ishermitian/issymmetric should accept optional atol keyword parameter

[GiggleLiu]

I was expecting the following code print two trues.

using LinearAlgebra
siz = 8
base_space = qr(randn(Float64, siz, siz)).Q
phases = rand(siz)
signs = Diagonal(phases)
A = base_space*signs*base_space'
println(A ≈ A')
println(ishermitian(A))

Instead, I get true and false.

The ishermitian turns out using ==. Isn't it better to use ≈ or add an atol keyword parameter?

julia/stdlib/LinearAlgebra/src/generic.jl

Line 971 in 4498d27

function ishermitian(A::AbstractMatrix)

[GiggleLiu]

I think this "feature" is not consistent with other julia methods such as eigen.
eigen(A) here will return real eigenvalues, which means the program take A as a hermitian matrix, although it has ~1e-16 error.

[KristofferC]

See the discussion in #10298.

[garrison]

I agree that the default behavior of ishermitian should not be changed (in line with previous discussion), but in my opinion an optional atol keyword argument would be a welcome feature.

[bjarthur]

simple enough to redefine ishermitian yourself if the julia devs have decided not to add a tolerance argument:

julia> import LinearAlgebra: ishermitian

julia> using LinearAlgebra

julia> function ishermitian(A::AbstractMatrix; isapprox=isapprox)
           indsm, indsn = axes(A)
           if indsm != indsn
               return false
           end
           for i = indsn, j = i:last(indsn)
               if !isapprox(A[i,j], adjoint(A[j,i]))
                   return false
               end
           end
           return true
       end
ishermitian (generic function with 14 methods)

julia> ishermitian([1 2; 2.00000000000001 1])
true

[timholy]

It's possibly less of a decision and more that no one has submitted a PR? But I suspect there is little point since lots of other routines will fail or produce surprising results unless it's exact.

[garrison]

It's possibly less of a decision and more that no one has submitted a PR?

If this is the case, it could be clarified by having an open issue regarding this missing feature. This could be accomplished most easily by opening and renaming this one, while perhaps adjusting the top comment (I no longer have the permission to do any of this), or alternatively by having a new issue elsewhere.

Suggested [edited] title: ishermitian should accept optional atol keyword parameter

[garrison]

But I suspect there is little point since lots of other routines will fail or produce surprising results unless it's exact.

In my own personal experience of dealing with quantum mechanics numerically, the most common use case for this missing feature has been as a sanity check that a matrix I have constructed is actually Hermitian before treating it conceptually as such. Without such a check, it is much easier to let a non-physical Hamiltonian or observable slip through.

Often, the matrix I have constructed will be non-Hermitian only by the tiniest amounts (e.g. because floating point addition is non-associative), and I'd like to perform a conceptual test of whether the matrix is "close enough" to Hermitian.

[timholy]

Reopened and renamed as suggested.

[bjarthur]

my use case is also as a sanity check to make sure that a numerically simulated matrix that should be symmetric actually is.

[bjarthur]

my re-definition above would be better if the added keyword argument were isequal=isequal and then folks like me and @garrison could use isequal=isapprox. for everyone else nothing changes and completely backward compatible.

[bjarthur]

@garrison see #36243 (comment). isapprox(A,A') suffices for me. hadn't thought of doing that.

[garrison]

Oh, interesting find regarding #36243. I had been thinking recently that anything decided here should apply to issymmetric as well.

[oscardssmith]

isapprox(A,A') will probably be about 2x slower, but that's probably not a major problem for many use cases.

[jlapeyre]

I wrote a package a while back to try to approach specifying measure of closeness in a uniform way: IsApprox.jl. I included code for many functions/methods including ishermitian.

Four subtypes of AbstractApprox are included: Equal, Approx, EachApprox, and UpToPhase.
IsApprox implements the interface at least partially for each of: isone, iszero, ishermitian, issymmetric, isreal, isinteger, istriu, istril, isbanded, isdiag, isposdef, ispossemidef, isunitary, isinvolution.