A least-squares fit of Lennard-Jones parameters and to simulated pressure data is…

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Problem

A least-squares fit of Lennard-Jones parameters

\epsilon

and

\sigma

to simulated pressure data is done by the normal equations, whose coefficient matrix has condition number

\kappa = 3.0\times10^{6}

. Two installations are built from an identical lockfile, which names every package with its exact version and hash, the compiled linear-algebra library included. They run on machines with different processors, and on each processor that library takes a different vectorised path through the products and sums of the solve. The two fitted values agree to eight significant figures and differ starting at the ninth. Binary64 arithmetic has machine epsilon

\epsilon_{\rm mach} = 2^{-52} \approx 2.22\times10^{-16}

. ((a)) Compute

\kappa\epsilon_{\rm mach}

. ((b)) State whether the observed ninth-digit disagreement is consistent with rounding alone, or points to a bug. ((c)) Name one thing the shared lockfile fixes and one thing it does not fix in this situation.

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