Upstream

Some implementations in Base/LinearAlgebra/SparseArrays have the following assumption:

1. `zero(T) isa T`
2. `one(T) isa T`
3. `-(::T) isa T`
4. `+(::T, ::T) isa T`
5. `*(::T, ::T) isa T`
6. `x * y = y * x`
7. Implements `==`
8. `*(x::T, y::S) = *(promote(x, y)...)`

However, these assumptions are violated for the following types where `MP` stands for MultivariatePolynomials and `MOI` stand for MathOptInterface.

| Type | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
|---------------------------------------|---|---|---|---|---|---|---|---|
| `Bool` | | | x | x | | | | |
| `JuMP.VariableRef` | x | x | x | x | x | | | x |
| `MOI.VariableRef` | x | x | x | x | x | | | x |
| `MOI.ScalarAffineExpression` | | | | | x | | x | x |
| `MP.AbstractVariable` | x | x | x | x | x | | | |
| `MP.AbstractMonomial` | x | x | x | x | | | | |
| `MP.AbstractTerm` | | | | x | | | | |
| `AbstractArray` | | | | | | x | | |
| Non-commutative `MP.AbstractVariable` | x | x | x | x | x | x | | |

The current approach is to create an `AbstractMutable` type and redefine methods for the methods defined in Base for these types.
The long term solution would be to fix the issue upstream. In the short term, as Julia v1.5 might be an LTS, that would allow the package to be simplified once we drop Julia v1.0 for all the fixes that are merged before Julia v1.5.

The following are the issues in `LinearAlgebra`.

- [ ] `Matrix(::Tridiagonal)`, `Matrix(::SymTridiagonal)`, `Matrix(::LowerTriangular)`, `Matrix(::UpperTriangular)`, `Matrix(::UnitLowerTriangular)`, `Matrix(::UnitUpperTriangular)` assumes 1: https://github.com/JuliaLang/julia/pull/36256
Current workaround:
https://github.com/JuliaOpt/MutableArithmetics.jl/blob/407a8fdfb5ef1810f3785a88e490291d37b9fb03/src/dispatch.jl#L231-L241
- [x] `LinearAlgebra._iszero` assumes 7: https://github.com/JuliaLang/julia/pull/36194
Current workaround:
https://github.com/JuliaOpt/MutableArithmetics.jl/blob/407a8fdfb5ef1810f3785a88e490291d37b9fb03/src/dispatch.jl#L267-L272
- [x] `convert(::Type{<:UniformScaling}, ::UniformScaling)` is not defined: https://github.com/JuliaLang/julia/pull/36193
Current workaround:
https://github.com/JuliaOpt/MutableArithmetics.jl/blob/407a8fdfb5ef1810f3785a88e490291d37b9fb03/src/MutableArithmetics.jl#L60-L64
- [x] `LinearAlgebra.diagm_container` assumes 1. : https://github.com/JuliaLang/julia/pull/35743
Current workaround:
https://github.com/JuliaOpt/MutableArithmetics.jl/blob/407a8fdfb5ef1810f3785a88e490291d37b9fb03/src/dispatch.jl#L27-L31
- [x] `-(::LinearAlgebra.Symmetric)` assumes 3. : https://github.com/JuliaLang/julia/pull/32375/
Current workaround:
https://github.com/JuliaOpt/MutableArithmetics.jl/blob/407a8fdfb5ef1810f3785a88e490291d37b9fb03/src/dispatch.jl#L204-L212
- [x] `LinearAlgebra.lu` assumes 4. and 5. : https://github.com/JuliaLang/julia/pull/26344
- [x] `*(::AbstractArray, ::AbstractArray)` assumes 4. : https://github.com/JuliaLang/julia/pull/18218/

The following are the issues in `SparseArrays`.

- [ ] `*(::AbstractSparseArray, ::AbstractSparseArray)` assumes 8.
Beginning of current workaround (many more follows):
https://github.com/JuliaOpt/MutableArithmetics.jl/blob/407a8fdfb5ef1810f3785a88e490291d37b9fb03/src/dispatch.jl#L105-L116