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Spherical metric error measurement in AAA

Investigate the benefits and feasibility of measuring AAA approximation errors using the Riemann sphere metric rather than the complex plane norm, and determine whether this improves pole/zero localization and symmetry of approximation quality around singularities.

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Background

The authors speculate that measuring errors on the Riemann sphere might balance approximation near poles versus near zeros, potentially addressing asymmetries observed in practice. They explicitly note that, to their knowledge, no AAA codes currently attempt this, leaving a gap between theory and implementation.

References

Perhaps the balance would be more even if AAA measured the $\infty$-norm of errors via a metric on the Riemann sphere rather than the complex plane, but so far as we know, no AAA codes have tried this.

— Applications of AAA rational approximation (2510.16237 - Nakatsukasa et al., 17 Oct 2025) in Section 4 Locating poles and zeros