An asymptotic expansion for the error term in the Brent-McMillan algorithm for Euler’s constant

The Brent-McMillan algorithm is the fastest known procedure for the high-precision computation of Euler’s constant γ and is based on the modified Bessel functions I_0(2x) and K_0(2x). An error estimate for this algorithm relies on the optimally truncated asymptotic expansion for the product I_0(2x)K_0(2x) when x assumes large positive integer values. An asymptotic expansion for this optimal error term is derived by exploiting the techniques developed in hyperasymptotics, thereby enabling more precise information on the error term than recently obtained bounds and estimates

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

Detection of a tropospheric ozone anomaly using a newly developed ozone retrieval algorithm for an up-looking infrared interferometer

Author Posting. © American Geophysical Union, 2009. This article is posted here by permission of American Geophysical Union for personal use, not for redistribution. The definitive version was published in Journal of Geophysical Research 114 (2009): D06304, doi:10.1029/2008JD010270.On 2 June 2003, the Baltimore Bomem Atmospheric Emitted Radiance Interferometer (BBAERI) recorded an infrared spectral time series indicating the presence of a tropospheric ozone anomaly. The measurements were collected during an Atmospheric Infrared Sounder (AIRS) validation campaign called the 2003 AIRS BBAERI Ocean Validation Experiment (ABOVE03) conducted at the United States Coast Guard Chesapeake Light station located 14 miles due east of Virginia Beach, Virginia (36.91°N, 75.71°W). Ozone retrievals were performed with the Kurt Lightner Ozone BBAERI Retrieval (KLOBBER) algorithm, which retrieves tropospheric column ozone, surface to 300 mbar, from zenith-viewing atmospheric thermal emission spectra. KLOBBER is modeled after the AIRS retrieval algorithm consisting of a synthetic statistical regression followed by a physical retrieval. The physical retrieval is implemented using the k-Compressed Atmospheric Radiative Transfer Algorithm (kCARTA) to compute spectra. The time series of retrieved integrated ozone column on 2 June 2003 displays spikes of about 10 Dobson units, well above the error of the KLOBBER algorithm. Using instrumentation at Chesapeake Light, satellite imaging, trace gas retrievals from satellites, and Potential Vorticity (PV) computations, it was determined that these sudden increases in column ozone likely were caused by a combination of midtropospheric biomass burning products from forest fires in Siberia, Russia, and stratospheric intrusion by a tropopause fold occurring over central Canada and the midwestern United States.NASA for its support through grant NAG5- 1156-7 for AIRS Validation and grant NNG04GN42G for development of AIRS trace gas products, and through a subcontract with JPL on the AIRS Project prime contract NAS7-03001 for continuing optimization and validation of AIRS trace gas products.

Modelling of the Effects of Stellar Feedback during Star Cluster Formation Using a Hybrid Gas and N-Body Method

Understanding the formation of stellar clusters requires following the interplay between gas and newly formed stars accurately. We therefore couple the magnetohydrodynamics code FLASH to the N-body code ph4 and the stellar evolution code SeBa using the Astrophysical Multipurpose Software Environment (AMUSE) to model stellar dynamics, evolution, and collisional N-body dynamics and the formation of binary and higher-order multiple systems, while implementing stellar feedback in the form of radiation, stellar winds and supernovae in FLASH. We here describe the algorithms used for each of these processes. We denote this integrated package Torch. We then use this novel numerical method to simulate the formation and early evolution of several examples of open clusters of ~1000 stars formed from clouds with a mass range of 10^3-10^5 M_sun. Analyzing the effects of stellar feedback on the gas and stars of the natal clusters, we find that in these examples, the stellar clusters are resilient to disruption, even in the presence of intense feedback. This can even slightly increase the amount of dense, Jeans unstable gas by sweeping up shells; thus, a stellar wind strong enough to trap its own H II region shows modest triggering of star formation. Our clusters are born moderately mass segregated, an effect enhanced by feedback, and retained after the ejection of their natal gas, in agreement with observations.Comment: Accepted to ApJ. 29 pages, 18 figures. Source code at https://bitbucket.org/torch-sf/torch/ and documentation at https://torch-sf.bitbucket.io

A Survey on Homomorphic Encryption Schemes: Theory and Implementation

Legacy encryption systems depend on sharing a key (public or private) among the peers involved in exchanging an encrypted message. However, this approach poses privacy concerns. Especially with popular cloud services, the control over the privacy of the sensitive data is lost. Even when the keys are not shared, the encrypted material is shared with a third party that does not necessarily need to access the content. Moreover, untrusted servers, providers, and cloud operators can keep identifying elements of users long after users end the relationship with the services. Indeed, Homomorphic Encryption (HE), a special kind of encryption scheme, can address these concerns as it allows any third party to operate on the encrypted data without decrypting it in advance. Although this extremely useful feature of the HE scheme has been known for over 30 years, the first plausible and achievable Fully Homomorphic Encryption (FHE) scheme, which allows any computable function to perform on the encrypted data, was introduced by Craig Gentry in 2009. Even though this was a major achievement, different implementations so far demonstrated that FHE still needs to be improved significantly to be practical on every platform. First, we present the basics of HE and the details of the well-known Partially Homomorphic Encryption (PHE) and Somewhat Homomorphic Encryption (SWHE), which are important pillars of achieving FHE. Then, the main FHE families, which have become the base for the other follow-up FHE schemes are presented. Furthermore, the implementations and recent improvements in Gentry-type FHE schemes are also surveyed. Finally, further research directions are discussed. This survey is intended to give a clear knowledge and foundation to researchers and practitioners interested in knowing, applying, as well as extending the state of the art HE, PHE, SWHE, and FHE systems.Comment: - Updated. (October 6, 2017) - This paper is an early draft of the survey that is being submitted to ACM CSUR and has been uploaded to arXiv for feedback from stakeholder

Computing Stieltjes constants using complex integration

The generalized Stieltjes constants $\gamma\_n(v)$ are, up to a simple scaling factor, the Laurent series coefficients of the Hurwitz zeta function $\zeta(s,v)$ about its unique pole $s = 1$. In this work, we devise an efficient algorithm to compute these constants to arbitrary precision with rigorous error bounds, for the first time achieving this with low complexity with respect to the order~$n$. Our computations are based on an integral representation with a hyperbolic kernel that decays exponentially fast. The algorithm consists of locating an approximate steepest descent contour and then evaluating the integral numerically in ball arithmetic using the Petras algorithm with a Taylor expansion for bounds near the saddle point. An implementation is provided in the Arb library. We can, for example, compute $\gamma\_n(1)$ to 1000 digits in a minute for any $n$ up to $n=10^{100}$. We also provide other interesting integral representations for $\gamma\_n(v)$, $\zeta(s)$, $\zeta(s,v)$, some polygamma functions and the Lerch transcendent

Rapid computation of special values of Dirichlet LL-functions

We consider computing the Riemann zeta function $\zeta(s)$ and Dirichlet $L$-functions $L(s,\chi)$ to $p$-bit accuracy for large $p$. Using the approximate functional equation together with asymptotically fast computation of the incomplete gamma function, we observe that $p^{3/2+o(1)}$ bit complexity can be achieved if $s$ is an algebraic number of fixed degree and with algebraic height bounded by $O(p)$. This is an improvement over the $p^{2+o(1)}$ complexity of previously published algorithms and yields, among other things, $p^{3/2+o(1)}$ complexity algorithms for Stieltjes constants and $n^{3/2+o(1)}$ complexity algorithms for computing the $n$th Bernoulli number or the $n$th Euler number exactly