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A Mathematical Basis for an Interval Arithmetic Standard

Bohlender, Gerd, Kulisch, Ulrich (2010)

Serdica Journal of Computing

Basic concepts for an interval arithmetic standard are discussed in the paper. Interval arithmetic deals with closed and connected sets of real numbers. Unlike floating-point arithmetic it is free of exceptions. A complete set of formulas to approximate real interval arithmetic on the computer is displayed in section 3 of the paper. The essential comparison relations and lattice operations are discussed in section 6. Evaluation of functions for interval arguments is studied in section 7. The desirability...

A necessary and sufficient criterion to guarantee feasibility of the interval Gaussian algorithm for a class of matrices

Günter Mayer, Lars Pieper (1993)

Applications of Mathematics

A necessary and sufficient to guarantee feasibility of the interval Gaussian algorithms for a class of matrices. We apply the interval Gaussian algorithm to an n × n interval matrix [ A ] the comparison matrix [ A ] of which is irreducible and diagonally dominant. We derive a new necessary and sufficient criterion for the feasibility of this method extending a recently given sufficient criterion.

A New Approach to Fuzzy Arithmetic

Popov, Antony (2010)

Serdica Journal of Computing

This work shows an application of a generalized approach for constructing dilation-erosion adjunctions on fuzzy sets. More precisely, operations on fuzzy quantities and fuzzy numbers are considered. By the generalized approach an analogy with the well known interval computations could be drawn and thus we can define outer and inner operations on fuzzy objects. These operations are found to be useful in the control of bioprocesses, ecology and other domains where data uncertainties exist.* This work...

A note on regularity and positive definiteness of interval matrices

Raena Farhadsefat, Taher Lotfi, Jiri Rohn (2012)

Open Mathematics

We present a sufficient regularity condition for interval matrices which generalizes two previously known ones. It is formulated in terms of positive definiteness of a certain point matrix, and can also be used for checking positive definiteness of interval matrices. Comparing it with Beeck’s strong regularity condition, we show by counterexamples that none of the two conditions is more general than the other one.

A novel interval arithmetic approach for solving differential-algebraic equations with VALENCIA-IVP

Andreas Rauh, Michael Brill, Clemens Günther (2009)

International Journal of Applied Mathematics and Computer Science

The theoretical background and the implementation of a new interval arithmetic approach for solving sets of differentialalgebraic equations (DAEs) are presented. The proposed approach computes guaranteed enclosures of all reachable states of dynamical systems described by sets of DAEs with uncertainties in both initial conditions and system parameters. The algorithm is based on VALENCIA-IVP, which has been developed recently for the computation of verified enclosures of the solution sets of initial...

A Solver for Complex-Valued Parametric Linear Systems

Popova, Evgenija, Kolev, Lyubomir, Krämer, Walter (2010)

Serdica Journal of Computing

This work reports on a new software for solving linear systems involving affine-linear dependencies between complex-valued interval parameters. We discuss the implementation of a parametric residual iteration for linear interval systems by advanced communication between the system Mathematica and the library C-XSC supporting rigorous complex interval arithmetic. An example of AC electrical circuit illustrates the use of the presented software.* This work was partly supported by the DFG grant GZ:...

A verified method for solving piecewise smooth initial value problems

Ekaterina Auer, Stefan Kiel, Andreas Rauh (2013)

International Journal of Applied Mathematics and Computer Science

In many applications, there is a need to choose mathematical models that depend on non-smooth functions. The task of simulation becomes especially difficult if such functions appear on the right-hand side of an initial value problem. Moreover, solution processes from usual numerics are sensitive to roundoff errors so that verified analysis might be more useful if a guarantee of correctness is required or if the system model is influenced by uncertainty. In this paper, we provide a short overview...

Algebraic Computations with Hausdorff Continuous Functions

Anguelov, Roumen (2007)

Serdica Journal of Computing

The paper has been presented at the 12th International Conference on Applications of Computer Algebra, Varna, Bulgaria, June, 2006.The set of Hausdorff continuous functions is the largest set of interval valued functions to which the ring structure of the set of continuous real functions can be extended. The paper deals with the automation of the algebraic operations for Hausdorff continuous functions using an ultra- arithmetical approach.

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