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engines/quan_loncar_v1/package/quan_loncar_optimizer/engine/quan_loncar_series_phase29/README.md · assembled 2026-07-29 15:57 UTC.

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Quan--Loncar series, Phase 29

Phase 29 differentiates the QNM point residue and its generalized mode volume along an arbitrary tangent of the full symmetric gap/radius_x/radius_y geometry vector.

At fixed frequency, the hole projections are differentiated analytically and composed into JVPs of the nonlinear operator A, observation map F, and source map B. The resonance shift uses the implicit pole rule. Right and left eigenvector derivatives use bordered singular solves with a fixed biorthogonal gauge. The derivative of y^H A_s x includes the mixed geometry-frequency derivative and A_ss; frequency derivatives are evaluated by centered analytic-frequency differences of the convergent contour engine.

This is a custom forward derivative, not differentiation through the Newton history. Its validation compares against independently re-solved poles and recomputed QNM residues on both sides of a dense 45-variable perturbation.