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1

A realistic tolerant solution of a system of interval linear equations with the use of multidimensional interval arithmetic

Piegat Andrzej, Pluciński Marcin

International Journal of Applied Mathematics and Computer Science

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2023

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Vol. 33, no. 2

229--247

EN

The paper presents a method of determining the robustness of solutions of systems of interval linear equations (ILEs). The method can be applied also for the ILE systems for which it has been impossible to find solutions so far or for which solutions in the form of improper intervals have been obtained (which cannot be implemented in practice). The research conducted by the authors has shown that for many problems it is impossible to arrive at ideal solutions that would be fully robust to data uncertainty. However, partially robust solutions can be obtained, and those with the highest robustness can be selected and put into practice. The paper shows that the degree of robustness to the uncertainty of the entire system can be calculated on the basis of the degrees of robustness of individual equations, which greatly simplifies calculations. The presented method is illustrated with a series of examples (also benchmark ones) that facilitate its understanding. It is an extension of the authors’ previously published method for first-order ILEs.

2

RDM interval method for solving quadratic interval equation

Landowski M.

Przegląd Elektrotechniczny

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2017

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R. 93, nr 1

65--68

EN

The main task of uncertainty theory is to find the solution with uncertain variable. The ways of uncertainty description are probability density distribution, possibility distribution or interval. To solve the problem with uncertainty variable the calculation on interval is needed. The article presents the usage of RDM interval arithmetic for solving quadratic interval equation. The results obtained from examples are compared with Moore’s standard interval arithmetic solutions.

PL

Głównym zadaniem teorii niepewnos´ci jest znalezienie rozwiazania ze zmienna˛ niepewna˛. Niepewnos´c´ moz˙na zapisac´ w postaci rozkładu ge˛stos´ci prawdopodobien´stwa, rozkładu moz˙liwos´ci lub przedziału. Do rozwia˛zania zadania ze zmienna˛ niepewna˛ potrzebne sa˛ obliczenia na przedziałach. Artykuł przedstawia wykorzystanie arytmetyki interwałowej RDM do rozwia˛zania interwałowych równan´ kwadratowych. Wyniki otrzymane z przykładów porównano z rozwia˛zaniami standardowej arytmetyki interwałowej Moore’a.

3

Is an interval the right result of arithmetic operations on intervals?

Piegat A., Landowski M.

International Journal of Applied Mathematics and Computer Science

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2017

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Vol. 27, no. 3

575--590

EN

For many scientists interval arithmetic (IA, I arithmetic) seems to be easy and simple. However, this is not true. Interval arithmetic is complicated. This is confirmed by the fact that, for years, new, alternative versions of this arithmetic have been created and published. These new versions tried to remove shortcomings and weaknesses of previously proposed options of the arithmetic, which decreased the prestige not only of interval arithmetic itself, but also of fuzzy arithmetic, which, to a great extent, is based on it. In our opinion, the main reason for the observed shortcomings of the present IA is the assumption that the direct result of arithmetic operations on intervals is also an interval. However, the interval is not a direct result but only a simplified representative (indicator) of the result. This hypothesis seems surprising, but investigations prove that it is true. The paper shows what conditions should be satisfied by the result of interval arithmetic operations to call it a “result”, how great its dimensionality is, how to perform arithmetic operations and solve equations. Examples illustrate the proposed method of interval computations.

4

Evaluation of the Accuracy of the Solution to the Heat Conduction Problem with the Interval Method of Crank-Nicolson Type

Jankowska M. A., Sypniewska-Kamińska G., Kamiński H.

Acta Mechanica et Automatica

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2012

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Vol. 6, no. 1

36-43

EN

The paper deals with the interval method of Crank-Nicolson type used for some initial-boundary value problem for the onedimensional heat conduction equation. The numerical experiments are directed at a short presentation of advantages of the interval solutions obtained in the floating-point interval arithmetic over the approximate ones. It is also shown how we can deal with errors that occur during computations in terms of interval analysis and interval arithmetic.

5

Zbadanie możliwości zastosowania arytmetyki interwałowej do wyznaczania sprawności silników indukcyjnych

Dąbała K.

Maszyny Elektryczne : zeszyty problemowe

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2009

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Nr 84

39-44

EN

There was done a literature insight into interval arithmetic: its historical development, interval notations used by different authors, definitions and theorems. The interval set with operations was classified as an appropriate algebra structure. It was given the traps of interval arithmetic. On the example of electric circuit there was shown a dependence of the interval arithmetic calculation result on the function form. There was done a comparative analysis of efficiency measurement uncertainty determination of induction motors using classical and interval methods.

6

Podstawy arytmetyki interwałowej

Lupa M.

Prace Naukowe Instytutu Matematyki i Informatyki Politechniki Częstochowskiej

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2005

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Vol. 4, nr 1

95-100

PL

Przedstawiono podstawy klasycznej arytmetyki interwałowej - omówiono działania na interwałach oraz własności tych działań, funkcje i macierze na interwałach oraz interwałowe układy równań.

7

Wpływ korelacji między błędami na współczynnik koherencji w ocenie niepewności za pomocą redukcyjnej arytmetyki interwałowej

Konopka K.

Zeszyty Naukowe. Elektryka / Politechnika Łódzka

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2001

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z. 98

335-344

PL

Zagadnienie dotyczy zastosowania redukcyjnej arytmetyki interwałowej do wyznaczania niepewności wyników pomiaru. Stosowany w niej współczynnik koherencji określa wzajemne związki zachodzące między zbiorami wartości błędu. W przypadku, gdy zbiory te nie są skorelowane, wartość współczynnika koherencji związana jest z kształtem rozkładu. Skorelowanie zbiorów powoduje pojawienie się dodatkowego składnika związanego ze współczynnikiem korelacji. W referacie pokazano w jaki sposób zależy wypadkowy współczynnik koherencji dla wybranych typów rozkładów od wartości współczynnika korelacji w zakresie od -1 do 1. W końcowej części przedstawiono wnioski dotyczące sposobu składania współczynnika korelacji i współczynnika koherencji związanego z kształtem rozkładu.

EN

The problem described applies to application of reducing interval arithmetic to determining the uncertainty of measuring results. The coherence factor used there describes mutual relations between error values sets. In case when these sets are not correlated the coherence factor is connected only with the form of distribution. When error sets are correlated, additional component appears. The paper shows how resultant coherence factor changes for different distribution forms and for correlation factor changing in range from -1 to 1.

8

Analysis of linear mechanical structures with uncertainties by means of interval methods

Kulpa Z., Pownuk A., Skalna I.

Computer Assisted Mechanics and Engineering Sciences

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1998

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Vol. 5, nr 4

443-477

EN

One of the simplest ways of representation of uncertain or inexact data, as well as inexact computations with them, is based on interval arithmetic. In this approach, an uncertain (real) number is represented by an interval (a continuous bounded subset) of real numbers which presumably contains the unknown exact value of the number in question. Despite its simplicity, it conforms very well to many practical situations, like tolerance handling or managing rounding errors in numerical computations. Also, the so-called alfa-cut method of handling fuzzy sets membership functions is based on replacing a fuzzy set problem with a set of interval problems.