Convergence-guaranteed Newton–Krylov solver for share inversion

Develop a Newton–Krylov solver that uses the exact matrix-free Jacobian–vector product to outperform the production damped diagonal iteration for large-scale contest share inversion.

Background

The paper provides a matrix-free Jacobian–vector product and discusses Newton–Krylov methods as a potentially more powerful alternative to the coordinatewise damped diagonal iteration currently used for inversion. Although the Newton–Krylov implementation is available, it is described as work in progress: it performs worse on benchmark cases and diverges on a constructed stall case. The authors therefore leave improving its performance relative to the diagonal solver as future solver work.

References

A Newton--Krylov solver would repeatedly ask it what coordinated change removes the current errors most efficiently. That more sophisticated solver is not what produced the reported million-runner result; the own-slope iteration did, and making Newton--Krylov beat it is future solver work.

— Scalable Inversion of Contests with Correlated Performances, Including Softmax and Multinomial Probit  (2609.01133 - Cotton, 1 Sep 2026) in Section 4.3, “Inversion algorithm”

Whether a race-market analogue of their conditions yields a convergence proof here is open.

— Scalable Inversion of Contests with Correlated Performances, Including Softmax and Multinomial Probit  (2609.01133 - Cotton, 1 Sep 2026) in Section 4.4, “Rationale for defaulting to a diagonal solver”