Add TaylorHomotopyContinuationJL: Taylor polynomialization + homotopy continuation + Newton polish for non-polynomial systems

[ChrisRackauckas-Claude]

Note

This PR should be ignored until reviewed by @ChrisRackauckas.

Summary

Adds TaylorHomotopyContinuationJL{AllRoots} to NonlinearSolveHomotopyContinuation: a solver for general (non-polynomial) nonlinear systems that

1. Taylor-expands the system around the initial guess into a polynomial surrogate of total degree degree (default 3), via TaylorSeries.jl jet transport — works on any generic Julia function, same operator-overloading requirement as ForwardDiff duals;
2. finds all roots of the surrogate with HomotopyContinuation.jl (no initial-guess dependence);
3. polishes every (near-)real path endpoint with a Newton iteration on the original system, reporting the distinct converged roots.

If the system is polynomial of total degree ≤ degree, the surrogate is exact and all roots are found. AllRoots = true returns an EnsembleSolution of every root (mirroring HomotopyContinuationJL{true}); the default single-root form returns the converged root closest to the initial guess. Supports oop/iip/scalar problems, explicit jacobians, and HomotopyNonlinearFunction transformations.

This complements the existing algorithm: HomotopyContinuationJL{true} requires a genuinely polynomial (HC-traceable) system; this works on transcendental ones at the cost of approximation, with the Newton polish recovering exactness of the reported roots.

Motivation / prototype results

Prototyped and benchmarked before this PR (details: https://github.com/ChrisRackauckas/InternalJunk/pull/69). On an 8-problem hard suite × 20 random starts (success = ‖f‖∞ < 1e-8): 100% success on every problem, including Freudenstein–Roth where TrustRegion stalls at the classic local-min of ‖f‖ (8/20), exp–sin where plain Newton scores 11/20, Bratu n=6 (returns both solution branches; Newton 17/20), Powell singular (multiplicity-4 root), and Kearfott's badly-scaled problem. Median solve times after tuning: 3–50 ms for n ≤ 5, ~0.4 s for n = 6 with 729 paths. The degree^n path count limits the method to roughly n ≤ 12 at degree 2, n ≤ 8 at degree 3 (documented in the docstring).

Implementation notes (perf/correctness findings baked in)

• compile = false for the HC solve — every call builds a System with fresh numeric coefficients, so HC's compiled straight-line programs never get cache hits; without this each solve costs ~3.5 s of codegen (700× slowdown at these sizes).
• All finite path endpoints are harvested, not just HC.solutions() — multiplicity-k roots (e.g. Powell singular) are classified as "excess" endpoints and dropped by the default accessors under every flag combination. As Newton guesses every finite endpoint is fair game; the polish filters junk.
• Polish success is judged by achieved residual, not step convergence — at singular roots Newton converges only linearly and would report MaxIters despite an excellent iterate.
• Total-degree start system for small path counts — dense Taylor truncations have mixed volume = Bézout number, so polyhedral mixed-cell computation is pure overhead (measured 47× on a trigexp system); polyhedral is used automatically above 200 paths.
• TaylorSeries jet spaces are cached per (numvars, order) behind a lock (variables! rebuilds multiplication tables on every call).

Tests

New test/taylor_polynomialize.jl wired into the Core group: scalar transcendental (all 3 roots of u = 2sin(u) + closest-root variant), exact recovery of all 9 Himmelblau critical points, transcendental oop (exp–sin) and iip (Bratu, both branches), explicit-jacobian path, no-real-roots retcodes, multiplicity-4 singular root, and HomotopyNonlinearFunction polynomialize/unpolynomialize round-trip.

Ran locally with Pkg.test() on Julia 1.11:

Test Summary: | Pass  Total     Time
AllRoots      |  169    169  3m52.6s
Test Summary: | Pass  Total     Time
Single Root   |   50     50  1m47.4s
Test Summary:        | Pass  Total     Time
Taylor Polynomialize |   40     40  8m50.0s
     Testing NonlinearSolveHomotopyContinuation tests passed

Runic has been run on all changed files. Version bumped 0.1.10 → 0.1.11; TaylorSeries (0.22.4+) added as a dependency; docstring added to the docs API page in the same PR.

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https://claude.ai/code/session_01AYuTWSRDz6kE8Bn8vAKh57

[ChrisRackauckas-Claude]

CI analysis after first full run:

Everything exercising this PR's change is green — sublibraries / lib/NonlinearSolveHomotopyContinuation Core tests on Julia 1 / lts / pre, QA (Aqua/ExplicitImports), downgrade-sublibraries for this sublib (validates the TaylorSeries 0.22.4 compat floor), Runic format check, spell check, and all downstream tests (ModelingToolkit, OrdinaryDiffEq, BoundaryValueDiffEq, DiffEqCallbacks).

The two failing jobs are pre-existing master failures, unrelated to this diff:

• Documentation: fails with ERROR: LoadError: Duplicate BibTeX entry key detected: 'yuan2015recent' — fails identically on the 2026-07-19 master push and all recent PRs; the build dies while loading the bibliography, before reaching this PR's added @docs entry.
• Downgrade / Downgrade Tests - Core (main package): fails on the 2026-07-19 master push and nearly all PRs since 2026-07-17. This PR does not touch the root project.

Both master failures are being investigated separately (bisect + fix PRs to follow).

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[ChrisRackauckas-Claude]

Follow-up on the two pre-existing master CI failures noted above — both are now root-caused with fixes up:

• Documentation: fixed by Remove duplicate 'yuan2015recent' BibTeX entry breaking docs build #1097 (duplicate BibTeX key introduced in 2a6f1a9, Jan 2024; its Documentation CI job passes). Once merged, this PR's docs job should go green on a rerun/rebase, rendering the new TaylorHomotopyContinuationJL docstring.
• Downgrade Tests - Core: fixed by Raise root LinearSolve compat floor to 5 to fix Downgrade CI #1095 (root LinearSolve compat floor, bisected to the revert in Require operator-capable NonlinearSolveBase #1079). The one remaining failure that fix unmasks is a machine-dependent robustness-test flake tracked in Machine-dependent failure: 23 Test Problems Broyden alg #4 on Brown almost linear (unmarked broken in #1051, fails on CI runners) #1096 (pre-existing on master's own Core jobs; unrelated to this PR).

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