[AUD-004][High] Return SVD failure when Golub-Kahan diagonalization stalls

[gabrielsantosphilips]

Audit finding

• ID: AUD-004
• Status: Verified defect
• Severity: High
• Confidence: High
• Audited revision: 2f479320d805a1f9f35ebe4afaaeeded48913a94

Problem

SingularValueDecomposition::Decompose() invokes a bounded, void diagonalization and always returns true, even when all iterations expire.

Source:

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

Lines 147 to 181 in 2f47932

	OPTIMIZE_FOR_SPEED bool SingularValueDecomposition<T, Rows, Cols>::Decompose(
	const math::Matrix<T, Rows, Cols>& a)
	{
	math::Matrix<T, Rows, Cols> bidiag = a;
	LeftVectors leftVecs{};
	RightVectors rightVecs{};
	Betas leftBeta{};
	Betas rightBeta{};
	Bidiagonalize(bidiag, leftVecs, leftBeta, rightVecs, rightBeta);
	ExtractBidiagonal(bidiag);
	AccumulateU(leftVecs, leftBeta);
	AccumulateV(rightVecs, rightBeta);
	DiagonalizeGolubKahan();
	MakeSingularValuesPositive();
	SortDescending();
	return true;
	}
	template<typename T, std::size_t Rows, std::size_t Cols>
	bool SingularValueDecomposition<T, Rows, Cols>::NextBlock(std::size_t& p, std::size_t& q)
	{
	for (std::size_t i = 0; i < Cols - 1; ++i)
	{
	T thresh = T{ 1e-6f } * (math::Abs(sigma.at(i, 0)) + math::Abs(sigma.at(i + 1, 0)));
	if (math::Abs(superdiag.at(i, 0)) <= thresh)
	superdiag.at(i, 0) = T{};
	}
	std::size_t right = Cols - 1;
	while (right > 0)
	{
	T thresh = T{ 1e-6f } * (math::Abs(sigma.at(right - 1, 0)) + math::Abs(sigma.at(right, 0)));

Reproduction

For a 3x3 matrix whose only nonzero entry is A(0,1)=1, the singular values must be (1,0,0). The implementation returns success with (0,0,0). The zero-diagonal/nonzero-superdiagonal block makes the QR sweep ineffective until the 270-iteration limit expires.

Acceptance criteria

• Diagonalization returns and propagates convergence status.
• Handle zero-diagonal bidiagonal blocks correctly.
• A failed decomposition cannot expose results as valid.
• Add reconstruction, residual, and status tests for this matrix and other bidiagonal edge forms.