Tridiagonal Toeplitz matrix complex eigenvalues instead of real

Specific Domains Numerics
2

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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