Svetoslav Markov

On directed interval arithmetic and its applications

We discuss two closely related interval arithmetic systems: i) the system of directed (generalized) intervals studied by E. Kaucher, and ii) the system of normal intervals together with the outer and inner interval operations. A relation between the two systems becomes feasible due to introduction of special notations and a so-called normal form of directed intervals. As an application, it has been shown that both interval systems can be used for the computation of tight inner and outer inclusions of ranges of functions and consequently for the development of software for automatic computation of ranges of functions.

The Set of Hausdorff Continuous Functions— The Largest Linear Space of Interval Functions

Hausdorff continuous (H-continuous) functions are special interval-valued functions which are commonly used in practice, e.g. histograms are such functions. However, in order to avoid arithmetic operations with intervals, such functions are traditionally treated by means of corresponding semi-continuous functions, which are real-valued functions. One difficulty in using H-continuous functions is that, if we add two H-continuous functions that have interval values at same argument using point-wise interval arithmetic, then we may obtain as a result an interval function which is not H-continuous. In this work we define addition so that the set of H-continuous functions is closed under this operation. Moreover, the set of H-continuous functions is turned into a linear space. It has been also proved that this space is the largest linear space of interval functions. These results make H-continuous functions an attractive tool in real analysis and provides a bridge between real and interval analysis.

An iterative method for algebraic solution to interval equations

Abstract The algebraic solution to systems of linear equations involving an interval square matrix and an interval right-hand side vector in terms of interval arithmetic is discussed. The basic concepts of interval arithmetic are given in a form suitable for our study. An iterative Jacobi type method is formulated and its convergence has been proved, under certain conditions on the interval matrix. In the special case when only the right-hand side is interval-valued we reduce the problem to two ordinary linear systems. An iterative numerical algorithm is proposed and numerically demonstrated.

The Contribution of T. Sunaga to Interval Analysis and Reliable Computing

The contribution of T. Sunaga to interval analysis and reliable computing is not well-known amongst specialists in the field. We present and comment Sunaga’s basic ideas and results related to the properties of intervals and their application.

On the Algebraic Properties of Intervals and Some Applications

The algebraic properties of interval vectors (boxes) are studied. Quasilinear spaces with group structure are studied. Some fundamental algebraic properties are developed, especially in relation to the quasidistributive law, leading to a generalization of the familiar theory of linear spaces. In particular, linear dependence and basis are defined. It is proved that a quasilinear space with group structure is a direct sum of a linear and a symmetric space. A detailed characterization of symmetric quasilinear spaces with group structure is found.

A self-validating numerical method for the matrix exponential

An algorithm is presented, which produces highly accurate and automatically verified bounds for the matrix exponential function. Our computational approach involves iterative defect correction, interval analysis and advanced computer arithmetic. The algorithm presented is based on the “scaling and squaring” scheme, utilizing Padé approximations and safe error monitoring. A PASCAL-SC program is reported and numerical results are discussed.ZusammenfassungEs wird ein Algorithmus vorgestellt, der hochgenaue und automatisch verifizierte Grenzen fuer die Exponentialfunktion einer Matrix liefert. Unser Verfahren benuetzt iterative Defektkorrektur, Intervall-Analysis und eine erweiterte Rechnerarithmetik. Der dargestellte Algorithmus basiert auf dem “scaling and squaring” Schema und benutzt Padé-Approximationen und safe-error-monitoring. Es werden ein PASCAL-SC Programm vorgestellt und numerische Resultate diskutiert.

On the algebraic Properties of Stochastic Arithmetic. Comparison to Interval Arithmetic

Interval arithmetic and stochastic arithmetic have been both developed for the same purpose, i. e. to control errors coming from floating point arithmetic of computers. Interval arithmetic delivers guaranteed bounds for numerical results whereas stochastic arithmetic provides confidence intervals with known probability. The algebraic properties of stochastic arithmetic are studied with an emphasis on the structure of the set of stochastic numbers. Some new properties of stochastic numbers are obtained based on the comparison with interval arithmetic in midpoint-radius form.

Theoretical and computational studies of some bioreactor models

We study certain classical basic models for bioreactor simulation in case of batch mode with decay. It is shown that in many cases the two-dimensional differential system describing the dynamics of the substrate and biomass concentrations can be reduced to an algebraic equation for the biomass together with a single differential equation for the substrate. Then from an analogy with the Henri-Michaelis-Menten enzyme kinetic mechanism a simple model is proposed for a bioreactor in batch mode with decay. Two more models are also proposed taking into account the phases of microbial growth. Some properties of these two models are studied and compared to classical Monod type models using computer simulations.

On the Hausdorff distance between the Heaviside step function and Verhulst logistic function

In this note we prove more precise estimates for the approximation of the step function by sigmoidal logistic functions. Numerical examples, illustrating our results are given, too.

Stochastic arithmetic: s-spaces and some applications

It has been recently shown that computation with stochastic numbers as regard to addition and multiplication by scalars can be reduced to computation in familiar vector spaces. In this work we show how this can be used for the algebraic solution of linear systems of equations with stochastic right-hand sides. On several examples we compare the algebraic solution with the simulated solution using the CADNA package.