Geodesic
Is an interval the right result of arithmetic operations on intervals?
International Journal of Applied Mathematics and Computer Science, Tome 27 (2017) no. 3, pp. 575-590.

Voir la notice de l'article provenant de la source Library of Science

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.
Keywords: interval arithmetic, one dimensional interval arithmetic, multidimensional interval arithmetic, RDM interval arithmetic
Mots-clés : arytmetyka interwałowa, arytmetyka interwałowa jednowymiarowa, arytmetyka interwałowa wielowymiarowa 
@article{IJAMCS_2017_27_3_a10,
     author = {Piegat, A. and Landowski, M.},
     title = {Is an interval the right result of arithmetic operations on intervals?},
     journal = {International Journal of Applied Mathematics and Computer Science},
     pages = {575--590},
     publisher = {mathdoc},
     volume = {27},
     number = {3},
     year = {2017},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IJAMCS_2017_27_3_a10/}
}
TY  - JOUR
AU  - Piegat, A.
AU  - Landowski, M.
TI  - Is an interval the right result of arithmetic operations on intervals?
JO  - International Journal of Applied Mathematics and Computer Science
PY  - 2017
SP  - 575
EP  - 590
VL  - 27
IS  - 3
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/IJAMCS_2017_27_3_a10/
LA  - en
ID  - IJAMCS_2017_27_3_a10
ER  - 
%0 Journal Article
%A Piegat, A.
%A Landowski, M.
%T Is an interval the right result of arithmetic operations on intervals?
%J International Journal of Applied Mathematics and Computer Science
%D 2017
%P 575-590
%V 27
%N 3
%I mathdoc
%U http://geodesic.mathdoc.fr/item/IJAMCS_2017_27_3_a10/
%G en
%F IJAMCS_2017_27_3_a10
Piegat, A.; Landowski, M. Is an interval the right result of arithmetic operations on intervals?. International Journal of Applied Mathematics and Computer Science, Tome 27 (2017) no. 3, pp. 575-590. http://geodesic.mathdoc.fr/item/IJAMCS_2017_27_3_a10/

[1] Bader, F. and Nipkow, T. (1998). Term Rewriting and All That, Cambridge University Press, Cambridge.

[2] Birkhoff, G. (1967). Lattice Theory, Vol. XXV, Colloquium Publications, American Mathematical Society, Providence, RI.

[3] Chalco-Cano, Y., Lodwick, W. and Bede, B. (2014). Single level constraint interval arithmetic, Fuzzy Sets and Systems 257: 146–168.

[4] Dabala, K. (2009). Research of possibilities of interval arithmetic application to induction motors efficiency determination, Zeszyty Problemowe: Maszyny Elektryczne (84): 39–44.

[5] Dymova, L. (2011). Soft Computing in Economics and Finance, Springer, Berlin/Heidelberg.

[6] Figuiredo, L. and Stolfi, J. (2004). Affine arithmetic: Concepts and applications, Numerical Algorithms 37(1): 147–158.

[7] Hanss, M. (2005). Applied Fuzzy Arithmetic, Springer, Berlin/Heidelberg.

[8] Hayes, B. (2003). A lucid interval, American Scientist 91(6): 484–488.

[9] Kaucher, E. (1980). Interval analysis in the extended interval space IR, Computing Supplement 2: 33–49.

[10] Kovalerchuk, B. and Kreinovich, V. (2016). Comparisons of applied tasks with intervals, fuzzy sets and probability approaches, Proceedings of the 2016 IEEE International Conference on Fuzzy Systems (FUZZ), Vancouver, Canada, pp. 1478–1483.

[11] Leontief, W. (1949). The Structure of the American Economy, 1919–1935, Oxford University Press, London.

[12] Leontief, W. (1966). Input-Output Economics, Oxford University Press, New York, NY.

[13] Lodwick, W. (1999). Constrained interval arithmetic, Technical Report CCM, University of Colorado at Denver, Denver, CO.

[14] Lodwick, W. and Dubois, D. (2015). Interval linear systems as a necessary step in fuzzy linear systems, Fuzzy Sets and Systems 281: 227–251.

[15] Lyashko, M. (2005). The optimal solution of an interval systems of linear algebraic equations, Reliable Computing 11(2): 227–251.

[16] Mazarhuiya, F., Mahanta, A. and Baruah, H. (2011). Solution of fuzzy equation a+x = b using method of superimposition, Applied Mathematics 2(8): 1039–1045.

[17] Moore, R. (1996). Interval Analysis, Prentice Hall, Englewood Cliffs, NJ.

[18] Moore, R., Baker, K. and Cloud, M. (2009). Introduction to Interval Analysis, SIAM, Philadelphia, PA.

[19] Moore, R. and Young, C. (1959). Interval analysis I, Technical Report LMSD285875, Lockheed Missiles and Space Division, Sunnyvale, CA.

[20] Neumaier, A. (1990). Interval Methods for Systems of Equations, Cambridge University Press, Cambridge.

[21] Pedrycz,W., Skowron, A. and Kreinovich, V. (2008). Handbook of Granular Computing, John Wiley, Chichester.

[22] Piegat, A. and Landowski, M. (2012). Is the conventional interval arithmetic correct?, Journal of Theoretical and Applied Computer Science 6(2): 27–44.

[23] Piegat, A. and Landowski, M. (2013). Two interpretations of multidimensional RDM interval arithmetic—multiplication and division, International Journal of Fuzzy Systems 15(4): 488–496.

[24] Piegat, A. and Pluciński, M. (2015). Computing with words with the use of inverse RDM models of membership functions, International Journal of Applied Mathematics and Computer Science 25(3): 675–688, DOI: 10.1515/amcs-2015-0049.

[25] Piegat, A. and Plucinski, M. (2017). Fuzzy number division and the multi-granularity phenomenon, Bulletin of the Polish Academy of Sciences: Technical Sciences 65(4): 497–511.

[26] Piegat, A. and Tomaszewska, K. (2013). Decision making under uncertainty using info-gap theory and a new multidimensional RDMinterval arithmetic, Przegląd Elektrotechniczny 89(8): 71–76.

[27] Pilarek, M. (2010). Solving systems of linear equations using the interval extended zero method and multimedia extensions, Scientific Research of the Institute of Mathematics and Computer Science 9(2): 203–212.

[28] Popova, E. (1998). Algebraic solutions to a class of interval equations, Journal of Universal Computer Science 4(1): 48–67.

[29] Sevastjanov, P. and Dymova, L. (2009). A new method for solving internal and fuzzy equations: Linear case, Information Sciences 179: 925–937.

[30] Shary, S. (1996). Algebraic approach to the interval linear static identification, tolerance and control problems, or one more application of Kaucher arithmetic, Reliable Computing 2(1): 3–33.

[31] Shary, S. (2002). A new technique in systems analysis under interval uncertainty and ambiguity, Reliable Computing 8: 321–418.

[32] Sunaga, T. (1958). Theory of an interval algebra and its application to numerical analysis, RAAG Memoirs 2: 547–564.

[33] Warmus, M. (1956). Calculus of approximations, Bulletin de l’Académie Polonaise des Sciences Cl. III 4(5): 253–259.