rdiv!(B, F::UpperHessenberg; shift::Number) does not return if shift=Inf

[olof3]

This issue was mentioned in #31738.

The problem seems to be that LinearAlgebra.givensAlgorithm gets stuck in an infinite loop if one of its input arguments is Inf.
Actually dlartg upon which that function is based does the same

T = Float64
ccall((:dlartg_64_, LinearAlgebra.LAPACK.liblapack), Cvoid,
       (Ref{T}, Ref{T}, Ref{T}, Ref{T}, Ref{T}),
      Ref{T}(1.0), Ref{T}(Inf), Ref{T}(), Ref{T}(), Ref{T}())

Even though the paper upon which the algorithms is based, On Computing Givens Rotations Reliably and Efficiently, states that "On a machine with correct IEEE floating point semantics with NaNs and +-inf, we insist that any standard conforming implementation must terminate and return some output values in all cases as described below.". More details are given in the section Exceptional Cases.

Not sure what the best approach to fix this is.

Should givensAlgorithm be modified?
(Also, the name is a bit confusing, perhaps it should be called something like_givens_rotation?)

Or should the the relevant !rdivand !ldiv be modified by adding something like the following in the beginning of the methods?

    if isinf(shift)
        if all(isfinite, F.data) && all(isfinite, B)
            return fill!(B, zero(eltype(B)))
        else
            return fill!(B, eltype(B)(NaN))
        end
    end

[stevengj]

Should givensAlgorithm be modified?

Probably.