Scalable matrix-exponential evaluation for MAPH E-steps

Develop and incorporate scalable matrix-exponential action or uniformization methods for evaluating the exact-event and right-censored MAPH expectation-maximization E-steps when the generator is large or stiff, replacing the dense block-matrix exponentials used in the implementation.

Background

The E-step currently evaluates required integral quantities using dense Van Loan block-matrix exponentials. The paper notes that uniformization and matrix-exponential actions based on Krylov-subspace or scaling-and-squaring methods may be preferable for larger systems, but these alternatives are not included in the accompanying software. Their implementation is explicitly deferred to future work.

References

For large $m$, forming the full $2m\times2m$ exponential can be avoided altogether by computing the action of the matrix exponential on a vector through Krylov-subspace or scaling-and-squaring methods \citep{sidje_1998, almohy_higham_2011}. We adopt the Van~Loan identity eq:vanloan-block for its simplicity at the modest phase counts of our examples; the alternatives above may be preferable at larger scale, and adding them to the companion software is left as future work.

— Multi-Absorbing Phase-Type Distributions for Right-Censored Competing Risks Data  (2609.19921 - Qiao et al., 17 Sep 2026) in Appendix, Section 7, Proofs: estimation and the EM algorithm