[AUD-024][High] Make condition-number detection scale invariant

[gabrielsantosphilips]

Audit finding

• ID: AUD-024
• Status: Verified defect
• Severity: High
• Confidence: High
• Audited revision: 2f479320d805a1f9f35ebe4afaaeeded48913a94

Problem

ConditionNumber() applies an absolute 1e-12 zero-row/norm threshold. This makes classification depend on matrix units rather than conditioning. It also screens only near-zero rows before calling assert-based Gaussian elimination.

Source:

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

Lines 15 to 51 in 2f47932

	{
	namespace detail
	{
	template<typename T, std::size_t N>
	bool HasZeroRow(const math::SquareMatrix<T, N>& a)
	{
	for (std::size_t i = 0; i < N; ++i)
	{
	bool allZero = true;
	for (std::size_t j = 0; j < N; ++j)
	{
	if (math::Abs(a.at(i, j)) > static_cast<T>(1e-12))
	{
	allZero = false;
	break;
	}
	}
	if (allZero)
	return true;
	}
	return false;
	}
	}
	template<typename T, std::size_t N>
	[[nodiscard]] OPTIMIZE_FOR_SPEED std::optional<T> ConditionNumber(const math::SquareMatrix<T, N>& a)
	{
	static_assert(std::is_floating_point_v<T>, "ConditionNumber supports floating-point types");
	if (detail::HasZeroRow(a))
	return std::nullopt;
	T normA = math::OneNorm(a);
	if (normA < static_cast<T>(1e-12))
	return std::nullopt;
	auto invA = SolveSystem(a, math::SquareMatrix<T, N>::Identity());
	return normA * math::OneNorm(invA);
	}
	}

Reproduction

diag(5e-13, 5e-13) is invertible with condition number one, but returns nullopt. A dependent-row matrix with no near-zero row passes the screen and reaches a zero pivot.

Acceptance criteria

• Rank/pivot detection is relative to matrix scale.
• Singular factorization returns status instead of asserting.
• Multiplying a nonsingular matrix by nonzero scalar does not change its reported condition number.
• Add multi-decade scale and dependent-row tests.

[gabrielfrasantos]

Fixed and merged in #325 (d71b05f on main).

Both reported cases reproduced: diag(5e-13, 5e-13) returned nullopt despite being perfectly conditioned, and a dependent-row matrix passed the near-zero-row screen and reached a zero pivot in assert-based Gaussian elimination.

The absolute 1e-12 screen is gone. Conditioning is now judged relative to matrix scale, and the solve goes through a new non-asserting TrySolveSystem that does its own elimination with a scale-relative pivot threshold (||A||_inf * eps * N) and returns std::nullopt on a singular or non-finite factorization instead of asserting.

Scale invariance measured across 24 decades on the same matrix:

matrix	condition number
[[3,1],[1,2]] * 5e-13	1.0
[[3,1],[1,2]]	3.2
[[3,1],[1,2]] * 5e11	1.0
diag(5e-13, 5e-13)	1.0
[[1,2],[2,4]] (dependent)	nullopt

The existing AllTinyRowReturnsNullopt test still passes, but now for the right reason: that matrix is genuinely near-singular rather than merely small.

Tests added: ScaleInvariantAcrossDecades and DependentRowsReturnNullopt.

TrySolveSystem is reusable and is what #299's H-infinity work uses to invert the augmented weight matrix without asserting.

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