Related papers: The complexity of multiple-precision arithmetic
A new mathematical notation is proposed for the iteration of functions. It facilitates the application of the iteration of functions in mathematical and logical expressions, definitions of sets, and formulations of algorithms. Illustrations…
We develop a framework to give upper bounds on the "practical" computational complexity of stability problems for a wide range of nonlinear continuous and hybrid systems. To do so, we describe stability properties of dynamical systems using…
A matrix approach to continuous iteration is proposed for general formal series. It leads, in particular, to an order{to{order iteration of the exponential function, and consequently to an algorithmic approach to tetration. Lower{order…
The aim of our paper is to formulate and solve problems concerning multitime multiple recurrence equations. We discuss in detail the generic properties and the existence and uniqueness of solutions. Among the general things, we discuss in…
Complexity theory is a useful tool to study computational issues surrounding the elicitation of preferences, as well as the strategic manipulation of elections aggregating together preferences of multiple agents. We study here the…
The era of huge data necessitates highly efficient machine learning algorithms. Many common machine learning algorithms, however, rely on computationally intensive subroutines that are prohibitively expensive on large datasets. Oftentimes,…
This paper introduces the multiplicative variant of the recently proposed asynchronous additive coarse-space correction method. Definition of an asynchronous extension of multiplicative correction is not straightforward, however, our…
Bayesian probabilistic numerical methods are a set of tools providing posterior distributions on the output of numerical methods. The use of these methods is usually motivated by the fact that they can represent our uncertainty due to…
We devise a simple but remarkably accurate iterative routine for calculating the roots of a polynomial of any degree. We demonstrate that our results have significant improvement in accuracy over those obtained by methods used in popular…
We describe algorithms and data structures to extend a neural network library with automatic precision estimation for floating point computations. We also discuss conditions to make estimations exact and preserve high computation…
Linear complexity is an important parameter for arrays that are used in applications related to information security. In this work we survey constructions of two and three dimensional arrays, and present new results on the multidimensional…
A moldable job is a job that can be executed on an arbitrary number of processors, and whose processing time depends on the number of processors allotted to it. A moldable job is monotone if its work doesn't decrease for an increasing…
Comparison to traditionally accurate computing, approximate computing focuses on the rapidity of the satisfactory solution, but not the unnecessary accuracy of the solution. Approximate bisimularity is the approximate one corresponding to…
Uncertainty is unavoidable in modeling dynamical systems and it may be represented mathematically by differential inclusions. In the past, we proposed an algorithm to compute validated solutions of differential inclusions; here we provide…
Incrementality is ubiquitous in human-human interaction and beneficial for human-computer interaction. It has been a topic of research in different parts of the NLP community, mostly with focus on the specific topic at hand even though…
Efficiently solving sparse linear algebraic equations is an important research topic of numerical simulation. Commonly used approaches include direct methods and iterative methods. Compared with the direct methods, the iterative methods…
We survey results on the formalization and independence of mathematical statements related to major open problems in computational complexity theory. Our primary focus is on recent findings concerning the (un)provability of complexity…
We present a new algorithm for computing upper bounds on the number of executions of each program instruction during any single program run. The upper bounds are expressed as functions of program input values. The algorithm is primarily…
The analysis of complex nonlinear systems is often carried out using simpler piecewise linear representations of them. A principled and practical technique is proposed to linearize and evaluate arbitrary continuous nonlinear functions using…
We develop a theory of multiplicative functions (with values inside or on the unit circle) in arithmetic progressions analogous to the well-known theory of primes in arithmetic progressions.