Claude/gaussian newton methods f p nsz#44
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Ray r(t)=O+t*d reduces F(x,y,z)=0 to g(t)=F(O+t*d); all intersections are the real roots of g on [t_min, t_max]. Pluggable backends behind RaySurface: - sampling: safeguarded Newton/bisection reference oracle - sturm: exact all-roots via interpolation, square-free, Sturm isolation; complex surfaces via gcd of Re g and Im g - interval/chebyshev: placeholders for later phases Reuses Solver.evaluate/get_derivative for g and grad F . d. Tests pending local extension build. https://claude.ai/code/session_015PRdmeBfg6J5JZdsAvXvRT
- Fix Newton divided-difference denominator (xs[i]-xs[i-j]) in interpolation - Add Number.__neg__, .precision, .is_complex, .parts(), .wrap() public API - Refactor backends to use the public Number API; pylint 10/10, isort clean - All 12 intersection tests pass; full suite 739 passed https://claude.ai/code/session_015PRdmeBfg6J5JZdsAvXvRT
g is fitted with a Chebyshev interpolant (Chebyshev-Gauss nodes), converted to a monomial proxy, isolated via the shared Sturm machinery (_isolate.py), then simple roots polished on the true g by Newton. auto now routes transcendental real surfaces here instead of sampling. Captures even-multiplicity roots. Extract _isolate.py (sturm_chain/variations/isolate/bisect/sign) shared by the sturm and chebyshev backends. pylint 10/10; full suite 742 passed. https://claude.ai/code/session_015PRdmeBfg6J5JZdsAvXvRT
- subdivision.py: adaptive Taylor-bound exclusion (|g(m)| <= |g'(m)|h + M2/2 h^2) with M2 estimated from g'; refines simple roots (rtsafe) and tangencies (g'=0), rejecting turning points where g != 0. Pure Python, real surfaces. - Chebyshev now self-validates: escalates degree until the spectral tail converges, so oscillatory walls a low degree would miss are captured. Fit reduced to O(m^2). - Move shared rtsafe into _isolate; sampling/subdivision reuse it. - Tests: corrugated capillary wall x^2+y^2-(1+0.3 sin 4z)^2 (12 roots at k*pi/4), Chebyshev self-validation parity, tangent double roots, polynomial parity, complex-rejection. 21 intersection tests; full suite 748 passed. https://claude.ai/code/session_015PRdmeBfg6J5JZdsAvXvRT
…rwise auto now reconciles both general backends for non-polynomial real surfaces: Chebyshev (precise on smooth roots) is backed up by subdivision (the safety net), returning their deduped union so neither method's blind spot drops a root. Drop the now-unused surface arg from get_backend. Test: auto on the corrugated wall returns all 12 roots. Full suite 749 passed. https://claude.ai/code/session_015PRdmeBfg6J5JZdsAvXvRT
Add a section covering the ray->g(t) reduction, the method table (auto/sturm/chebyshev/subdivision/sampling), tangent-root recovery, complex support via sturm, and the practical-vs-rigorous caveat for subdivision. https://claude.ai/code/session_015PRdmeBfg6J5JZdsAvXvRT
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