NUMERICAL ANALYSIS

Hilbert-Matrix Determinant: Where FP64 Flips Sign

The Hilbert-matrix determinant is provably positive for every n. Computed in FP64, the sign comes out negative for n=14, 15, 16, and 20. This is not a library bug — it is what happens when the condition number exceeds the precision FP64 has to work with.

The Evidence

Mathematical Fact

The Hilbert matrix H_n has entries H_ij = 1/(i+j-1). Its determinant is positive for every n — a proven theorem. So any negative sign a numerical routine returns is a precision artifact, not the true value.

Matrix Size	FP64 Sign (NumPy)	VLA Sign
n = 13	+	+
n = 14	-	+
n = 15	-	+
n = 16	-	+
n = 17-19	+	+
n = 20	-	+

FP64 sign errors (n=14,15,16,20)

VLA sign errors

Why This Happens

The Hilbert Matrix is Notoriously Ill-Conditioned

The condition number grows exponentially with n. For n=15, the condition number exceeds 10¹⁷ — meaning FP64's 15-16 significant digits are completely consumed.

NumPy uses FP64, which carries ~15 decimal digits. At n=15 the Hilbert matrix needs more precision than that to resolve the determinant's sign, so the computed sign can flip — a result the true value rules out.

VLA's Advantage

VLA uses 512-bit integer arithmetic providing 77 decimal digits of precision. Even for n=20 Hilbert matrices (condition number ~10²⁸), VLA has enough precision to compute exact results.

Implications

Control Systems

Hilbert matrices appear in polynomial fitting and control system design. Wrong signs can lead to unstable controllers.

Optimization

Many optimization algorithms involve ill-conditioned matrices. If your solver gives wrong answers on Hilbert matrices, it will give wrong answers elsewhere too.

Validation

The Hilbert matrix is a standard ill-conditioning benchmark. If FP64 loses the sign here, it is worth checking where else in a pipeline conditioning quietly exceeds its precision.

Can Your Calculations Be Trusted?

Let us validate your numerical pipelines with exact arithmetic.

Get in touch