For the familiar Fibonacci sequence | deened by f 1 = f 2 = 1, and f n = f n?1 + f n?2 for n > 2 | f n increases exponentially with n at a rate given by the golden ratio (1 + p 5)=2 = 1:61803398 : : :. But for a simple modiication with both additions and subtractions | the random Fibonacci sequences deened by t 1 = t 2 = 1, and for n > 2, t n = t n?1 t n?2 , where each sign is independent and e...

In this paper, we study the ε-lemniscate of the characteristic polynomial in relation to the pseudospectrum of the associated matrix. It is natural to investigate this question because these two sets can be seen as generalizations of eigenvalues. The question of numerical determination of the ε-lemniscate raises the problem of computing the characteristic polynomial p. We can express the coeffi...

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...

Polarity estimation in large-scale and multi-topic domains is a difficult issue. Most state-of-the-art solutions essentially rely on frequencies of sentiment-carrying words (e.g., taken from a lexicon) when analyzing the sentiment conveyed by natural language text. These approaches ignore the structural aspects of a document, which contain valuable information. Rhetorical Structure Theory (RST)...

This paper proposes a memristor SPICE model using the Tukey window (or tapered cosine window) function. Compared with the previously proposed models based on the boundary function, the proposed model is resistant to numerical errors. From the SPICE result of a relaxation oscillator, we observe that the proposed model can be effectively used to minimise the effect of round-off errors.

Background: The fastest Ironman race times in ‘Ironman Hawaii’ were achieved in very recent years. This study investigated the change in sex difference in both race performance and the age of peak performance across years in the top ten athletes for split disciplines and overall race time in the ‘Ironman Hawaii’ between 1983 and 2012. Methods: Changes in split times, overall race times, and age...

Given a univariate polynomial P with a k-fold multiple root or a k-fold root cluster near some e z, we discuss various different methods to compute a disc near e z which either contains exactly or contains at least k roots of P . Many of the presented methods are known, some are new. We are especially interested in rigorous methods, that is taking into account all possible effects of rounding e...

Interval analysis is a relatively new mathematical tools that allows one to deal with problems that may have to be solved numerically with a computer. Examples of such problems are system solving and global optimization but numerous other problems may be addressed as well. This approach has the following general advantages: • it allows to find solutions of a problem only within some finite doma...

Examples of weakly infeasible semidefinite programs are useful to test whether semidefinite solvers can detect infeasibility. However, finding non trivial such examples is notoriously difficult. This note shows how to use Lasserre’s semidefinite programming relaxations for polynomial optimization in order to generate examples of weakly infeasible semidefinite programs. Such examples could be us...

Division-free arithmetic computations can be boiled down to summation due to Dekker/Veltkamp’s algorithm of 1971. The known double-precision numerical algorithms for summation are highly effective but limited by rounding errors. Our new summation algorithms relaxe this limitation, although they still almost entirely amount to double-precision additions. The efficiency of the algorithms is confi...