[AUD-005][High] Correct Durand-Kerner for non-monic polynomials

[gabrielsantosphilips]

Audit finding

• ID: AUD-005
• Status: Verified defect
• Severity: High
• Confidence: High
• Audited revision: 2f479320d805a1f9f35ebe4afaaeeded48913a94

Problem

The Durand-Kerner correction divides by the root-difference product but omits the polynomial leading coefficient. Polynomial scaling must not change roots.

Source:

numerical-toolbox-cpp/numerical/solvers/DurandKerner.hpp

Lines 69 to 86 in 2f47932

	}
	template<typename T, std::size_t MaxOrder>
	bool DurandKerner<T, MaxOrder>::Iterate(Roots& roots,
	std::span<const T> coefficients, std::size_t order, T tolerance)
	{
	bool converged = true;
	for (std::size_t r = 0; r < order; ++r)
	{
	auto pVal = EvaluatePolynomial(coefficients, roots[r]);
	auto denominator = ComputeDenominator(roots, r, order);
	auto correction = pVal / denominator;
	roots[r] -= correction;
	if (math::Abs(correction) > tolerance)
	converged = false;
	}

Reproduction

10(x-1)(x-2), represented as {10, -30, 20}, returned two NaN roots. All existing nonlinear tests use monic polynomials.

The solver also returns initial/last iterates without convergence status and uses a tolerance-based sort comparator that is not a strict weak ordering.

Acceptance criteria

• Normalize coefficients or include the leading coefficient in each correction.
• Root results are invariant under nonzero coefficient scaling.
• Return explicit convergence/failure status.
• Replace the comparator with a strict weak ordering.
• Add non-monic residual and scale-invariance TEST_F coverage.