Convergence analysis of a nonlinear eigensolver based on rational approximation of the resolvent

What the math gives to ML

The paper gives a constructive route from resolvent evaluations to rational approximations whose poles encode spectral information, together with perturbation and backward-stability guarantees. The transferable asset is not the nonlinear eigensolver itself, but the combination of block probing, barycentric rational representations, and adaptive localization of interpolation nodes. In neural networks this suggests rational spectral layers that use feature channels as simultaneous probes, and rational activations or filters whose nodes are zoomed toward the spectral or preactivation region actually occupied by the data. The main engineering risks are shifted-solve cost and denominator ill-conditioning, so initial tests should use small graph operators, state-space matrices, or compact MLPs.