Forward-error analysis of the transformed recursion

Develop a forward-error analysis for the transformed quasilinear recursion for multiplication in the standard real Cayley–Dickson tower in order to characterize its numerical behavior.

Background

The paper presents a quasilinear multiplication algorithm based on a two-call recursion for the alternating product on imaginary Cayley–Dickson elements. Although the algorithm is numerically tested against an independent implementation, no formal stability or forward-error analysis is provided. The authors explicitly identify such an analysis as an unresolved practical question because it could clarify how errors propagate through the coefficient transformation, recursive evaluations, and recombination steps.

References

Several practical questions are likewise left open. A forward-error analysis would clarify the numerical behavior of the transformed recursion, while cache-aware scheduling and accelerator implementations could change the observed crossover points.

— Quasilinear multiplication in the real Cayley--Dickson tower  (2609.11588 - Lemley, 10 Sep 2026) in Section 5, “Outlook and future work”