How to execute schur! for a pair of submatrices?

[andreasvarga]

schur! accepts as arguments only strided matrices, but I have to call it on submatrices. Is it any trick to overcome this limitation?

The following call with views fails:

julia> a = rand(Complex{BigFloat},5,5); b = rand(Complex{BigFloat},5,5);

julia> LinearAlgebra.schur!(a[1:2,1:2],b[1:2,1:2]);  #OK

julia> LinearAlgebra.schur!(view(a,1:2,1:2),view(b,1:2,1:2));
ERROR: MethodError: Cannot `convert` an object of type Matrix{Complex{BigFloat}} to an object of type SubArray{Complex{BigFloat}, 2, Matrix{Complex{BigFloat}}, Tuple{UnitRange{Int64}, UnitRange{Int64}}, false}
The function `convert` exists, but no method is defined for this combination of argument types.

Closest candidates are:
  SubArray{T, N, P, I, L}(::Any, ::Any, ::Any, ::Any) where {T, N, P, I, L}
   @ Base subarray.jl:19
  convert(::Type{T}, ::T) where T
   @ Base Base.jl:126
  convert(::Type{T}, ::Factorization) where T<:AbstractArray
   @ LinearAlgebra C:\Users\Andreas\.julia\juliaup\julia-1.11.5+0.x64.w64.mingw32\share\julia\stdlib\v1.11\LinearAlgebra\src\factorization.jl:104
  ...

Stacktrace:
 [1] GeneralizedSchur{…}(S::SubArray{…}, T::Matrix{…}, alpha::Vector{…}, beta::Vector{…}, Q::Matrix{…}, Z::Matrix{…})
   @ LinearAlgebra C:\Users\Andreas\.julia\juliaup\julia-1.11.5+0.x64.w64.mingw32\share\julia\stdlib\v1.11\LinearAlgebra\src\schur.jl:320
 [2] GeneralizedSchur(S::SubArray{Complex{…}, 2, Matrix{…}, Tuple{…}, false}, T::Matrix{Complex{…}}, alpha::Vector{Complex{…}}, beta::Vector{Complex{…}}, Q::Matrix{Complex{…}}, Z::Matrix{Complex{…}})
   @ LinearAlgebra C:\Users\Andreas\.julia\juliaup\julia-1.11.5+0.x64.w64.mingw32\share\julia\stdlib\v1.11\LinearAlgebra\src\schur.jl:326
 [3] ggschur!(A::SubArray{Complex{…}, 2, Matrix{…}, Tuple{…}, false}, B::SubArray{Complex{…}, 2, Matrix{…}, Tuple{…}, false}; wantQ::Bool, wantZ::Bool, scale::Bool, kwargs::@Kwargs{})
   @ GenericSchur C:\Users\Andreas\.julia\packages\GenericSchur\8grtV\src\generalized.jl:75
 [4] ggschur!(A::SubArray{Complex{BigFloat}, 2, Matrix{Complex{BigFloat}}, Tuple{UnitRange{Int64}, UnitRange{Int64}}, false}, B::SubArray{Complex{BigFloat}, 2, Matrix{Complex{BigFloat}}, Tuple{UnitRange{Int64}, UnitRange{Int64}}, false})
   @ GenericSchur C:\Users\Andreas\.julia\packages\GenericSchur\8grtV\src\generalized.jl:24
 [5] schur!(A::SubArray{Complex{BigFloat}, 2, Matrix{Complex{…}}, Tuple{UnitRange{…}, UnitRange{…}}, false}, B::SubArray{Complex{BigFloat}, 2, Matrix{Complex{…}}, Tuple{UnitRange{…}, UnitRange{…}}, false}; kwargs::@Kwargs{})
   @ GenericSchur C:\Users\Andreas\.julia\packages\GenericSchur\8grtV\src\generalized.jl:16
 [6] schur!(A::SubArray{Complex{BigFloat}, 2, Matrix{Complex{BigFloat}}, Tuple{UnitRange{Int64}, UnitRange{Int64}}, false}, B::SubArray{Complex{BigFloat}, 2, Matrix{Complex{BigFloat}}, Tuple{UnitRange{Int64}, UnitRange{Int64}}, false})
   @ GenericSchur C:\Users\Andreas\.julia\packages\GenericSchur\8grtV\src\generalized.jl:14
 [7] top-level scope
   @ REPL[56]:1
Some type information was truncated. Use `show(err)` to see complete types.

[andreasvarga]

I found that

LinearAlgebra.schur!(collect(view(a,1:2,1:2)),collect(view(b,1:2,1:2)))

works. This involves I guess a copying of the subarrays.

It would be possible to provide schur! also for AbstractMatrix arguments to cover the case of submatrices? Why StridedMatrix is favoured if no LAPACK calls are performed?

[RalphAS]

views are actually included in the StridedMatrix hierarchy, so I don't think that is the trouble.

It seems that the underlying issue is that the Schur and GeneralizedSchur types are parametrized so that all the component matrix fields must be of the same type. Since the schur! functions (in LinearAlgebra too, not just here) construct ordinary Matrix types for the new components (Schur vectors), there is a mismatch when the triangular factor is a view.

There may be a good way to fix this, but it should be addressed in the LinearAlgebra package first.

[andreasvarga]

Indeed, the following also fails:

julia> a = rand(5,5); b = rand(5,5);

julia> schur!(view(a,1:3,1:3),view(b,1:3,1:3))
ERROR: MethodError: Cannot `convert` an object of type Matrix{Float64} to an object of type SubArray{Float64, 2, Matrix{Float64}, Tuple{UnitRange{Int64}, UnitRange{Int64}}, false}
The function `convert` exists, but no method is defined for this combination of argument types.

Closest candidates are:
  SubArray{T, N, P, I, L}(::Any, ::Any, ::Any, ::Any) where {T, N, P, I, L}
   @ Base subarray.jl:19
  convert(::Type{T}, ::T) where T
   @ Base Base.jl:126
  convert(::Type{T}, ::Factorization) where T<:AbstractArray
   @ LinearAlgebra C:\Users\Andreas\.julia\juliaup\julia-1.11.5+0.x64.w64.mingw32\share\julia\stdlib\v1.11\LinearAlgebra\src\factorization.jl:104
  ...

Stacktrace:
 [1] GeneralizedSchur{Float64, SubArray{…}, Vector{…}, Vector{…}}(S::SubArray{Float64, 2, Matrix{…}, Tuple{…}, false}, T::SubArray{Float64, 2, Matrix{…}, Tuple{…}, false}, alpha::Vector{ComplexF64}, beta::Vector{Float64}, Q::Matrix{Float64}, Z::Matrix{Float64})  
   @ LinearAlgebra C:\Users\Andreas\.julia\juliaup\julia-1.11.5+0.x64.w64.mingw32\share\julia\stdlib\v1.11\LinearAlgebra\src\schur.jl:320
 [2] GeneralizedSchur(S::SubArray{Float64, 2, Matrix{…}, Tuple{…}, false}, T::SubArray{Float64, 2, Matrix{…}, Tuple{…}, false}, alpha::Vector{ComplexF64}, beta::Vector{Float64}, Q::Matrix{Float64}, Z::Matrix{Float64})
   @ LinearAlgebra C:\Users\Andreas\.julia\juliaup\julia-1.11.5+0.x64.w64.mingw32\share\julia\stdlib\v1.11\LinearAlgebra\src\schur.jl:326
 [3] schur!(A::SubArray{Float64, 2, Matrix{Float64}, Tuple{UnitRange{Int64}, UnitRange{Int64}}, false}, B::SubArray{Float64, 2, Matrix{Float64}, Tuple{UnitRange{Int64}, UnitRange{Int64}}, false})
   @ LinearAlgebra C:\Users\Andreas\.julia\juliaup\julia-1.11.5+0.x64.w64.mingw32\share\julia\stdlib\v1.11\LinearAlgebra\src\schur.jl:352
 [4] top-level scope
   @ REPL[18]:1
Some type information was truncated. Use `show(err)` to see complete types.

Is this a known problem which will be fixed soon or should we open an issue to report it?