Numerical simulation is used more and more frequently in the analysis of physical phenomena. A simulation requires several phases. The first phase consists of constructing a physical model based on the results of experimenting with the phenomena. Next, the physical model is approximated by a mathematical model. Generally, the mathematical model contains algebraic expressions, ordinary or partia...

Arithmetic Transform (AT) has been known under various names, including Inverse Integer Reed-Muller (IRM) transform. We outline the developments that place AT in the center of very practical arithmetic circuit applications, and explain the conditions for making the AT practical. We show that AT allows the only known scheme of treating all the approximation and imprecision sources of arithmetic ...

A chip architecture that integrates an optical sensor and a pixel level processing element based on binary stochastic arithmetic is proposed. The optical sensor is formed by an array of fully connected pixels, and each pixel contains a sensing element and a Pulse Frequency Modulator (PFM) converting the incident light to bit streams of identical pulses. The processing element is based on binary...

Given a real matrix, we analyze an open interval, called row exclusion interval, such that the real eigenvalues do not belong to it. We characterize when the row exclusion interval is nonempty. In addition to the exclusion interval, inclusion intervals for the real eigenvalues, alternative to those provided by the Gerschgorin disks, are also considered for matrices whose off-diagonal entries pr...

A chip architecture that integrates an optical sensor and a pixel level processing element based on binary stochastic arithmetic is proposed. The optical sensor is formed by an array of fully connected pixels, and each pixel contains a sensing element and a Pulse Frequency Modulator (PFM) converting the incident light to bit streams of identical pulses. The processing element is based on binary...

Compensated summation algorithms are designed to improve the accuracy of ill-conditioned sums. They are based on algorithms, such as FastTwoSum, which are proved to provide, with rounding to nearest, the sum of two floating-point numbers and the associated rounding error. Discrete stochastic arithmetic enables one to estimate rounding error propagation in numerical codes. It requires a random r...

The analysis of hybrid systems exhibiting probabilistic behaviour is notoriously difficult. To enable mechanised analysis of such systems, we extend the reasoning power of arithmetic satisfiability-modulo-theory solving (SMT) by a comprehensive treatment of randomized (a.k.a. stochastic) quantification over discrete variables within the mixed Boolean-arithmetic constraint system. This provides ...

The aim of this work, is to evaluate the value of a fuzzy integral by applying the Newton-Cotes integration rules via a reliable scheme. In order to perform the numerical examples, the CADNA (Control of Accuracy and Debugging for Numerical Applications) library and the CESTAC (Controle et Estimation Stochastique des Arrondis de Calculs) method are applied based on the stochastic arithmetic. By ...