An effective method for division of rectangular intervals

Edrees M. Nori Mahmood; Gultekin Soylu; Edrees M. Nori Mahmood; Gultekin Soylu

doi:10.3934/math.2020409

AIMS Mathematics

2020, Volume 5, Issue 6: 6355-6372. doi: 10.3934/math.2020409

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Research article

An effective method for division of rectangular intervals

Edrees M. Nori Mahmood ¹,
Gultekin Soylu ^{2
,
,}

1.
Department of Operations Research and Intelligent Techniques, University of Mosul, Iraq
2.
Department of Mathematics, Akdeniz University, Antalya, 07058, Turkey

Received: 05 June 2020 Accepted: 30 July 2020 Published: 10 August 2020
MSC : 65G30, 65G40, 65Y04

This paper focuses on the division of intervals in rectangular form. The particular case where the intervals are in the complex plane is considered. For two rectangular complex intervals $Z_{1}$ and $Z_{2}$ finding the smallest rectangle containing the exact set $\left\{ z_{1}\ast z_{2}:z_{1}\in Z_{1}, z_{2}\in Z_{2}\right\} $ of the operation $\ast\in\{+, -, \cdot, \diagup\}$ is the main objective of complex interval arithmetic. For the operations addition, subtraction and multiplication, the optimal solution can be easily found. In the case of division the solution requires rather complicated calculations. This is due to the fact that space of rectangular intervals is not closed under division. The quotient of two rectangular intervals is an irregular shape in general. This work introduces a new method for the determination of the smallest rectangle containing the result in the case of division. The method obtains the optimal solution with less computational cost compared to the algorithms currently available.
- interval arithmetic,
- interval division,
- complex interval,
- rectangular interval,
- global optimization
Citation: Edrees M. Nori Mahmood, Gultekin Soylu. An effective method for division of rectangular intervals[J]. AIMS Mathematics, 2020, 5(6): 6355-6372. doi: 10.3934/math.2020409

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Abstract

This paper focuses on the division of intervals in rectangular form. The particular case where the intervals are in the complex plane is considered. For two rectangular complex intervals $Z_{1}$ and $Z_{2}$ finding the smallest rectangle containing the exact set $\left\{ z_{1}\ast z_{2}:z_{1}\in Z_{1}, z_{2}\in Z_{2}\right\} $ of the operation $\ast\in\{+, -, \cdot, \diagup\}$ is the main objective of complex interval arithmetic. For the operations addition, subtraction and multiplication, the optimal solution can be easily found. In the case of division the solution requires rather complicated calculations. This is due to the fact that space of rectangular intervals is not closed under division. The quotient of two rectangular intervals is an irregular shape in general. This work introduces a new method for the determination of the smallest rectangle containing the result in the case of division. The method obtains the optimal solution with less computational cost compared to the algorithms currently available.

References

[1]	M. Petkovic, M. Petković, M. S. Petkovic, et al. Complex Interval Arithmetic and Its Applications, Berlin: Wiley-VCH, 1998.
[2]	R. B. Boche, Complex interval arithmetic with some applications, Technical Report 4-22-66-1, Lockheed Missiles and Space Company, Sunnyvale-CA, 1966.
[3]	G. Alefeld, J. Herzberger, Introduction to interval computations, New York: Academic Press, 1983.
[4]	I. Gargantini, P. Henrici, Circular arithmetic and the determination of polynomial zeros, Numerische Mathematik, 18 (1971), 305-320. doi: 10.1007/BF01404681
[5]	R. Klatte, C. Ullrich, Complex sector arithmetic, Computing, 24 (1980), 139-148. doi: 10.1007/BF02281720
[6]	U. Kulisch, Computer arithmetic and validity: theory, implementation, and applications, Berlin: Walter de Gruyter, 2013.
[7]	J. Rokne, P. Lancaster, Complex interval arithmetic, Commun. ACM, 14 (1971), 111-112. doi: 10.1145/362515.362563
[8]	R. Lohner, J. W. V. Gudenberg, Complex interval division with maximum accuracy, In: 1985 IEEE 7th Symposium on Computer Arithmetic (ARITH), IEEE, 1985, 332-336.
[9]	R. E. Moore, Interval Analysis, Englewood Cliffs: Prentice-Hall, 1966.
[10]	R. E. Moore, R. B. Kearfott, M. J. Cloud, Introduction to interval analysis, Siam, 110, 2009.
[11]	A. Neumaier, Interval methods for systems of equations, Cambridge: Cambridge university press, 1990.
[12]	L. Jaulin, M. Kieffer, O. Didrit, et al. Interval analysis, In: Applied interval analysis, London: Springer, 2001.
[13]	E. Hansen, G. W. Walster, Global optimization using interval analysis, New York: Marcel Dekker, 2003.
[14]	J. Hu, Z. Qiu, Non-probabilistic convex models and interval analysis method for dynamic response of a beam with bounded uncertainty, Appl. Math. Model., 34 (2010), 725-734. doi: 10.1016/j.apm.2009.06.013
[15]	Z. Qiu, J. Hu, J. Yang, et al. Exact bounds for the sensitivity analysis of structures with uncertain-but-bounded parameters, Appl. Math. Model., 32 (2007), 1143-1157, DOI: 10.1016/j.apm.2007.03.004.
[16]	Z. Qiu, X. Wang, Comparison of dynamic response of structures with uncertain-but-bounded parameters using non-probabilistic interval analysis method and probabilistic approach, Int. J. Solids Struct., 40 (2003), 5423-5439, DOI: 10.1016/S0020-7683(03)00282-8. doi: 10.1016/S0020-7683(03)00282-8
[17]	J. Wu, Y. Q. Zhao, S. H. Chen, An improved interval analysis method for uncertain structures, Struct. Eng. Mech., 20 (2005), 713-726, DOI: 10.12989/SEM.2005.20.6.713. doi: 10.12989/sem.2005.20.6.713
[18]	C. Jiang, G. R. Liu, X. Han, A novel method for uncertainty inverse problems and application to Material characterization of composites, Exp. Mech., 48 (2008), 539-548, DOI: 10.1007/s11340-007-9081-5. doi: 10.1007/s11340-007-9081-5
[19]	J. Liu, H. Cai, C. Jiang, et al. An interval inverse method based on high dimensional model representation and affine arithmetic, Appl. Math. Modeling, 63 (2018), 732-743, DOI: 10.1016/j.apm.2018.07.009 doi: 10.1016/j.apm.2018.07.009
[20]	J. Liu, X. Han, C. Jiang, et al. Dynamic load identification for uncertain structures based on interval analysis and regularization method, Int. J. Comput. Methods., 8 (2011), 667-683, DOI: 10.1142/S0219876211002757. doi: 10.1142/S0219876211002757
[21]	W. Zhang, J. Liu, X. Han, et al. A computational inverse technique for determination of encounter condition on projectile penetrating multilayer medium, Inverse Problems Sci. Eng., 20 (2012), 1195-1213, DOI: 10.1080/17415977.2012.659733. doi: 10.1080/17415977.2012.659733
[22]	X. Feng, K. Zhuo, J. Wu, et al. A New Interval Inverse Analysis Method and Its Application in Vehicle Suspension Design, SAE Int. J. Mater. Manf., 9 (2016), 315-320, DOI: 10.4271/2016-01-0277. doi: 10.4271/2016-01-0277
[23]	C. Jiang, Q. F. Zhang, X. Han, et al. Multidimensional parallelepiped model-a new type of nonprobabilistic convex model for structural uncertainty analysis, Int. J. Numer. Meth. Engng., 103 (2015), 31-59, DOI: 10.1002/nme.4877. doi: 10.1002/nme.4877
[24]	V. H. Truong, J. Liu, X. Meng, et al. Uncertainty analysis on vehicle-bridge system with correlative interval variables based on multidimensional parallelepiped model, Int. J. Comput. Methods, 15 (2018), 1850030, DOI:10.1142/S0219876218500305.

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