eigen() broken for non-Hermitian matrices

[sverek]

Both in 1.6.2 and 1.7b3, see examples below with BigFloat.

julia> versioninfo()
Julia Version 1.6.2
Commit 1b93d53fc4 (2021-07-14 15:36 UTC)
Platform Info:
  OS: macOS (x86_64-apple-darwin18.7.0)
  CPU: Intel(R) Core(TM) i7-8569U CPU @ 2.80GHz
  WORD_SIZE: 64
  LIBM: libopenlibm
  LLVM: libLLVM-11.0.1 (ORCJIT, skylake)

julia> using LinearAlgebra,GenericLinearAlgebra

julia> eigen(Hermitian(rand(BigFloat,5,5)))
Eigen{BigFloat, BigFloat, Matrix{BigFloat}, Vector{BigFloat}}
values:
5-element Vector{BigFloat}:
 -0.4887698284034353629278662405847996486390203845256632669531308786790234904305252
 -0.23957178126044890148706578948820548968416904833626474092213791610934499449746
  0.2420315098582791585836827261159398231227201754519835421916618918814673416784328
  1.027553335067571884556797097988170714414183181354401756940097138295884750065342
  2.4179948357615710761208524931932816638859780417157267213822396311464899653017
vectors:
5×5 Matrix{BigFloat}:
  0.643244   0.056325  -0.703372   0.15764   0.251957
  0.137664  -0.823489   0.111178  -0.380953  0.381352
  0.335786   0.350451   0.593617   0.159468  0.621788
 -0.395071   0.390543  -0.309542  -0.627235  0.449617
 -0.546308  -0.208215  -0.211448   0.641226  0.449792

julia> eigen(rand(BigFloat,5,5))
ERROR: MethodError: no method matching eigen!(::Matrix{BigFloat}; permute=true, scale=true, sortby=LinearAlgebra.eigsortby)
Closest candidates are:
  eigen!(::StridedMatrix{T}; permute, scale, sortby) where T<:Union{Float32, Float64} at /Users/julia/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.6/LinearAlgebra/src/eigen.jl:148
  eigen!(::StridedMatrix{T}; permute, scale, sortby) where T<:Union{ComplexF32, ComplexF64} at /Users/julia/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.6/LinearAlgebra/src/eigen.jl:171
  eigen!(::StridedMatrix{T}, ::StridedMatrix{T}; sortby) where T<:Union{Float32, Float64} at /Users/julia/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.6/LinearAlgebra/src/eigen.jl:427 got unsupported keyword arguments "permute", "scale"
  ...
Stacktrace:
 [1] eigen(A::Matrix{BigFloat}; permute::Bool, scale::Bool, sortby::typeof(LinearAlgebra.eigsortby))
   @ LinearAlgebra /Users/julia/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.6/LinearAlgebra/src/eigen.jl:237
 [2] eigen(A::Matrix{BigFloat})
   @ LinearAlgebra /Users/julia/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.6/LinearAlgebra/src/eigen.jl:235
 [3] top-level scope
   @ REPL[6]:1

julia> versioninfo()
Julia Version 1.7.0-beta3.0
Commit e76c9dad42 (2021-07-07 08:12 UTC)
Platform Info:
  OS: macOS (x86_64-apple-darwin18.7.0)
  CPU: Intel(R) Core(TM) i7-8569U CPU @ 2.80GHz
  WORD_SIZE: 64
  LIBM: libopenlibm
  LLVM: libLLVM-12.0.0 (ORCJIT, skylake)

julia> using LinearAlgebra,GenericLinearAlgebra

julia> eigen(Hermitian(rand(BigFloat,5,5)))
Eigen{BigFloat, BigFloat, Matrix{BigFloat}, Vector{BigFloat}, Vector{BigFloat}}
values:
5-element Vector{BigFloat}:
 -0.3013651960644129641532610658746961406964259498102245867410423605298043880153657
  0.120203860860071466734019164981546073833764155606255715475030075776944488735232
  0.6240177079049022904005106766910157195384973670823691563637390563798449544924699
  0.7367367930040014998095951360312239998683781592003136115029612450139105705095599
  2.332417706482406886949287685629486433665965295696010610051049348107239712251295
vectors:
5×5 Matrix{BigFloat}:
 -0.0392047  -0.197348     0.861579   0.316458   0.34213
 -0.0925792   0.961961     0.137852  -0.0203742  0.215967
 -0.716137   -0.16382     -0.297161  -0.0398325  0.608621
  0.360393   -0.0939928    0.133732  -0.821625   0.410281
  0.5892     -0.00360311  -0.363992   0.472001   0.545487

julia> eigen(rand(BigFloat,5,5))
ERROR: MethodError: no method matching eigen!(::Matrix{BigFloat}; permute=true, scale=true, sortby=LinearAlgebra.eigsortby, jvl=false, jvr=true, jce=false, jcv=false)
Closest candidates are:
  eigen!(::StridedMatrix{T}; permute, scale, sortby, jvl, jvr, jce, jcv) where T<:Union{Float32, Float64} at /Users/julia/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.7/LinearAlgebra/src/eigen.jl:163
  eigen!(::StridedMatrix{T}; permute, scale, sortby, jvl, jvr, jce, jcv) where T<:Union{ComplexF32, ComplexF64} at /Users/julia/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.7/LinearAlgebra/src/eigen.jl:206
  eigen!(::StridedMatrix{T}, ::StridedMatrix{T}; sortby) where T<:Union{Float32, Float64} at /Users/julia/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.7/LinearAlgebra/src/eigen.jl:478 got unsupported keyword arguments "permute", "scale", "jvl", "jvr", "jce", "jcv"
  ...
Stacktrace:
 [1] eigen(A::Matrix{BigFloat}; permute::Bool, scale::Bool, sortby::typeof(LinearAlgebra.eigsortby), jvl::Bool, jvr::Bool, jce::Bool, jcv::Bool)
   @ LinearAlgebra /Users/julia/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.7/LinearAlgebra/src/eigen.jl:276
 [2] eigen(A::Matrix{BigFloat})
   @ LinearAlgebra /Users/julia/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.7/LinearAlgebra/src/eigen.jl:274
 [3] top-level scope
   @ REPL[6]:1

[andreasnoack]

This is actually never worked. Most of the pieces are there but they are not assembled. I.e. you can do

schur(rand(BigFloat,5,5))

but I haven't yet written the diagonalization routine for (quasi) triangular matrices.