arXiv subjects

Fredrik Johansson

Publications and source records attributed to Fredrik Johansson.

36 records · Page 2Linked to original sources

arXiv2017

Nemo/Hecke: Computer Algebra and Number Theory Packages for the Julia Programming Language

We introduce two new packages, Nemo and Hecke, written in the Julia programming language for computer algebra and number theory. We demonstrate that high performance generic algorithms can be implemented in Julia, without the need to resort to a low-level C implementation. For specialised algorithms, we use Julia's efficient native C interface to wrap existing C/C++ libraries such as Flint, Arb, Antic and Singular. We give examples of how to use Hecke and Nemo and discuss some algorithms that we have implemented to provide high performance basic arithmetic.

cs.MS↗

arXiv2017

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.

cs.MS↗

arXiv2016

Arb: Efficient Arbitrary-Precision Midpoint-Radius Interval Arithmetic

Arb is a C library for arbitrary-precision interval arithmetic using the midpoint-radius representation, also known as ball arithmetic. It supports real and complex numbers, polynomials, power series, matrices, and evaluation of many special functions. The core number types are designed for versatility and speed in a range of scenarios, allowing performance that is competitive with non-interval arbitrary-precision types such as MPFR and MPC floating-point numbers. We discuss the low-level number representation, strategies for precision and error bounds, and the implementation of efficient polynomial arithmetic with interval coefficients.

cs.MS↗

arXiv2016

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.

cs.MS↗

arXiv2016

A Passage-of-time Model of the Cerebellar Purkinje Cell

The cerebellar Purkinje cell controlling eyeblinks can learn, remember and reproduce the interstimulus interval in a classical conditioning paradigm. Given temporally separated inputs, the cerebellar Purkinje cell learns to pause its tonic inhibition of a motor pathway with high temporal precision so that an overt blink occurs at the right time. Most models relegate the Purkinje cell's passage-of-time representation to afferent granule cells, a subpopulation of which is supposedly selected for synaptic depression in order to make the Purkinje cell pause. However, granule cell models have recently faced two crucial challenges: 1) bypassing granule cells and directly stimulating the Purkinje cell's pre-synaptic fibers during training still produces a well-timed pause, and 2) the Purkinje cell can reproduce the learned pause, invariant to the temporal structure of probe stimulation. Here, we present a passage-of-time model which is internal to the Purkinje cell and is invariant to probe structure. The model accurately simulates the Purkinje cell learning mechanism and makes testable electrophysiological predictions. Importantly, the model is a numerical proof-of-principle for a biological learning mechanism which does not rely on changes of synaptic strength.

q-bio.NC↗

arXiv2015

Efficient implementation of elementary functions in the medium-precision range

We describe a new implementation of the elementary transcendental functions exp, sin, cos, log and atan for variable precision up to approximately 4096 bits. Compared to the MPFR library, we achieve a maximum speedup ranging from a factor 3 for cos to 30 for atan. Our implementation uses table-based argument reduction together with rectangular splitting to evaluate Taylor series. We collect denominators to reduce the number of divisions in the Taylor series, and avoid overhead by doing all multiprecision arithmetic using the mpn layer of the GMP library. Our implementation provides rigorous error bounds.

cs.MS↗

arXiv2013

A fast algorithm for reversion of power series

We give an algorithm for reversion of formal power series, based on an efficient way to implement the Lagrange inversion formula. Our algorithm requires $O(n^{1/2}(M(n) + MM(n^{1/2})))$ operations where $M(n)$ and $MM(n)$ are the costs of polynomial and matrix multiplication respectively. This matches the asymptotic complexity of an algorithm of Brent and Kung, but we achieve a constant factor speedup whose magnitude depends on the polynomial and matrix multiplication algorithms used. Benchmarks confirm that the algorithm performs well in practice.

cs.SC↗

arXiv2013

A bound for the error term in the Brent-McMillan algorithm

The Brent-McMillan algorithm B3 (1980), when implemented with binary splitting, is the fastest known algorithm for high-precision computation of Euler's constant. However, no rigorous error bound for the algorithm has ever been published. We provide such a bound and justify the empirical observations of Brent and McMillan. We also give bounds on the error in the asymptotic expansions of functions related to modified Bessel functions.

math.NA↗

arXiv2013

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.

cs.SC↗

arXiv2013

Using functional equations to enumerate 1324-avoiding permutations

We consider the problem of enumerating permutations with exactly r occurrences of the pattern 1324 and derive functional equations for this general case as well as for the pattern avoidance (r=0) case. The functional equations lead to a new algorithm for enumerating length n permutations that avoid 1324. This approach is used to enumerate the 1324-avoiders up to n=31. We also extend those functional equations to account for the number of inversions and derive analogous algorithms.

math.CO↗

arXiv2013

Rigorous high-precision computation of the Hurwitz zeta function and its derivatives

We study the use of the Euler-Maclaurin formula to numerically evaluate the Hurwitz zeta function $ζ(s,a)$ for $s, a \in \mathbb{C}$, along with an arbitrary number of derivatives with respect to $s$, to arbitrary precision with rigorous error bounds. Techniques that lead to a fast implementation are discussed. We present new record computations of Stieltjes constants, Keiper-Li coefficients and the first nontrivial zero of the Riemann zeta function, obtained using an open source implementation of the algorithms described in this paper.

cs.SC↗

arXiv2013

Ore Polynomials in Sage

We present a Sage implementation of Ore algebras. The main features for the most common instances include basic arithmetic and actions; gcrd and lclm; D-finite closure properties; natural transformations between related algebras; guessing; desingularization; solvers for polynomials, rational functions and (generalized) power series. This paper is a tutorial on how to use the package.

cs.SC↗

arXiv2013

Finding Hyperexponential Solutions of Linear ODEs by Numerical Evaluation

We present a new algorithm for computing hyperexponential solutions of ordinary linear differential equations with polynomial coefficients. The algorithm relies on interpreting formal series solutions at the singular points as analytic functions and evaluating them numerically at some common ordinary point. The numerical data is used to determine a small number of combinations of the formal series that may give rise to hyperexponential solutions.

cs.SC↗

arXiv2012

Efficient implementation of the Hardy-Ramanujan-Rademacher formula

We describe how the Hardy-Ramanujan-Rademacher formula can be implemented to allow the partition function $p(n)$ to be computed with softly optimal complexity $O(n^{1/2+o(1)})$ and very little overhead. A new implementation based on these techniques achieves speedups in excess of a factor 500 over previously published software and has been used by the author to calculate $p(10^{19})$, an exponent twice as large as in previously reported computations. We also investigate performance for multi-evaluation of $p(n)$, where our implementation of the Hardy-Ramanujan-Rademacher formula becomes superior to power series methods on far denser sets of indices than previous implementations. As an application, we determine over 22 billion new congruences for the partition function, extending Weaver's tabulation of 76,065 congruences.

math.NT↗

arXiv2009

No observational constraints from hypothetical collisions of hypothetical dark halo primordial black holes with galactic objects

It was suggested by several authors that hypothetical primordial black holes (PBHs) may contribute to the dark matter in our Galaxy. There are strong constraints based on the Hawking evaporation that practically exclude PBHs with masses m~1e15-1e16g and smaller as significant contributors to the Galactic dark matter. Similarly, PBHs with masses greater than about 1e26g are practically excluded by the gravitational lensing observation. The mass range between 10e16g<m<10e26g is unconstrained. In this paper, we examine possible observational signatures in the unexplored mass range, investigating hypothetical collisions of PBHs with main sequence stars, red giants, white dwarfs, and neutron stars in our Galaxy. This has previously been discussed as possibly leading to an observable photon eruption due to shock production during the encounter. We find that such collisions are either too rare to be observed (if the PBH masses are typically larger than about 1e20g), or produce too little power to be detected (if the masses are smaller than about 1e20g).

astro-ph↗

arXiv2009

Scaling limits of anisotropic Hastings-Levitov clusters

We consider a variation of the standard Hastings-Levitov model HL(0), in which growth is anisotropic. Two natural scaling limits are established and we give precise descriptions of the effects of the anisotropy. We show that the limit shapes can be realised as Loewner hulls and that the evolution of harmonic measure on the cluster boundary can be described by the solution to a deterministic ordinary differential equation related to the Loewner equation. We also characterise the stochastic fluctuations around the deterministic limit flow.

math.PR↗

arXiv2009

Optimal Holder exponent for the SLE path

We prove an upper bound on the optimal Hölder exponent for the chordal SLE path parameterized by capacity and thereby establish the optimal exponent as conjectured by J. Lind. We also give a new proof of the lower bound. Our proofs are based on the sharp estimates of moments of the derivative of the inverse map. In particular, we improve an estimate of the second author.

math.PR↗

arXiv2008

Rescaled Levy-Loewner hulls and random growth

We consider radial Loewner evolution driven by unimodular Lévy processes. We rescale the hulls of the evolution by capacity, and prove that the weak limit of the rescaled hulls exists. We then study a random growth model obtained by driving the Loewner equation with a compound Poisson process. The process involves two real parameters: the intensity of the underlying Poisson process and a localization parameter of the Poisson kernel which determines the jumps. A particular choice of parameters yields a growth process similar to the Hastings-Levitov $\rm{HL}(0)$ model. We describe the asymptotic behavior of the hulls with respect to the parameters, showing that growth tends to become localized as the jump parameter increases. We obtain deterministic evolutions in one limiting case, and Loewner evolution driven by a unimodular Cauchy process in another. We show that the Hausdorff dimension of the limiting rescaled hulls is equal to 1. Using a different type of compound Poisson process, where the Poisson kernel is replaced by the heat kernel, as driving function, we recover one case of the aforementioned model and $\rm{SLE}(κ)$ as limits.

math.CV↗