Use of the `info` field in the `Smith` and `AbstractHermite` types over throwing errors?

[brainandforce]

Like other LinearAlgebra.Factorization types, the Smith, RowHermite, and ColumnHermite types have an info::LinearAlgebra.BlasInt field that is zero when the factorization is a success. This could be used to convey to a user why a factorization failed without throwing an exception. This is what factorization types in the linear algebra standard library do (for instance, LinearAlgebra.QR).

I don't know the details of how this value is intended to be interpreted, but I do think that we can use this value to convey information about why a factorization failed. For this, we'd need some standards, and my idea is to set each bit of info to indicate something about the factorization failure:

• The first bit should be 1 if the Smith/Hermite normal form matrix does not take the expected form (diagonal for SNF, trapezoidal for HNF).
• The second bit should be 1 if the left unimodular factor is not unimodular.
• The third bit should be 1 if the right unimodular factor is not unimodular.

This information can be exploited by the show(::IO, ::MIME"text/plain", ::T) method to explain to a user why a factorization failed in interactive use. My issue is that users writing non-interactive code would have to manually check for successful factorizations with LinearAlgebra.issuccess. Of course, this would be a breaking change if implemented.