$f,g\colon\mathbb{R}\longrightarrow\mathbb{R}$, it is natural to define $f+g$ as the function that maps $x\in\mathbb{R}$ to $f(x) + g(x)$. However, in base R, objects of class function do not have arithmetic methods defined, so idiom such as "f + g" returns an error, even though it has a perfectly reasonable expectation. The vfunc package offers this functionality. Other similar features are provided, which lead to compact and readable idiom. A wide class of coding bugs is eliminated.
An efficient method is proposed for computing the structure of Jordan blocks of a matrix of integers or rational numbers by exact computation. We have given a method for computing Jordan chains of a matrix with exact computation. However, for deriving just the structure of Jordan chains, the algorithm can be reduced to increase its efficiency. We propose a modification of the algorithm for that purpose. Results of numerical experiments are given.
The usual formulation of efficient division uses Newton iteration to compute an inverse in a related domain where multiplicative inverses exist. On one hand, Newton iteration allows quotients to be calculated using an efficient multiplication method. On the other hand, working in another domain is not always desirable and can lead to a library structure where arithmetic domains are interdependent. This paper uses the concept of a whole shifted inverse and modified Newton iteration to compute quotients efficiently without leaving the original domain. The iteration is generic to domains having a suitable shift operation, such as integers or polynomials with coefficients that do not necessarily commute.
Manuel Kauers, Christoph Koutschan, Thibaut Verron
Although in theory we can decide whether a given D-finite function is transcendental, transcendence proofs remain a challenge in practice. Typically, transcendence is certified by checking certain incomplete sufficient conditions. In this paper we propose an additional such condition which catches some cases on which other tests fail.
Twenty years after the discovery of the F5 algorithm, Gröbner bases with signatures are still challenging to understand and to adapt to different settings. This contrasts with Buchberger's algorithm, which we can bend in many directions keeping correctness and termination obvious. I propose an axiomatic approach to Gröbner bases with signatures with the purpose of uncoupling the theory and the algorithms, and giving general results applicable in many different settings (e.g. Gröbner for submodules, F4-style reduction, noncommutative rings, non-Noetherian settings, etc.).
In this paper, we examine the structure of systems that are weighted homogeneous for several systems of weights, and how it impacts the computation of Gröbner bases. We present several linear algebra algorithms for computing Gröbner bases for systems with this structure, either directly or by reducing to existing structures. We also present suitable optimization techniques. As an opening towards complexity studies, we discuss potential definitions of regularity and prove that they are generic if non-empty. Finally, we present experimental data from a prototype implementation of the algorithms in SageMath.
It is known for linear operators with polynomial coefficients annihilating a given D-finite function that there is a trade-off between order and degree. Raising the order may give room for lowering the degree. The relationship between order and degree is typically described by a hyperbola known as the order-degree curve. In this paper, we add the height into the picture, i.e., a measure for the size of the coefficients in the polynomial coefficients. For certain situations, we derive relationships between order, degree, and height that can be viewed as order-degree-height surfaces.
A general overview of the existing difference ring theory for symbolic summation is given. Special emphasis is put on the user interface: the translation and back translation of the corresponding representations within the term algebra and the formal difference ring setting. In particular, canonical (unique) representations and their refinements in the introduced term algebra are explored by utilizing the available difference ring theory. Based on that, precise input-output specifications of the available tools of the summation package Sigma are provided.
Our research concerns generating imperative programs from Answer Set Programming Specifications. ASP is highly declarative and is ideal for writing specifications. Further with negation-as-failure it is easy to succinctly represent combinatorial search problems. We are currently working on synthesizing imperative programs from ASP programs by turning the negation into useful computations. This opens up a novel way to synthesize programs from executable specifications.
As a typical application, the Lenstra-Lenstra-Lovasz lattice basis reduction algorithm (LLL) is used to compute a reduced basis of the orthogonal lattice for a given integer matrix, via reducing a special kind of lattice bases. With such bases in input, we propose a new technique for bounding from above the number of iterations required by the LLL algorithm. The main technical ingredient is a variant of the classical LLL potential, which could prove useful to understand the behavior of LLL for other families of input bases.
A paper by Bruno Salvy and the author introduced measured multiseries and gave an algorithm to compute these for a large class of elementary functions, modulo a zero-equivalence method for constants. This gave a theoretical background for the implementation that Salvy was developing at that time. The main result of the present article is an algorithm to calculate measured multiseries for integrals of functions of the form h*sin G, where h and G belong to a Hardy field. The process can reiterated with the resulting algebra, and also applied to solutions of a second order differential equation of a particular form.
In this paper, we solve the existence problem of telescopers for rational functions in three discrete variables. We reduce the problem to that of deciding the summability of bivariate rational functions, which has been solved recently. The existence criteria we present is needed for detecting the termination of Zeilberger's algorithm to the function classes studied in this paper.
Multiplicative order of an element $a$ of group $G$ is the least positive integer $n$ such that $a^n=e$, where $e$ is the identity element of $G$. If the order of an element is equal to $|G|$, it is called generator or primitive root. This paper describes the algorithms for computing multiplicative order and primitive root in $\mathbb{Z}^*_{p}$, we also present a logarithmic improvement over classical algorithms.
Highly efficient and even nearly optimal algorithms have been developed for the classical problem of univariate polynomial root-finding (see, e.g., \cite{P95}, \cite{P02}, \cite{MNP13}, and the bibliography therein), but this is still an area of active research. By combining some powerful techniques developed in this area we devise new nearly optimal algorithms, whose substantial merit is their simplicity, important for the implementation.
Manuel Kauers, Maximilian Jaroschek, Fredrik Johansson
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.
In this paper, we address the problem of safety verification of interval hybrid systems in which the coefficients are intervals instead of explicit numbers. A hybrid symbolic-numeric method, based on SOS relaxation and interval arithmetic certification, is proposed to generate exact inequality invariants for safety verification of interval hybrid systems. As an application, an approach is provided to verify safety properties of non-polynomial hybrid systems. Experiments on the benchmark hybrid systems are given to illustrate the efficiency of our method.