Numerical behavior of bordered system near EPs and diverging eigenvalue sensitivity

[cc202408]

Hi Benoît,

Thank you for the excellent work! I'm currently working on the location of EPs and I’ve been studying your JCP 2020 paper with great interest.

I noticed that your method defines an analytical function h(μ) = (λ₊ - λ₋)² and solves for EPs via its Taylor expansion. Therefore, we have to evaluate the derivatives of eigenvalue $\lambda$ with respect to the parameter $\nu$ near the EPs via the bordered system.

However, I’m curious about one point: near the EPs, the derivative of eigenvalue $\lambda$ with respect to the parameter $\nu$ (parameter sensitivity $d\lambda/d\nu$) tends to infinite (and some methods use this diverge property to roughly locate the EPs with $1/(d\lambda/d\nu)=0$). Would the bordered system become numerically ill-conditioned or very hard to solve (I would solve the bordered system with Schur complement) ?

I would really appreciate your insights about this, as I am currently working on locating EPs in a thermoacoustic problem with very high sensitivity to parameters.

Thanks again for sharing the code!

[nennigb]

The bordered matrix behaves generally well when the parameter $\nu_0$ is 'far enough' from an EP. The numerical stability depends on the distance, the problem and on the normalization condition. In practice, $\nu_0\approx \nu_{EP} \alpha$, with $|\alpha|\in [0.98, 1.02]$, ie few percent from the EP is generally fine but it could be also larger. Currently the implemented condition is using a unit vector. We are working to propose several kind of normalization condition.

The dλ / dν is effectively diverging, but not dh/dν after the recombination. Since $h(\nu)$ is analytic in a vicinity of $\nu_0$, it allows to compute higher order derivative and to be more precise on the EP location (staying not to close from the EP). In other word, using higher order derivative and the regularization avoid to compute the derivative (and the eigenvalues) closer and closer from the EP (like newton approach will do) so it should be find with your Schur complement.

Let me know if you have other question.

ps: For better regularization, you can read https://arxiv.org/abs/2505.06141