fix: give up on Bareiss solves that grow past what is workable - #77
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Bareiss is fraction-free, which is what lets it sidestep the nested-fraction growth of `symbolic_linear_solve`, but its entries are exact polynomials whose degree grows with every pivot. On a dense system in many atoms that growth is combinatorial: the harmonic equations of a van der Pol oscillator in three harmonics never finish, and one multiplication can exhaust memory outright. Bound both the predicted width of a product, checked before multiplying since a single multiply can OOM before its result can be measured, and the width of a finished entry. The systems this solve handles well are far below the bounds: 10 monomials for a Duffing oscillator, 28 for a parametron, 344 for a van der Pol with two harmonics. Raise `BareissTooLarge` rather than `BareissFailure`. The two are not the same condition: `BareissFailure` means the system is not polynomial in the unknowns and `rearrange!` answers it with the LU fallback, whereas the LU fallback grows without bound on exactly the systems that are too large for Bareiss. Size has to reach a caller that has a cheaper route instead.
SymbolicUtils 3 folded this identity away on its own and SymbolicUtils 4 does not, which is why the determinant-carrying denominators of a symbolic linear solve stopped simplifying. `collapse_pythagorean` applies it directly, in one bottom-up walk rather than the exponential round trip `trig_reduce` needs, and `rearrange!` uses it on the solutions. No measurable speedup to report: the Krylov-Bogoliubov equations, the caller whose systems actually carry trig, move from 3.61s to 3.43s at order 2, which is noise. The resulting equations are unchanged. Worth having as the identity being restored, not as a performance fix.
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The limit cycle tutorial hangs. A van der Pol ansatz in three harmonics gives a
dense 6x6 whose Bareiss entries grow combinatorially, and one multiplication can
exhaust memory before its result can even be measured. Bound the predicted width
of a product before multiplying, and the width of a finished entry after. The
systems this solve handles well sit far below the bounds: 10 monomials for a
Duffing oscillator, 28 for a parametron, 344 for a van der Pol with two harmonics.
Size raises
BareissTooLarge, notBareissFailure. They're different conditions:BareissFailuremeans the system isn't polynomial in the unknowns andrearrange!answers it with the LU fallback, but that fallback grows withoutbound on exactly the systems that are too large for Bareiss. Size has to reach a
caller with a cheaper route instead, which HarmonicBalance's
add_jacobian!nowis.
Second commit collapses
cos^2 + sin^2, which SymbolicUtils 4 stopped folding.It's correct and the equations are unchanged, but I measured no benefit: Krylov
order 2 goes 3.61s -> 3.43s, which is noise. Split out so it can be dropped.