Unnecessary restiction of bandwidths in generalized eigenvalues

[KlausC]

In the generalized eigenvalue problem eigvals(A, B) with banded matrices, the bandwidth of B must not exceed that of A. There is no mathematical justification for that. It would be easy to extend the bandwidth in the used workspace to the maximum of both.

julia> A = Symmetric(BandedMatrix(0=>ones(3)))
3×3 Symmetric{Float64,BandedMatrix{Float64,Array{Float64,2},Base.OneTo{Int64}}}:
 1.0   ⋅    ⋅ 
  ⋅   1.0   ⋅ 
  ⋅    ⋅   1.0

julia> B = Symmetric(BandedMatrix(0=>ones(3), 1=>ones(2)))
3×3 Symmetric{Float64,BandedMatrix{Float64,Array{Float64,2},Base.OneTo{Int64}}}:
 1.0  1.0   ⋅ 
 1.0  1.0  1.0
  ⋅   1.0  1.0

julia> eigvals(A, B)
ERROR: LAPACKException(2)
Stacktrace:
 [1] chklapackerror at /buildworker/worker/package_linux64/build/usr/share/julia/stdlib/v1.5/LinearAlgebra/src/lapack.jl:38 [inlined]
 [2] pbstf!(::Char, ::Int64, ::Int64, ::SubArray{Float64,2,Array{Float64,2},Tuple{UnitRange{Int64},Base.Slice{Base.OneTo{Int64}}},false}) at /home/crusius/.julia/dev/BandedMatrices/src/lapack.jl:282
 [3] eigvals!(::Symmetric{Float64,BandedMatrix{Float64,Array{Float64,2},Base.OneTo{Int64}}}, ::Symmetric{Float64,BandedMatrix{Float64,Array{Float64,2},Base.OneTo{Int64}}}) at /home/crusius/.julia/dev/BandedMatrices/src/symbanded/symbanded.jl:151
 [4] eigvals(::Symmetric{Float64,BandedMatrix{Float64,Array{Float64,2},Base.OneTo{Int64}}}, ::Symmetric{Float64,BandedMatrix{Float64,Array{Float64,2},Base.OneTo{Int64}}}) at /home/crusius/.julia/dev/BandedMatrices/src/symbanded/symbanded.jl:162
 [5] top-level scope at REPL[29]:1

julia> eigvals(B, A)
3-element Array{Float64,1}:
 -0.41421356237309503
  0.9999999999999999
  2.414213562373095

[dlfivefifty]

You are right, can you produce a PR? Probably in the eigvals overload