Not following the ARPACK guidelines for then generalized problem

[andreasnoack]

The following problem was provided on julia users

julia> n = 10
10

julia> A = zeros(n,n);

julia> A[1,1] = 100;

julia> A[10,10] = -100;

julia> C = eye(n);

julia> eigs(A,C)
ERROR: Base.LinAlg.ARPACKException("unspecified ARPACK error: -9999")
 in aupd_wrapper(::Type{T}, ::Base.LinAlg.#matvecA!#69{Array{Float64,2}}, ::Base.LinAlg.##66#73{Array{
Float64,2}}, ::Base.LinAlg.##67#74, ::Int64, ::Bool, ::Bool, ::String, ::Int64, ::Int64, ::String, ::F
loat64, ::Int64, ::Int64, ::Array{Float64,1}) at ./linalg/arpack.jl:53
 in #_eigs#62(::Int64, ::Int64, ::Symbol, ::Float64, ::Int64, ::Void, ::Array{Float64,1}, ::Bool, ::Ba
se.LinAlg.#_eigs, ::Array{Float64,2}, ::Array{Float64,2}) at ./linalg/arnoldi.jl:271
 in _eigs(::Array{Float64,2}, ::Array{Float64,2}) at ./linalg/arnoldi.jl:158
 in #eigs#56(::Array{Any,1}, ::Function, ::Array{Float64,2}, ::Array{Float64,2}) at ./linalg/arnoldi.j
l:80
 in eigs(::Array{Float64,2}, ::Array{Float64,2}) at ./linalg/arnoldi.jl:80

Right now, we are using mode 2 when solving symmetric generalized problems but according to the source we shouldn't. In dsaupd.f line 291

c  3. If M can be factored into a Cholesky factorization M = LL`
c     then Mode = 2 should not be selected.  Instead one should use
c     Mode = 1 with  OP = inv(L)*A*inv(L`).  Appropriate triangular
c     linear systems should be solved with L and L` rather
c     than computing inverses.  After convergence, an approximate
c     eigenvector z of the original problem is recovered by solving
c     L`z = x  where x is a Ritz vector of OP.

[ViralBShah]

Seems like the fix for this would also need to be backported to 0.5.

[rleegates]

I'm encountering the same problem. If I may ask, what is the status on this issue?

[andreasnoack]

I don't think anybody is currently working on this but eventually, it will get fixed. ARPACK is just a bit of a time sink and right now other tasks have higher priority. However, feel free to work on it if you need this fixed more urgently.

[ViralBShah]

Is the fix to essentially do what the snippet above says - Mode = 1 with OP = inv(L)*A*inv(L')?

[timholy]

Cross-linking sbromberger/LightGraphs.jl#736 (comment).
sbromberger/LightGraphs.jl#736 (comment) is a different problem but may be worth mentioning.

[ViralBShah]

I might be able to take a look at this next week, unless someone beats me to it.

[ViralBShah]

Closing in favour of #488. Please reopen if the issue is different.