[WIP] Power Iteration Method for Numerical Stability Analysis of the Non-Linear System

[Bot-Enigma-0]

Proposed Changes

Give a brief overview of your contribution here in a few sentences. We have a non-linear system described by x_{n+1} = F(x_n), where x_n and x_{n+1} are the solution states at time steps n and n+1, and F represents our nonlinear solver iteration. The spectral radius of the Jacobian matrix J_F = dF/dx determines the stability of our iterative scheme.

The power iteration method allows us to estimate the dominant eigenvalue (spectral radius) without explicitly forming the Jacobian matrix. The basic idea is:

1. Start with a random vector v
2. Repeatedly compute w = J_F * v (using forward-mode AD)
3. Normalize w to get the new v
4. The norm of w converges to the dominant eigenvalue of the system

The current version has some bugs leading to seg faults which i'm trying to debug. I would appreciate any comments.

Related Work

Resolve any issues (bug fix or feature request), note any related PRs, or mention interactions with the work of others, if any.

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