An Introduction to Newton’s Method on Semirings

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" summary

Claims a generalization of Newton's method to polynomial fixpoint systems over omega-continuous semirings that always converges from 0 and whose i‑th approximant equals the yield of all derivation trees with Strahler number ≤ i. Method unfolds the induced context‑free grammar, solves the resulting linear equations (use Arden’s lemma over the language semiring), and often gets exponential convergence over the reals. For idempotent commutative semirings termination occurs after n+1 iterations, giving a constructive Parikh‑style regular approximation.

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" chapters (11)

1. 0:00Introduction to semi-rings
2. 1:38Kleene iteration method
3. 2:38Linear systems analysis
4. 6:41Limitations of Kleene iteration
5. 11:47Generalizing Newton's method
6. 12:54Derivation tree analysis
7. 17:28Strahler numbers in trees
8. 24:28Grammar unfolding algorithm
9. 31:55Convergence and probability
10. 34:03Applications and Parikh's thm
11. 44:00Concluding Q&A

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