University of Texas at El Paso DigitalCommons@UTEP Departmental Technical Reports (CS) Department of Computer Science 10-1-2001 Interval Mathematics: Algebraic Aspects Sompong Dhompongsa Vladik Kreinovich University of Texas at El Paso, [email protected] Hung T. Nguyen Follow this and additional works at: http://digitalcommons.utep.edu/cs_techrep Part of the Computer Engineering Commons Comments: UTEP-CS-01-29. Published in the Proceedings of the Second International Conference on Intelligent Technologies InTech'2001, Bangkok, Thailand, November 27-29, 2001, pp. 30-38. Recommended Citation Dhompongsa, Sompong; Kreinovich, Vladik; and Nguyen, Hung T., "Interval Mathematics: Algebraic Aspects" (2001). Departmental Technical Reports (CS). Paper 416. http://digitalcommons.utep.edu/cs_techrep/416 This Article is brought to you for free and open access by the Department of Computer Science at DigitalCommons@UTEP. It has been accepted for inclusion in Departmental Technical Reports (CS) by an authorized administrator of DigitalCommons@UTEP. For more information, please contact [email protected]. Interval Mathematics: Algebraic Aspects Sompong Dhompongsa Vladik Kreinovich Hung T. Nguyen Chiang Mai University Dept. Computer Science Mathematics Dept. 239 Huay Kaew Road University of Texas New Mexico State University Chiang Mai 50200, Thailand El Paso, TX 79968, USA Las Cruces, NM 88003, USA [email protected] [email protected] [email protected] Abstract: Many application-oriented mathematical models deal with real numbers. In real life, due to the inevitable measurement inaccuracy, we do not know the exact values of the measured quantities, we know, at best, the intervals of possible values. It is thus desirable to analyze how the corresponding mathematical results will look if we replace numbers by intervals. Keywords: interval mathematics 1 Introduction: Why interval mathe- 2.2 Linear equations with interval coeffi- ! matics? cients: polytope " A natural idea is to consider the matrix ! ¡ Mathematical models normally deal with real num- ¡ and the vector with interval coefficients bers. In real life, due to the inevitable measurement ¡ and . Then, it is natural to define the solution inaccuracy, we do not know the exact values of the ! # £ set as the set of all solutions of the system measured quantities, we know, at best, the intervals ¡ £$ ¢ $ for all %$¨ and all , i.e., for which of possible values. Thus, with applications in mind, ¡ $ & ' it is desirable to revisit computational formulas and and £ for all and . related mathematical results and see what they look Definition 2.1. Let be an interval matrix, and let like if we use intervals instead of real numbers. ( be an interval vector. We say that a vector is a ¡ ¡ ) solution for the system of interval linear In particular, if we know only the intervals , for equations if there exists a matrix %$¥ and a vector ¡ £ two numbers ¢ and , then the set of possible values £ £$ for which . ¢¨§©£ of ¢¥¤¦£ (correspondingly, of ) also forms an ¡ interval. It is natural to call this interval the sum ¤ ¡ Theorem 2.1 [1]. For every system of interval linear (corr., product) of the original intervals and . equations, its set of solutions is a polytope. In other words, for every set of interval linear equa- 2 First algebraic aspect: Adding inter- tions, the border of its solution set is piece-wise lin- val data. How do intervals change ear (hyperplanar). simple algebraic shapes? 2.1 Simplest shape: (hyper)plane 2.3 Symmetric linear equations with inter- val coefficients: piece-wise quadratic The simplest possible shape is a hyperplane, which shape * can be defined as a solution set for a system of linear When we consider symmetric interval matrices , equations i.e., matrices for which , then it is natural to restrict ourselves only to symmetric matrices $+ . § £ ¢ Definition 2.2. Let be a symmetric interval ma- ( £ ¢ i.e., in matrix form, , where , trix, and let be an interval vector. We say that a ¡ £ £ ) , and . vector is a s-solution for the system of interval linear equations if there exists a symmetric Definition 2.5. A set is called semialgebraic if it can ¡ £$ # £ matrix $¥ and a vector for which . be represented as a finite union of subsets, each of which is defined by a finite system of polynomial §§/§ ¦ *,+.- 102 Theorem 2.2 [2, 3, 6]. For every system of interval equations and inequalities of the ¡ §/§/§ ¦ §§/§ ¦ *435- *879- 102"6 102;: linear equations , its set of s-solutions has a types and piece-wise quadratic border. (for some polynomials * ). In other words, the border of this set consists of $>= Definition 2.6. For every set < , and for ev- §/§§ §/§§ finitely many pieces each of which can be described A ? & @0 # ery subset & , by a by a quadratic equation: it projection B1CD-E<F2 we mean the set of all vectors §/§§ 0 £¢ ¦¨§ G - 2 $I= ¡ H that can be extended to a vector § § ¤ § ¤¥¤ §§/§ - 2 $J< . It is known that a projection of a semialgebraic set is 2.4 Linear equations with interval coeffi- also semialgebraic. Now, we are ready for the main cients and general linear dependence results: between © and : arbitrary algebraic ! * sets ! of a system, is also semialgebraic, and that, vice It ¢ The symmetry relation ¢ is a particular case is known that for every semialgebraic set < , and for ¢ §§/§ §/§/§ of linear dependence between the coefficients . A & &@0 # every subset ? , the cor- £ The corresponding equations # can be de- * responding projection of < , i.e., the set of all vectors ¢ & ' §/§§ scribed if we fix the values for , and then 0 G - 2 $K= H that can be extended to a solu- ¢ ¢ §/§/§ take . 2 tion - of a system, is also semialgebraic, Symmetry is just one of such relations. It is natural and that, vice to consider other relations such as antisymmetry, etc. In general, we arrive at the following definition: Theorem 2.3 [4]. For every system of interval lin- ear equations with interval coefficients, the set of its Definition 2.3. By a system of interval linear equa- solutions is semialgebraic. tions with dependent coefficients, we mean a system of the type Theorem 2.4 [4]. Every semialgebraic set can be represented as a projection of the set of all solutions ¢ £ of a system of interval linear equations with interval coefficients. where ( In this representation, we allow intervals to be ar- ¢ ¢ ¤ ¢ bitrarily wide. In terms of measurements, wide in- tervals correspond to low measurement accuracy. It is natural to ask the following question: if we only £ £ ¤ £ consider narrow intervals, which correspond to high ! measurement accuracy, will we still get all possible ¢ £ £ ¢ , , , and are given real numbers, ( semialgebraic shapes or a narrower class of shapes? ¡ ! '"$# !&%' & ), and coefficients ON ¦ can take arbitrary values from the given intervals ( . Definition 2.7 [10]. For a given LM6 , an interval P P ( ;Q¥R ¤KRTS is called absolutely L -narrow if P Definition 2.4. We say that a vector is a solution L RU L§DV ,V RU L , and is relatively -narrow if . to a given system of linear interval equations with £ dependent coefficients if it solves a system ¦ L$6 )( for some values $ . Theorem 2.5 [5]. For every , every semial- gebraic set can be represented as a projection of the To describe the set of such solutions, we must recall solution set of some system of interval linear equa- two notions: the notion of a semialgebraic set and tions with dependent coefficients, whose intervals are the notion of a projection: both absolutely and relatively L -narrow. 3 Second algebraic aspect: Adding in- To describe the class of all interval-rational func- terval operations Interval + rational = tions, we must recall the definition of an algebraic function: algebraic Definition 3.2. An analytic function ¦ §/§/§ 3.1 Rational functions - 2 is called algebraic if The set of rational functions can be defined as the §§/§ §§/§ ¦ smallest set of functions that contains constants and * - T- 2 2 variables and is closed under arithmetic operations §/§/§ ¤ Q § * - 2 ( ). for some polynomial for which 1* the partial derivative is not identically 0 ¦ ! * ( ). We say that the corresponding §§/§ 3.2 What happens if we add intervals: a * - 2 polynomial defines the algebraic §§/§ question T- 2 function . A natural question is: what class of functions will be " get if we add to this list the “interval” operation, i.e., Definition 3.3. Assume that is an open domain in # $#%"'& = an operation that transforms a function of vari- = . We say that an algebraic function §/§§ ( ( ables and given intervals into the bounds can be locally represented by an interval-rational §/§§ ¦ ¨¦ 2%$ for the range - function if for almost every point ( " there exists a neighborhood and an interval- §§/§ ( ( §§/§ - 2 - 2 rational function such that for all §§/§ ( - 2 - 2 - 2 §/§§ §/§§ § from , . ( ( - 2 V $ $ Here, almost every is understood in the usual math- 3.3 Definitions and the main result ematical sense: a property is true for almost every point if the set of all points in which it is not true has Definition 3.1. By interval-rational functions of Lebesgue measure 0. § § § several real variables , we mean functions from the following class: Theorem 3.1 [8]. Every interval-rational function is algebraic. ¡ Constants and variables themselves are Theorem 3.2 [8]. Every algebraic function can be interval-rational