fmcubic 0.1.0

Reference implementation of the Fernández Molina series method for solving cubic equations (AIP Advances 12, 045002, 2022).

fmcubic

A serious, honest Python implementation of the Fernández Molina series method for cubic equations.

Reference: Fernández Molina, R. A., Sigalotti, L. Di G., Rendón, O., & Mejías, A. J. (2022). A rapidly convergent method for solving third-order polynomials. AIP Advances 12, 045002. https://doi.org/10.1063/5.0073851

Authors of the mathematical method: Ramón A. Fernández Molina · Leonardo Di G. Sigalotti · Otto Rendón · Antonio J. Mejías

Reference Python implementation of the method. Explicit about what is faithful to the paper and what is a practical fallback in IEEE-754 double precision.

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Install

pip install -e .
# or just drop the `fmcubic/` folder next to your code

Requires Python ≥ 3.11 (uses math.cbrt).

Usage

from fmcubic import solve_cubic

res = solve_cubic(1, -6, 11, -6)
print(res.method)        # → "Fernández Molina (trigonometric fallback, Δ<0)"
print(res.branch)        # → "trig-viete"
print(res.iterations)    # → 1
for r in res.roots:
    print(r)             # → 1, 2, 3
print(res.notes)

solve_cubic returns a SolveResult with:

field	meaning
roots	three Complex(re, im) roots
method	human-readable branch label
branch	machine tag (series-A-Eq17, trig-viete, …)
iterations	series terms or Newton iterations
convergence_trace	per-step approximation of x₁
p, q, delta	depressed-cubic invariants
ratio	q²/(4Δ) — the series convergence ratio
notes	free-form caveats

Branch tree

Tag	Condition	Origin in paper	Status
triple-root	p≈0 ∧ q≈0	trivial	paper-faithful
p-axis-Eq57	p≈0, q≠0	Eq. (57)	paper-faithful
delta-zero	Δ≈0	closed form	paper-faithful
curve-Eq37-58	p³ + 27q²/2 ≈ 0	Eq. (37) + (58)	paper-faithful
series-A-Eq17	Δ>0,	q²/(4Δ)	< 1
series-B-Eq30	Δ>0,	q²/(4Δ)	> 1
buffer-newton			ratio
trig-viete	Δ < 0	Viète	numerical equivalent

The complex-branch series proposed in the paper for Δ<0 is not implemented faithfully — we use Viète's trigonometric form, which is mathematically exact but is a substitution, not a faithful execution.

Numerical honesty

The solve_cubic docstring spells this out, and the Patel-Teja benchmark demonstrates it:

• Δ = q²/4 + p³/27 suffers catastrophic cancellation when q²/4 ≈ −p³/27 (e.g. Patel-Teja case V: ~10⁵¹ + ~10⁵¹ → ~10³⁷). In double precision this leaves ~2 significant digits of Δ.
• The series method consumes the ratio 4Δ/q², which is far less sensitive — but any caller using Δ directly should be aware.
• Some polynomials evaluate to f(x) == 0.0 exactly in IEEE-754 at a point that does not satisfy Vieta's relations (spurious zero). Use vieta_residuals(A, B, C, D, roots) to detect this.

Patel-Teja benchmark

from fmcubic import run_patel_teja_validation

report = run_patel_teja_validation()
print("OK?" , report.overall_pass)
for cr in report.cases:
    print(cr.case.id, "Δ=", cr.delta, "pass=", cr.pass_overall)
    for m in cr.methods:
        print(" ", m.method, m.picked, "vieta-sum=", m.vieta["sum"])

Case	Conditioning	Validation criterion
Z	severe (Δ ≈ 10⁻³⁴)	FM↔Cardano consensus
ρ	well conditioned	FM matches paper to <1e-7 %
V	extreme (10⁵¹+10⁵¹)	FM↔Cardano consensus + Vieta check

Tests

python -m pip install pytest
python -m pytest -q

Covers: triple root, p=0/q≠0, Δ=0, three real roots, complex conjugates, extreme coefficients, buffer zone, Patel-Teja Z/ρ/V, and Vieta consistency.

What's faithful and what isn't (one-page summary)

Faithful to the paper:

• Depression to y³ + py + q = 0 and shift recovery.
• All four closed-form special cases (B1-B4) including the special curve p³ + 27q²/2 = 0 (Eqs. 37 + 58).
• Series A (Eq. 17) and Series B (Eq. 30) with incremental binomial recurrence c_{i+1} = c_i · (1/3 − i)/(i+1).
• Stabilized Eq. (51)-(53) deflation Γ for the other two roots.

Practical / equivalent:

• Trigonometric (Viète) form for Δ<0 — paper proposes a complex series; we substitute the closed form.
• A small buffer zone around |ratio|=1 that switches to Newton-Raphson because the series convergence is impractically slow there.

Not yet implemented:

• The complex-branch series for Δ<0.
• Extended-precision (≥128-bit) mode required to reproduce Patel-Teja cases Z and V exactly.

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Mathematical method © Fernández Molina, Sigalotti, Rendón & Mejías (2022). Please cite: AIP Advances 12, 045002. https://doi.org/10.1063/5.0073851