Matrix methods are increasingly popular for polynomial root-finding. The idea is to approximate the roots as the eigenvalues of the companion or generalized companion matrix associated with an input polynomial. The algorithms also solve secular equation. QR algorithm is the most customary method for eigen-solving, but we explore the inverse Rayleigh quotient iteration instead, which turns out t...

Abstract A number of theoretical and computational problems for matrix polynomials are solved by passing to linearizations. Therefore a perturbation theory, that relates perturbations in the linearization equivalent corresponding polynomial, is needed. In this paper we develop an algorithm finds which coefficients polynomial corresponds given entire pencil. Moreover find transformation matrices...

Hamkins and Löwe proved that the modal logic of forcing is S4.2. In this paper, we consider its modal companion, the intermediate logic KC and relate it to the fatal Heyting algebra HZFC of forcing persistent sentences. This Heyting algebra is equationally generic for the class of fatal Heyting algebras. Motivated by these results, we further analyse the class of fatal Heyting algebras.

We present an operator approach to Rogers-type formulas and Mehler’s formulas for the Al-Salam-Carlitz polynomials Un(x, y, a; q). By using the q-exponential operator, we obtain a Rogers-type formula which leads to a linearization formula. With the aid of a bivariate augmentation operator, we get a simple derivation of Mehler’s formula due to AlSalam and Carlitz. By means of the Cauchy companio...

A common way of computing the roots of a polynomial is to find the eigenvalues of a linearization, such as the companion (when the polynomial is expressed in the monomial basis), colleague (Chebyshev basis) or comrade matrix (general orthogonal polynomial basis). For the monomial case, many studies exist on the stability of linearization-based rootfinding algorithms. By contrast, little seems t...

The most common way of solving the quadratic eigenvalue problem (QEP) (λM+λD+K)x = 0 is to convert it into a linear problem (λX+Y )z = 0 of twice the dimension and solve the linear problem by the QZ algorithm or a Krylov method. In doing so, it is important to understand the influence of the linearization process on the accuracy and stability of the computed solution. We discuss these issues fo...

The classical approach to investigating polynomial eigenvalue problems is linearization, where the polynomial is converted into a larger matrix pencil with the same eigenvalues. For any polynomial there are infinitely many linearizations with widely varying properties, but in practice the companion forms are typically used. However, these companion forms are not always entirely satisfactory, an...

In 1990 Krivine (1990b) introduced the notion of storage operators. They are λ-terms which simulate call-by-value in the call-by-name strategy. Krivine (1990b) has shown that there is a very simple type in the AF2 type system for storage operators using Gődel translation from classical to intuitionistic logic. Parigot (1993a) and Krivine (1994) have shown that storage operators play an importan...

We describe a BDI (Belief, Desire, Intention) goal-oriented architecture for a conversational virtual companion embodied as a child’s Toy, designed to be both entertaining and capable of carrying out collaborative tasks. We argue that the goal-oriented approach supports both structured conversational activities (e.g., story-telling, collaborative games) as well as more “free-flowing” engaging d...

Older adults with strong social connections are at a reduced risk for health problems and mortality. We describe two field studies to inform the development of a virtual agent designed to provide long-term, continuous social support to isolated older adults. Findings include the topics that older adults would like to discuss with a companion agent, in addition to overall reactions to interactin...