Compute selected eigenvectors with eigvecs

[Robert DJ]

I would like to compute the eigenvector of a particular eigenvalue and according to the docs, eigvecs is up for the job.

However, I get errors. The following call should give the same output as eigvecs(A):

julia> A = rand(3,3)

3x3 Array{Float64,2}:

 0.165111  0.995671  0.393572

 0.496099  0.640361  0.595876

 0.632881  0.3909    0.723987

julia> v = eigvals(A)

3-element Array{Float64,1}:

  1.68982

 -0.230301

  0.0699443

julia> eigvecs(A, v)

ERROR: MethodError: `eigfact` has no method matching eigfact(::Array{Float64,2}, ::Array{Float64,1})

Closest candidates are:

  eigfact{T}(::Union{DenseArray{T,2},SubArray{T,2,A<:DenseArray{T,N},I<:Tuple{Vararg{Union{Colon,Int64,Range{Int64}}}},LD}})

  eigfact{TA,TB}(::AbstractArray{TA,2}, ::AbstractArray{TB,2})

 in eigvecs at linalg/eigen.jl:70

julia> eigvecs(A, v)

I have

julia> versioninfo()

Julia Version 0.4.0

Commit 0ff703b (2015-10-08 06:20 UTC)

Platform Info:

  System: Darwin (x86_64-apple-darwin14.5.0)

  CPU: Intel(R) Core(TM)2 Duo CPU     P7350  @ 2.00GHz

  WORD_SIZE: 64

  BLAS: libopenblas (USE64BITINT DYNAMIC_ARCH NO_AFFINITY Penryn)

  LAPACK: libopenblas64_

  LIBM: libopenlibm

  LLVM: libLLVM-3.3

[Dan]

Linear algebra tells us Symmetric matrices have real eigenvalues and general matrices have complex eigenvalues in general. `eigvecs` can be called with an upper and lower bound on the eigenvalues of the eigenvectors to be found, but upper and lower bounds are applicable only to Real numbers, which means we need to use the Symmetric matrix version of `eigvals` and `eigvecs` and then things work out better.

In code:

julia> A = rand(3,3)

3x3 Array{Float64,2}:

 0.63668    0.252549  0.467667 

 0.0372453  0.288091  0.0918441

 0.383552   0.478298  0.148384 

julia> B = Symmetric(A+A')

3x3 Symmetric{Float64,Array{Float64,2}}:

 1.27336   0.289794  0.851219

 0.289794  0.576181  0.570142

 0.851219  0.570142  0.296767

julia> eigvals(B)

3-element Array{Float64,1}:

 -0.349693

  0.516073

  1.97993 

julia> eigvecs(B,1.9,2.0)

3x1 Array{Float64,2}:

 -0.772992

 -0.369141

 -0.515963

Because of numerical errors, when looking for a specific eigenvalue it might be prudent to use enlarge the range by some small factor. This info is the outcome of using `help` and `methods` in the REPL, so more information might be available with a little more investigation.

[Robert DJ]

Ahh! I see that I did not read the docs for eigvecs correctly! 

Thanks a lot!