Tridiagonal Toeplitz matrix complex eigenvalues instead of real

The spectrum of tridiagonal Toeplitz matrices is known analytically (Tridiagonal-Toeplitz-Matrix – Wikipedia) and is (for non-zero off-diagonals) simple and real. Julia’s eigen function returns for small non-symmetric Toeplitz matrices the correct real spectrum, but for larger sizes fails to do so. E.g.

n = 200
p = 0.7
q = 0.4
A = Tridiagonal([q for _ in 1:(n-1)], zeros(n), [p for _ in 1:(n-1)])
A = Array(A)
maximum(imag.(eigvals(A)))

Shows non-zero imaginary parts significantly larger than machine precision. The same happens in numpy. Changing to BigFloat helped for sizes of n around 200. I did not test for larger n, but would guess that at sufficiently large n it will fail as well.

Is there a way to detect the failure of eigen?

EDIT: In python an example was given here: python - How to get real eigenvalues and eigenvectors for a tridiagonal Toeplitz matrix? - Stack Overflow
and here
python - Eigenvectors are complex but only for large matrices - Stack Overflow

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The eigenvalues of these matrices are super ill conditioned. I spent some time on these matrices with @sverek a while ago. Sven-Erik knows a lot more about these than I do.

The condition number of the eigenvalue is roughly reciprocal of the inner product between the right and the left (normalized) eigenvectors for that value. (I think I have the condition number from Nick HIgham’s book). So you have

julia> n = 20;

julia> A = Tridiagonal([q for _ in 1:(n-1)], zeros(n), [p for _ in 1:(n-1)]);

julia> F = eigen(Array(A));

julia> vr = (A - F.values[end]*I)\randn(n) |> t -> t/norm(t);

julia> vl = (A' - F.values[end]*I)\randn(n) |> t -> t/norm(t);

julia> abs(inv(vl'vr))
6.737730466564851

julia> n = 200;

julia> A = Tridiagonal([q for _ in 1:(n-1)], zeros(n), [p for _ in 1:(n-1)]);

julia> F = eigen(Array(A));

julia> vr = (A - F.values[end]*I)\randn(n) |> t -> t/norm(t);

julia> vl = (A' - F.values[end]*I)\randn(n) |> t -> t/norm(t);

julia> abs(inv(vl'vr))
7.360334831197046e19

so n doen’t have to get large before small perturbations will be heavily amplified in the eigenvalues.

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I can send you some current references regarding this (and some ways how to deal with it). By the way, the spectrum does not have to be real (e.g., just set q=-0.4).

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By the way, when I run your code

using LinearAlgebra
n = 200
p = 0.7
q = 0.4
A = Tridiagonal([q for _ in 1:(n-1)], zeros(n), [p for _ in 1:(n-1)])
A = Array(A)

to build the matrix A of the size 200×200 in Julia 1.6.3 and then I compute the eigenvalues, I get just real eigenvalues for this size

julia> λ = eigvals(A)
200-element Vector{Float64}:
 -1.058223000149799
 -1.0576000083075798
 -1.0572843720725433
 -1.056332059729569
 -1.0544327076739584
 -1.0543669353419873
 -1.0515818708879903
 -1.0503080759555965
 -1.0477502207464444
 -1.045428090444631
  ⋮
  1.0477851444774355
  1.050211862207378
  1.051655860894167
  1.0543900728172997
  1.0543919786468086
  1.0563388912151381
  1.0572818987403343
  1.0576011289599778
  1.058222590801755

I only get complex eigvenvalues for n=207 (and larger).

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Fyi, that is not the case here on Win10 for Julia 1.6.3 or v"1.7.0-rc1" …

NB:
the maximum imaginary part is ~ 0.05% of the corresponding real part

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How come? Julia is the same, the rules for floating point arithmetics are the same. Is it that it is linked to different BLAS or LAPACK or what?

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As @andreasnoack mentions, the matrix is ill-conditioned. For such matrices, computing pseudospectrum instead of spectrum (eigenvalues) gives some/more insight. Julia has a package for computing Pseudospectra too.

For your matrix with n=1000 it looks like this

pseudospectrum_of_tridiag_toeplitz

The code to produce it this:

using LinearAlgebra
using Plots
using Pseudospectra

n = 1000
p = 0.7
q = 0.4
A = Tridiagonal([q for _ in 1:(n-1)], zeros(n), [p for _ in 1:(n-1)])
A = Array(A)
λ = eigvals(A)

spectralportrait(A)

For n=200, the pseudospectrum is this
pseudospectrum_of_tridiag_toeplitz

2 Likes

Thank you for all your great answers. I was not aware that finding the eigenvalues of tridiagonal Toeplitz matrices is such an ill-conditioned problem. I will mark @andreasnoack post as answer, but I am very grateful for all your comments, especially @zdenek_hurak pointing to the pseudo spectrum.

@zdenek_hurak @rafael.guerra I did my tests in Windows 10. On WSL 2, Ubuntu 20.04 I get non-zero imaginary parts for n=200 as well. My specs are for
Windows

Julia Version 1.6.2
Commit 1b93d53fc4 (2021-07-14 15:36 UTC)
Platform Info:
  OS: Windows (x86_64-w64-mingw32)
  CPU: Intel(R) Core(TM) i7-8665U CPU @ 1.90GHz
  WORD_SIZE: 64
  LIBM: libopenlibm
  LLVM: libLLVM-11.0.1 (ORCJIT, skylake)
Environment:
  JULIA_NUM_THREADS = 4

and WSL 2 Ubuntu:

julia> versioninfo()
Julia Version 1.6.3
Commit ae8452a9e0 (2021-09-23 17:34 UTC)
Platform Info:
 OS: Linux (x86_64-pc-linux-gnu)
 CPU: Intel(R) Core(TM) i7-8665U CPU @ 1.90GHz
 WORD_SIZE: 64
 LIBM: libopenlibm
 LLVM: libLLVM-11.0.1 (ORCJIT, skylake)

Could it be something processor specific?