Evaluating parametric holonomic sequences using rectangular splitting

We adapt the rectangular splitting technique of Paterson and Stockmeyer to the problem of evaluating terms in holonomic sequences that depend on a parameter. This approach allows computing the $n$-th term in a recurrent sequence of suitable type using $O(n^{1/2})$ "expensive" operations at the cost of an increased number of "cheap" operations. Rectangular splitting has little overhead and can perform better than either naive evaluation or asymptotically faster algorithms for ranges of $n$ encountered in applications. As an example, fast numerical evaluation of the gamma function is investigated. Our work generalizes two previous algorithms of Smith.Comment: 8 pages, 2 figure

Computing hypergeometric functions rigorously

We present an efficient implementation of hypergeometric functions in arbitrary-precision interval arithmetic. The functions ${}_0F_1$, ${}_1F_1$, ${}_2F_1$ and ${}_2F_0$ (or the Kummer $U$-function) are supported for unrestricted complex parameters and argument, and by extension, we cover exponential and trigonometric integrals, error functions, Fresnel integrals, incomplete gamma and beta functions, Bessel functions, Airy functions, Legendre functions, Jacobi polynomials, complete elliptic integrals, and other special functions. The output can be used directly for interval computations or to generate provably correct floating-point approximations in any format. Performance is competitive with earlier arbitrary-precision software, and sometimes orders of magnitude faster. We also partially cover the generalized hypergeometric function ${}_pF_q$ and computation of high-order parameter derivatives.Comment: v2: corrected example in section 3.1; corrected timing data for case E-G in section 8.5 (table 6, figure 2); adjusted paper siz

Computing the Lambert W function in arbitrary-precision complex interval arithmetic

We describe an algorithm to evaluate all the complex branches of the Lambert W function with rigorous error bounds in interval arithmetic, which has been implemented in the Arb library. The classic 1996 paper on the Lambert W function by Corless et al. provides a thorough but partly heuristic numerical analysis which needs to be complemented with some explicit inequalities and practical observations about managing precision and branch cuts.Comment: 16 pages, 4 figure

Convergence rates for loop-erased random walk and other Loewner curves

We estimate convergence rates for curves generated by Loewner's differential equation under the basic assumption that a convergence rate for the driving terms is known. An important tool is what we call the tip structure modulus, a geometric measure of regularity for Loewner curves parameterized by capacity. It is analogous to Warschawski's boundary structure modulus and closely related to annuli crossings. The main application we have in mind is that of a random discrete-model curve approaching a Schramm-Loewner evolution (SLE) curve in the lattice size scaling limit. We carry out the approach in the case of loop-erased random walk (LERW) in a simply connected domain. Under mild assumptions of boundary regularity, we obtain an explicit power-law rate for the convergence of the LERW path toward the radial SLE$_2$ path in the supremum norm, the curves being parameterized by capacity. On the deterministic side, we show that the tip structure modulus gives a sufficient geometric condition for a Loewner curve to be H\"{o}lder continuous in the capacity parameterization, assuming its driving term is H\"{o}lder continuous. We also briefly discuss the case when the curves are a priori known to be H\"{o}lder continuous in the capacity parameterization and we obtain a power-law convergence rate depending only on the regularity of the curves.Comment: Published in at http://dx.doi.org/10.1214/13-AOP872 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Political E-Mail: Protected Speech or Unwelcome Spam?

Candidates for political office are using unsolicited bulk e-mails to reach the electorate. Commonly known as political spam, this campaign tactic is an inexpensive supplement to television, radio, and print ads. Advocates claim that campaigning via the internet reduces candidates\u27 dependence on fundraising, but critics detest political spam as the latest nuisance. This iBrief examines the legal basis for political spam, distinguishes political spam from analogous regulated speech, and argues that political spam serves an interest worth protecting