Max-min algebra is an algebraic structure in which classical addition and multiplication are replaced by $\oplus$ and $\otimes$, where $a\oplus b=max\{a,b\},a\otimes b=min\{a,b\}$. The notation $𝐀\otimes 𝐱=𝐛$ represents an interval system of linear equations, where $𝐀=[\underline{A},\overline{A}]$, $𝐛=[\underline{b},\overline{b}]$ are given interval matrix and interval vector, respectively, and a solution is from a given interval vector $𝐱=[\underline{x},\overline{x}]$. We define six types of solvability of max-min interval systems with bounded solution and give necessary and sufficient conditions for them.

We discuss the effects that the usual set theoretic and arithmetic operations with fuzzy sets and fuzzy numbers have with respect to the energies and entropies of the fuzzy sets connected and of the resulting fuzzy sets, and we also compare the entropies and energies of the results of several of those operations.

Due to the versatility as well as its ease of implementation, the Fast Transversal Filters algorithm is attractive for many adaptive filtering applications. However, it is not widely used because of its undesirable tendency to diverge when operating in finite precision arithmetic. To compensate, modifications to the algorithm have been introduced that are either occasional (performed when a predefined condition(s) is violated) or structured as part of the normal update iteration. However, in neither...

With the emergence of mixed precision hardware, mixed precision GMRES-based iterative refinement schemes for solving linear systems $Ax=b$ have recently been developed. However, in certain settings, GMRES may require too many iterations per refinement step, making it potentially more expensive than the alternative of recomputing the LU factors in a higher precision. In this work, we incorporate the idea of Krylov subspace recycling, a well-known technique for reusing information across sequential invocations,...

The interval eigenproblem in max-min algebra is studied. A classification of interval eigenvectors is introduced and six types of interval eigenvectors are described. Characterization of all six types is given for the case of strictly increasing eigenvectors and Hasse diagram of relations between the types is presented.

Numerical integration is an important operation for scientific computations. Although the different quadrature methods have been well studied from a mathematical point of view, the analysis of the actual error when performing the quadrature on a computer is often neglected. This step is however required for certified arithmetics.
We study the Newton-Cotes quadrature scheme in the context of multiple-precision arithmetic and give enough details on the algorithms and the error bounds to enable software...

Nikdy neztotožnujme realitu s jejím matematickým či numerickým modelem. (Věnováno Emilu Vitáskovi k jeho 80. narozeninám.)