Implement ordschur with arbitrary reordering of eigenvalues

[aravindh-krishnamoorthy]

Potential fix for #1047

function ordschur!(S::LinearAlgebra.Schur, p::AbstractVector{<:Integer})

Code implements the algorithm for arbitrary reordering of eigenvalues along the lines of Daniel Kressner, "Block algorithms for reordering standard and generalized Schur forms," ACM Transactions on Mathematical Software Volume 32 Issue 4 pp 521-532 https://doi.org/10.1145/1186785.1186787

• First implementation
• Optimised implementation
• Documentation
• Documentation compilation check
• Unit test
• Unit test check

[aravindh-krishnamoorthy]

Optimisation Tracker

(This comment will be continually updated with the current optimisation status)

$1024\times1024$ matrix single reordering: about $3\times$ LAPACK

julia> using LinearAlgebra
julia> using BenchmarkTools
julia> N = 1024;
julia> X = randn(N,N);
julia> S = schur(X);
julia> idx = findlast(x->isreal(x), S.values);
julia> p = circshift(1:N,N-idx+1);
julia> b = falses(N);
julia> b[idx:end] .= true;
julia> S1 = ordschur(S,b);
julia> S2 = ordschur!(copy(S),p);
julia> norm(S1.values - S2.values)
1.5089416132685045e-13
julia> norm(S1.values - S.values)
1096.6594764777658
julia> norm(S1.Z*S1.T*S1.Z' - X)
1.892660374773724e-11
julia> norm(S2.Z*S2.T*S2.Z' - X)
1.8935267963444116e-11
julia> @btime ordschur(S,b);
  5.880 ms (27 allocations: 16.05 MiB)
julia> @btime ordschur!(copy(S),p);
  18.555 ms (6808 allocations: 46.56 MiB)