In this article, we present an efficient algorithm to compute the faithful rounding of the l 2 -norm of a floating-point vector. This means that the result is accurate to within 1 bit of the underlying floating-point type. This algorithm does not generate overflows or underflows spuriously, but does so when the final result calls for such a numerical exception to be raised. Moreover, the algorithm is well suited for parallel implementation and vectorization. The implementation runs up to 3 times faster than the netlib version on current processors.

A new class of chaotic systems is discovered that are generated in a practical, nonlinear, mutually coupled phase-locked loop (PLL) circuit. Presented theoretical results make it possible to understand experimental results of mutually coupled PLL's on the onset of chaos using the geometry of the invariant manifolds, while the resultant simple geometry and complex dynamics is expected to have applications in other areas, e.g., power systems or interacting bar magnets. Motivated by the numerical study of this system, the topological horseshoe is proven to be generated in the codimension 3 unfolding of a degenerated orbit-flip homoclinic point for this system. Qualitatively different type of bifurcation phenomena are also observed to appear depending on the phase detector (PD) characteristics.