Simple and Nearly Optimal Polynomial Root-finding by Means of Root Radii Approximation

We propose a new simple but nearly optimal algorithm for the approximation of all sufficiently well isolated complex roots and root clusters of a univariate polynomial. Quite typically the known root-finders at first compute some crude but reasonably good approximations to well-conditioned roots (that is, those isolated from the other roots) and then refine the approximations very fast, by using Boolean time which is nearly optimal, up to a polylogarithmic factor. By combining and extending some old root-finding techniques, the geometry of the complex plane, and randomized parametrization, we accelerate the initial stage of obtaining crude to all well-conditioned simple and multiple roots as well as isolated root clusters. Our algorithm performs this stage at a Boolean cost dominated by the nearly optimal cost of subsequent refinement of these approximations, which we can perform concurrently, with minimum processor communication and synchronization. Our techniques are quite simple and elementary; their power and application range may increase in their combination with the known efficient root-finding methods.Comment: 12 pages, 1 figur

Multi-Wavelength Properties of Barred Galaxies in the Local Universe. I: Virgo Cluster

We study in detail how the barred galaxy fraction varies as a function of luminosity, HI gas mass, morphology and color in the Virgo cluster in order to provide a well defined, statistically robust measurement of the bar fraction in the local universe spanning a wide range in luminosity (factor of ~100) and HI gas mass. We combine multiple public data-sets (UKIDSS near-infrared imaging, ALFALFA HI gas masses, GOLDMine photometry). After excluding highly inclined systems, we define three samples where galaxies are selected by their B-band luminosity, H-band luminosity, and HI gas mass. We visually assign bars using the high resolution H-band imaging from UKIDSS. When all morphologies are included, the barred fraction is ~17-24% while for morphologically selected discs, we find that the barred fraction in Virgo is ~29-34%: it does not depend strongly on how the sample is defined and does not show variations with luminosity or HI gas mass. The barred fraction depends most strongly on the morphological composition of the sample: when the disc populations are separated into lenticulars (S0--S0/a), early-type spirals (Sa--Sb), and late-type spirals (Sbc--Sm), we find that the early-type spirals have a higher barred fraction (~45-50%) compared to the lenticulars and late-type spirals (~22-36%). This difference may be due to the higher baryon fraction of early-type discs which makes them more susceptible to bar instabilities. We do not find any evidence of barred galaxies being preferentially blue.Comment: 13 pages, 14 figures. Submitted to Ap