorithm, default_lq_algorithm, default_svd_algorithm, default_eigh_algorithm
 import MatrixAlgebraKit: _gpu_geqrf!, _gpu_ungqr!, _gpu_unmqr!, _gpu_gesvd!, _gpu_Xgesvdp!, _gpu_gesvdj!
 import MatrixAlgebraKit: _gpu_heevj!, _gpu_heevd!, _gpu_heev!, _gpu_heevx!
@@ -128,10 +128,17 @@ function MatrixAlgebraKit._project_hermitian_diag!(A::StridedROCMatrix, B::Strid
 end
 
 MatrixAlgebraKit.ishermitian_exact(A::StridedROCMatrix) = all(A .== adjoint(A))
-MatrixAlgebraKit.ishermitian_exact(A::Diagonal{T, <:StridedROCVector{T}}) where {T} = all(A.diag .== adjoint(A.diag))
+MatrixAlgebraKit.ishermitian_exact(A::Diagonal{T, <:StridedROCVector{T}}) where {T} =
+    all(A.diag .== adjoint(A.diag))
+MatrixAlgebraKit.ishermitian_approx(A::StridedROCMatrix; kwargs...) =
+    @invoke MatrixAlgebraKit.ishermitian_approx(A::Any; kwargs...)
 
-MatrixAlgebraKit.isantihermitian_exact(A::StridedROCMatrix) = all(A .== -adjoint(A))
-MatrixAlgebraKit.isantihermitian_exact(A::Diagonal{T, <:StridedROCVector{T}}) where {T} = all(A.diag .== -adjoint(A.diag))
+MatrixAlgebraKit.isantihermitian_exact(A::StridedROCMatrix) =
+    all(A .== -adjoint(A))
+MatrixAlgebraKit.isantihermitian_exact(A::Diagonal{T, <:StridedROCVector{T}}) where {T} =
+    all(A.diag .== -adjoint(A.diag))
+MatrixAlgebraKit.isantihermitian_approx(A::StridedROCMatrix; kwargs...) =
+    @invoke MatrixAlgebraKit.isantihermitian_approx(A::Any; kwargs...)
 
 function MatrixAlgebraKit._avgdiff!(A::StridedROCMatrix, B::StridedROCMatrix)
     axes(A) == axes(B) || throw(DimensionMismatch())

ext/MatrixAlgebraKitCUDAExt/MatrixAlgebraKitCUDAExt.jl

@@ -4,7 +4,7 @@ using MatrixAlgebraKit
 using MatrixAlgebraKit: @algdef, Algorithm, check_input
 using MatrixAlgebraKit: one!, zero!, uppertriangular!, lowertriangular!
 using MatrixAlgebraKit: diagview, sign_safe
-using MatrixAlgebraKit: LQViaTransposedQR, TruncationByValue
+using MatrixAlgebraKit: LQViaTransposedQR, TruncationByValue, AbstractAlgorithm
 using MatrixAlgebraKit: default_qr_algorithm, default_lq_algorithm, default_svd_algorithm, default_eig_algorithm, default_eigh_algorithm
 import MatrixAlgebraKit: _gpu_geqrf!, _gpu_ungqr!, _gpu_unmqr!, _gpu_gesvd!, _gpu_Xgesvdp!, _gpu_Xgesvdr!, _gpu_gesvdj!, _gpu_geev!
 import MatrixAlgebraKit: _gpu_heevj!, _gpu_heevd!
@@ -134,11 +134,19 @@ function MatrixAlgebraKit._project_hermitian_diag!(A::StridedCuMatrix, B::Stride
     return nothing
 end
 
-MatrixAlgebraKit.ishermitian_exact(A::StridedCuMatrix) = all(A .== adjoint(A))
-MatrixAlgebraKit.ishermitian_exact(A::Diagonal{T, <:StridedCuVector{T}}) where {T} = all(A.diag .== adjoint(A.diag))
-
-MatrixAlgebraKit.isantihermitian_exact(A::StridedCuMatrix) = all(A .== -adjoint(A))
-MatrixAlgebraKit.isantihermitian_exact(A::Diagonal{T, <:StridedCuVector{T}}) where {T} = all(A.diag .== -adjoint(A.diag))
+MatrixAlgebraKit.ishermitian_exact(A::StridedCuMatrix) =
+    all(A .== adjoint(A))
+MatrixAlgebraKit.ishermitian_exact(A::Diagonal{T, <:StridedCuVector{T}}) where {T} =
+    all(A.diag .== adjoint(A.diag))
+MatrixAlgebraKit.ishermitian_approx(A::StridedCuMatrix; kwargs...) =
+    @invoke MatrixAlgebraKit.ishermitian_approx(A::Any; kwargs...)
+
+MatrixAlgebraKit.isantihermitian_exact(A::StridedCuMatrix) =
+    all(A .== -adjoint(A))
+MatrixAlgebraKit.isantihermitian_exact(A::Diagonal{T, <:StridedCuVector{T}}) where {T} =
+    all(A.diag .== -adjoint(A.diag))
+MatrixAlgebraKit.isantihermitian_approx(A::StridedCuMatrix; kwargs...) =
+    @invoke MatrixAlgebraKit.isantihermitian_approx(A::Any; kwargs...)
 
 function MatrixAlgebraKit._avgdiff!(A::StridedCuMatrix, B::StridedCuMatrix)
     axes(A) == axes(B) || throw(DimensionMismatch())

src/common/defaults.jl

@@ -19,3 +19,13 @@ function default_pullback_gaugetol(a)
     n = norm(a, Inf)
     return eps(eltype(n))^(3 / 4) * max(n, one(n))
 end
+
+"""
+    default_hermitian_tol(A)
+
+Default tolerance for deciding to warn if the provided `A` is not hermitian.
+"""
+function default_hermitian_tol(A)
+    n = norm(A, Inf)
+    return eps(eltype(n))^(3 / 4) * max(n, one(n))
+end

src/common/matrixproperties.jl

@@ -69,31 +69,32 @@ is_right_isometric(A; kwargs...) = is_left_isometric(A'; kwargs...)
 Test whether a linear map is Hermitian, i.e. `A = A'`.
 The `isapprox_kwargs` can be used to control the tolerances of the equality.
 """
-function ishermitian(A; atol::Real = 0, rtol::Real = 0, norm = LinearAlgebra.norm, kwargs...)
+function ishermitian(A; atol::Real = 0, rtol::Real = 0, kwargs...)
     if iszero(atol) && iszero(rtol)
         return ishermitian_exact(A; kwargs...)
     else
-        return 2 * norm(project_antihermitian(A; kwargs...)) ≤ max(atol, rtol * norm(A))
+        return ishermitian_approx(A; atol, rtol, kwargs...)
     end
 end
-function ishermitian_exact(A)
-    return A == A'
-end
-function ishermitian_exact(A::StridedMatrix; kwargs...)
-    return strided_ishermitian_exact(A, Val(false); kwargs...)
+
+ishermitian_exact(A) = A == A'
+ishermitian_exact(A::StridedMatrix; kwargs...) = strided_ishermitian_exact(A, Val(false); kwargs...)
+function ishermitian_approx(A; atol, rtol, kwargs...)
+    return norm(project_antihermitian(A; kwargs...)) ≤ max(atol, rtol * norm(A))
 end
+ishermitian_approx(A::StridedMatrix; kwargs...) = strided_ishermitian_approx(A, Val(false); kwargs...)
 
 """
     isantihermitian(A; isapprox_kwargs...)
 
 Test whether a linear map is anti-Hermitian, i.e. `A = -A'`.
 The `isapprox_kwargs` can be used to control the tolerances of the equality.
 """
-function isantihermitian(A; atol::Real = 0, rtol::Real = 0, norm = LinearAlgebra.norm, kwargs...)
+function isantihermitian(A; atol::Real = 0, rtol::Real = 0, kwargs...)
     if iszero(atol) && iszero(rtol)
         return isantihermitian_exact(A; kwargs...)
     else
-        return 2 * norm(project_hermitian(A; kwargs...)) ≤ max(atol, rtol * norm(A))
+        return isantihermitian_approx(A; atol, rtol, kwargs...)
     end
 end
 function isantihermitian_exact(A)
@@ -102,6 +103,10 @@ end
 function isantihermitian_exact(A::StridedMatrix; kwargs...)
     return strided_ishermitian_exact(A, Val(true); kwargs...)
 end
+function isantihermitian_approx(A; atol, rtol, kwargs...)
+    return norm(project_hermitian(A; kwargs...)) ≤ max(atol, rtol * norm(A))
+end
+isantihermitian_approx(A::StridedMatrix; kwargs...) = strided_ishermitian_approx(A, Val(true); kwargs...)
 
 # blocked implementation of exact checks for strided matrices
 # -----------------------------------------------------------
@@ -139,3 +144,51 @@ function _ishermitian_exact_offdiag(Al, Au, ::Val{anti}) where {anti}
     end
     return true
 end
+
+
+function strided_ishermitian_approx(
+        A::AbstractMatrix, anti::Val;
+        blocksize = 32, atol::Real = default_hermitian_tol(A), rtol::Real = 0
+    )
+    n = size(A, 1)
+    ϵ² = abs2(zero(eltype(A)))
+    ϵ²max = oftype(ϵ², rtol > 0 ? max(atol, rtol * norm(A)) : atol)^2
+    for j in 1:blocksize:n
+        jb = min(blocksize, n - j + 1)
+        ϵ² += _ishermitian_approx_diag(view(A, j:(j + jb - 1), j:(j + jb - 1)), anti)
+        ϵ² < ϵ²max || return false
+        for i in 1:blocksize:(j - 1)
+            ib = blocksize
+            ϵ² += 2 * _ishermitian_approx_offdiag(
+                view(A, i:(i + ib - 1), j:(j + jb - 1)),
+                view(A, j:(j + jb - 1), i:(i + ib - 1)),
+                anti
+            )
+            ϵ² < ϵ²max || return false
+        end
+    end
+    return true
+end
+
+function _ishermitian_approx_diag(A, ::Val{anti}) where {anti}
+    n = size(A, 1)
+    ϵ² = abs2(zero(eltype(A)))
+    @inbounds for j in 1:n
+        @simd for i in 1:j
+            val = _project_hermitian(A[i, j], A[j, i], !anti)
+            ϵ² += abs2(val) * (1 + Int(i < j))
+        end
+    end
+    return ϵ²
+end
+function _ishermitian_approx_offdiag(Al, Au, ::Val{anti}) where {anti}
+    m, n = size(Al) # == reverse(size(Al))
+    ϵ² = abs2(zero(eltype(Al)))
+    @inbounds for j in 1:n
+        @simd for i in 1:m
+            val = _project_hermitian(Al[i, j], Au[j, i], !anti)
+            ϵ² += abs2(val)
+        end
+    end
+    return ϵ²
+end

src/implementations/eigh.jl

@@ -8,30 +8,40 @@ copy_input(::typeof(eigh_trunc), A) = copy_input(eigh_full, A)
 
 copy_input(::typeof(eigh_full), A::Diagonal) = copy(A)
 
-function check_input(::typeof(eigh_full!), A::AbstractMatrix, DV, ::AbstractAlgorithm)
+check_hermitian(A, ::AbstractAlgorithm) = check_hermitian(A)
+check_hermitian(A, alg::Algorithm) = check_hermitian(A; atol = get(alg.kwargs, :hermitian_tol, default_hermitian_tol(A)))
+function check_hermitian(A; atol::Real = default_hermitian_tol(A), rtol::Real = 0)
     m, n = size(A)
     m == n || throw(DimensionMismatch("square input matrix expected"))
+    ishermitian(A; atol, rtol) ||
+        throw(DomainError(A, "Hermitian matrix was expected. Use `project_hermitian` to project onto the nearest hermitian matrix."))
+    return nothing
+end
+
+function check_input(::typeof(eigh_full!), A::AbstractMatrix, DV, alg::AbstractAlgorithm)
+    check_hermitian(A, alg)
     D, V = DV
+    m = size(A, 1)
     @assert D isa Diagonal && V isa AbstractMatrix
     @check_size(D, (m, m))
     @check_scalar(D, A, real)
     @check_size(V, (m, m))
     @check_scalar(V, A)
     return nothing
 end
-function check_input(::typeof(eigh_vals!), A::AbstractMatrix, D, ::AbstractAlgorithm)
-    m, n = size(A)
-    m == n || throw(DimensionMismatch("square input matrix expected"))
+function check_input(::typeof(eigh_vals!), A::AbstractMatrix, D, alg::AbstractAlgorithm)
+    check_hermitian(A, alg)
+    m = size(A, 1)
     @assert D isa AbstractVector
-    @check_size(D, (n,))
+    @check_size(D, (m,))
     @check_scalar(D, A, real)
     return nothing
 end
 
-function check_input(::typeof(eigh_full!), A::AbstractMatrix, DV, ::DiagonalAlgorithm)
-    m, n = size(A)
-    @assert m == n && isdiag(A)
-    @assert (eltype(A) <: Real && issymmetric(A)) || ishermitian(A)
+function check_input(::typeof(eigh_full!), A::AbstractMatrix, DV, alg::DiagonalAlgorithm)
+    check_hermitian(A, alg)
+    @assert isdiag(A)
+    m = size(A, 1)
     D, V = DV
     @assert D isa Diagonal && V isa Diagonal
     @check_size(D, (m, m))
@@ -40,12 +50,12 @@ function check_input(::typeof(eigh_full!), A::AbstractMatrix, DV, ::DiagonalAlgo
     @check_scalar(V, A)
     return nothing
 end
-function check_input(::typeof(eigh_vals!), A::AbstractMatrix, D, ::DiagonalAlgorithm)
-    m, n = size(A)
-    @assert m == n && isdiag(A)
-    @assert (eltype(A) <: Real && issymmetric(A)) || ishermitian(A)
+function check_input(::typeof(eigh_vals!), A::AbstractMatrix, D, alg::DiagonalAlgorithm)
+    check_hermitian(A, alg)
+    @assert isdiag(A)
+    m = size(A, 1)
     @assert D isa AbstractVector
-    @check_size(D, (n,))
+    @check_size(D, (m,))
     @check_scalar(D, A, real)
     return nothing
 end

src/implementations/projections.jl

@@ -85,14 +85,16 @@ function project_hermitian_native!(A::AbstractMatrix, B::AbstractMatrix, anti::V
     return B
 end
 
+@inline function _project_hermitian(Aij::Number, Aji::Number, anti::Bool)
+    return anti ? (Aij - Aji') / 2 : (Aij + Aji') / 2
+end
 function _project_hermitian_offdiag!(
         Au::AbstractMatrix, Al::AbstractMatrix, Bu::AbstractMatrix, Bl::AbstractMatrix, ::Val{anti}
     ) where {anti}
-
     m, n = size(Au) # == reverse(size(Au))
     return @inbounds for j in 1:n
         @simd for i in 1:m
-            val = anti ? (Au[i, j] - adjoint(Al[j, i])) / 2 : (Au[i, j] + adjoint(Al[j, i])) / 2
+            val = _project_hermitian(Au[i, j], Al[j, i], anti)
             Bu[i, j] = val
             aval = adjoint(val)
             Bl[j, i] = anti ? -aval : aval
@@ -104,7 +106,7 @@ function _project_hermitian_diag!(A::AbstractMatrix, B::AbstractMatrix, ::Val{an
     n = size(A, 1)
     @inbounds for j in 1:n
         @simd for i in 1:(j - 1)
-            val = anti ? (A[i, j] - adjoint(A[j, i])) / 2 : (A[i, j] + adjoint(A[j, i])) / 2
+            val = _project_hermitian(A[i, j], A[j, i], anti)
             B[i, j] = val
             aval = adjoint(val)
             B[j, i] = anti ? -aval : aval

src/yalapack.jl

@@ -10,7 +10,7 @@ module YALAPACK # Yet another lapack wrapper
 
 using LinearAlgebra: BlasFloat, BlasReal, BlasComplex, BlasInt, Char, LAPACK,
     LAPACKException, SingularException, PosDefException, checksquare, chkstride1,
-    require_one_based_indexing, triu!, issymmetric, ishermitian, isposdef, adjoint!
+    require_one_based_indexing, triu!, isposdef, adjoint!
 
 using LinearAlgebra.BLAS: @blasfunc, libblastrampoline
 using LinearAlgebra.LAPACK: chkfinite, chktrans, chkside, chkuplofinite, chklapackerror
@@ -984,16 +984,12 @@ for (heev, heevx, heevr, heevd, hegvd, elty, relty) in
                 A::AbstractMatrix{$elty},
                 W::AbstractVector{$relty} = similar(A, $relty, size(A, 1)),
                 V::AbstractMatrix{$elty} = A;
-                uplo::AbstractChar = 'U'
+                uplo::AbstractChar = 'U',
+                kwargs...
             ) # shouldn't matter but 'U' seems slightly faster than 'L'
             require_one_based_indexing(A, V, W)
             chkstride1(A, V, W)
             n = checksquare(A)
-            if $elty <: Real
-                issymmetric(A) || throw(ArgumentError("A must be symmetric"))
-            else
-                ishermitian(A) || throw(ArgumentError("A must be Hermitian"))
-            end
             chkuplofinite(A, uplo)
             n == length(W) || throw(DimensionMismatch("length mismatch between A and W"))
             if length(V) == 0
@@ -1063,11 +1059,6 @@ for (heev, heevx, heevr, heevd, hegvd, elty, relty) in
             require_one_based_indexing(A, V, W)
             chkstride1(A, V, W)
             n = checksquare(A)
-            if $elty <: Real
-                issymmetric(A) || throw(ArgumentError("A must be symmetric"))
-            else
-                ishermitian(A) || throw(ArgumentError("A must be Hermitian"))
-            end
             chkuplofinite(A, uplo)
             if haskey(kwargs, :irange)
                 il = first(kwargs[:irange])
@@ -1175,11 +1166,6 @@ for (heev, heevx, heevr, heevd, hegvd, elty, relty) in
             require_one_based_indexing(A, V, W)
             chkstride1(A, V, W)
             n = checksquare(A)
-            if $elty <: Real
-                issymmetric(A) || throw(ArgumentError("A must be symmetric"))
-            else
-                ishermitian(A) || throw(ArgumentError("A must be Hermitian"))
-            end
             chkuplofinite(A, uplo)
             if haskey(kwargs, :irange)
                 il = first(irange)
@@ -1289,16 +1275,12 @@ for (heev, heevx, heevr, heevd, hegvd, elty, relty) in
                 A::AbstractMatrix{$elty},
                 W::AbstractVector{$relty} = similar(A, $relty, size(A, 1)),
                 V::AbstractMatrix{$elty} = A;
-                uplo::AbstractChar = 'U'
+                uplo::AbstractChar = 'U',
+                kwargs...
             ) # shouldn't matter but 'U' seems slightly faster than 'L'
             require_one_based_indexing(A, V, W)
             chkstride1(A, V, W)
             n = checksquare(A)
-            if $elty <: Real
-                issymmetric(A) || throw(ArgumentError("A must be symmetric"))
-            else
-                ishermitian(A) || throw(ArgumentError("A must be Hermitian"))
-            end
             uplo = 'U' # shouldn't matter but 'U' seems slightly faster than 'L'
             chkuplofinite(A, uplo)
             n == length(W) || throw(DimensionMismatch("length mismatch between A and W"))

test/projections.jl

@@ -2,7 +2,7 @@ using MatrixAlgebraKit
 using Test
 using TestExtras
 using StableRNGs
-using LinearAlgebra: LinearAlgebra, Diagonal, norm
+using LinearAlgebra: LinearAlgebra, Diagonal, norm, normalize!
 
 const BLASFloats = (Float32, Float64, ComplexF32, ComplexF64)
 
@@ -43,6 +43,21 @@ const BLASFloats = (Float32, Float64, ComplexF32, ComplexF64)
         @test isantihermitian(Ba)
         @test Ba ≈ Aa
     end
+
+    # test approximate error calculation
+    A = normalize!(randn(rng, T, m, m))
+    Ah = project_hermitian(A)
+    Aa = project_antihermitian(A)
+
+    Ah_approx = Ah + noisefactor * Aa
+    ϵ = norm(project_antihermitian(Ah_approx))
+    @test !ishermitian(Ah_approx; atol = (999 // 1000) * ϵ)
+    @test ishermitian(Ah_approx; atol = (1001 // 1000) * ϵ)
+
+    Aa_approx = Aa + noisefactor * Ah
+    ϵ = norm(project_hermitian(Aa_approx))
+    @test !isantihermitian(Aa_approx; atol = (999 // 1000) * ϵ)
+    @test isantihermitian(Aa_approx; atol = (1001 // 1000) * ϵ)
 end
 
 @testset "project_isometric! for T = $T" for T in BLASFloats