Related papers: Exact $p$-adic computation in Magma
We study algorithms for the fast computation of modular inverses. Newton-Raphson iteration over $p$-adic numbers gives a recurrence relation computing modular inverse modulo $p^m$, that is logarithmic in $m$. We solve the recurrence to…
The height of an algebraic number $\alpha$ is a measure of how arithmetically complicated $\alpha$ is. We say $\alpha$ is totally $p$-adic if the minimal polynomial of $\alpha$ splits completely over the field $\mathbb{Q}_p$ of $p$-adic…
In this work, we develop a new iterative method for computing the digits of $\pi$ by argument reduction of the tangent function. This method combines a modified version of the iterative formula for $\pi$ with squared convergence that we…
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…
We report on implementations for algorithms treating algebraic and arithmetic properties of hypergeometric functions in the computer algebra system SageMath. We treat hypergeometric series over the rational numbers, over finite fields, and…
In [Pollack-Stevens 2011], efficient algorithms are given to compute with overconvergent modular symbols. These algorithms then allow for the fast computation of $p$-adic $L$-functions and have further been applied to compute rational…
Continued fraction expansions provide a well-established bridge between algebraic properties of numbers and combinatorics on words. In this article, we investigate the algebraicity of $p$-adic numbers whose continued fractions arise from…
The recent development in analog computation is reviewed in this paper. Analog computation was used in many applications where power and energy efficiency is of paramount importance. It is shown that by using innovative architecture and…
We introduce an effective algorithmic method for the computation of a lower bound for uniform expansion in one-dimensional dynamics. The approach employs interval arithmetic and thus provides a rigorous numerical result (computer-assisted…
Special kinds of continued fractions have been proved to converge to transcendental real numbers by means of the celebrated Subspace Theorem. In this paper we study the analogous $p$--adic problem. More specifically, we deal with Browkin…
Our paper presents an attempt to axiomatise signal processing. Our long-term goal is to formulate signal processing algorithms for an ideal world of exact computation and prove properties about them, then interpret these ideal formulations…
We describe algorithms to compute elliptic functions and their relatives (Jacobi theta functions, modular forms, elliptic integrals, and the arithmetic-geometric mean) numerically to arbitrary precision with rigorous error bounds for…
We have focused on Answer Set Programming (ASP), more specifically, answer set counting, exploring both exact and approximate methodologies. We developed an exact ASP counter, sharpASP, which utilizes a compact encoding for propositional…
We show that Hermite's approximations to values of the exponential function at given algebraic numbers are nearly optimal when considered from an adelic perspective. We achieve this by taking into account the ratio of these values whenever…
These notes survey some basic results in toric varieties over a field with examples and applications. A computer algebra package (written by the second author) is described which deals with both affine and projective toric varieties in any…
In this work, we consider the properties of the two-term Machin-like formula and develop an algorithm for computing digits of $\pi$ by using its rational approximation. In this approximation, both terms are constructed by using a…
Using the theory of $(\phi,\Gamma)$-modules and the formalism of Selmer complexes we construct the p-adic height for p-adic representations with coefficients in an affinoid algebra over $Q_p$.
The mathematical basis of p-adic Higgs mechanism discussed in papers [email protected] 9410058-62 is considered in this paper. The basic properties of p-adic numbers, of their algebraic extensions and the so called canonical…
Since the elimination algorithm of Fourier and Motzkin, many different methods have been developed for solving linear programs. When analyzing the time complexity of LP algorithms, it is typically either assumed that calculations are…
This application for learning APPROXIMATION ALGORITHM has been designed in Java which will make user comfortable in learning the very complex subject "NP-Completeness" and the solution to NP-Complete problem using approximation algorithm.