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Multiplication distributivity of proper and improper intervals
 RELIABLE COMPUTING
, 2001
"... The arithmetic on an extended set of proper and improper intervals presents algebraic completion of the conventional interval arithmetic allowing thus efficient solution of some interval algebraic problems. In this paper we summarize and present all distributive relations, known by now, on multiplic ..."
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The arithmetic on an extended set of proper and improper intervals presents algebraic completion of the conventional interval arithmetic allowing thus efficient solution of some interval algebraic problems. In this paper we summarize and present all distributive relations, known by now, on multiplication and addition of generalized (proper and improper) intervals.
Inclusion Isotone Extended Interval Arithmetic  A Toolbox Update
, 1996
"... In this report we deal with the correct formulation of a special extended interval arithmetic in the context of interval Newton like methods. We first demonstrate some of the problems arising from selected older definitions. Then we investigate the basic aim, concept, and properties important for de ..."
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Cited by 6 (0 self)
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In this report we deal with the correct formulation of a special extended interval arithmetic in the context of interval Newton like methods. We first demonstrate some of the problems arising from selected older definitions. Then we investigate the basic aim, concept, and properties important for defining a correct extended interval division. Finally, we give a proper way for defining the extended interval operations needed in our special context, and we prove their inclusion isotonicity. Additionally, we give some sample applications. We conclude with two updated implementations of our extended interval operations in the toolbox environments [2] and [3].
Algebraic Solutions to a Class of Interval Equations
 J. UNIVERSAL COMPUTER SCIENCE
, 1998
"... The arithmetic on the extended set of proper and improper intervals is an algebraic completion of the conventional interval arithmetic and thus facilitates the explicit solution of certain interval algebraic problems. Due to the existence of inverse elements with respect to addition and multiplicati ..."
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Cited by 4 (4 self)
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The arithmetic on the extended set of proper and improper intervals is an algebraic completion of the conventional interval arithmetic and thus facilitates the explicit solution of certain interval algebraic problems. Due to the existence of inverse elements with respect to addition and multiplication operations certain interval algebraic equations can be solved by elementary algebraic transformations. The conditionally distributive relations between extended intervals allow that complicated interval algebraic equations, multiincident on the unknown variable, be reduced to simpler ones. In this paper we give the general type of "pseudolinear" interval equations in the extended interval arithmetic. The algebraic solutions to a pseudolinear interval equation in one variable are studied. All numeric and parametric algebraic solutions, as well as the conditions for nonexistence of the algebraic solution to some basic types pseudolinear interval equations in one variable are found. Some examples leading to algebraic solution of the equations under consideration and the extra functionalities for performing true symbolicalgebraic manipulations on interval formulae in a Mathematica package are discussed.
Directed Interval Arithmetic in Mathematica: Implementation and Applications
, 1996
"... This report presents an experimental Mathematica ..."
A Generalization of BIAS Specifications
 J. UCS
, 1995
"... : This paper generalizes the specification of Basic Interval Arithmetic Subroutines (BIAS) to support interval arithmetic on directed (i.e. proper and improper) intervals. This is due to our understanding that the arithmetic involving improper intervals will be increasingly used in future applicatio ..."
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Cited by 4 (3 self)
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: This paper generalizes the specification of Basic Interval Arithmetic Subroutines (BIAS) to support interval arithmetic on directed (i.e. proper and improper) intervals. This is due to our understanding that the arithmetic involving improper intervals will be increasingly used in future applications and the corresponding interval arithmetic implementations require no additional cost. We extend BIAS specification to be sufficiently precise and complete, to include everything a user needs, such as subroutine's purpose, name, method of invocation and details of its behaviour and communication with the environment. The specified interval arithmetic subroutines for directed intervals are consistent with conventional interval arithmetic and IEEE floatingpoint arithmetic. Key Words: specification, interval arithmetic Category: D.2.1, D.3., K.6.3 1 Introduction Interval arithmetic [Alefeld and Herzberger 1974], [Moore 1966] is widely recognized as a valuable computing technique. Numerou...
Diagrammatic representation for interval arithmetic
 LINEAR ALGEBRA AND ITS APPLICATIONS
, 2001
"... The paper presents a diagrammatic representation of a standard interval space (the socalled MRdiagram), and shows how to represent and perform interval arithmetic and derive its various properties using the diagram. The representation is an extension and refinement of the ISdiagram representation ..."
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The paper presents a diagrammatic representation of a standard interval space (the socalled MRdiagram), and shows how to represent and perform interval arithmetic and derive its various properties using the diagram. The representation is an extension and refinement of the ISdiagram representation devised earlier by the author to represent interval relations. First, the MRdiagram is defined together with appropriate graphical notions and constructions for basic interval relations and operations. Second, diagrammatic constructions for all standard arithmetic operations are presented. Several examples of the use of these constructions to aid reasoning about various simple, though nontrivial, properties of interval arithmetic are included in order to show how the representation facilitates both deeper understanding of the subject matter and reasoning about its properties.
A Generalized Interval LU Decomposition for the Solution of Interval Linear Systems
, 2007
"... Generalized intervals (intervals whose bounds are not constrained to be increasingly ordered) extend classical intervals providing better algebraic properties. In particular, the generalized interval arithmetic is a group for addition and for multiplication of zero free intervals. These properties ..."
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Generalized intervals (intervals whose bounds are not constrained to be increasingly ordered) extend classical intervals providing better algebraic properties. In particular, the generalized interval arithmetic is a group for addition and for multiplication of zero free intervals. These properties allow one constructing a LU decomposition of a generalized interval matrix A: the two computed generalized interval matrices L and U satisfy A = LU with equality instead of the weaker inclusion obtained in the context of classical intervals. Some potential applications of this generalized interval LU decomposition are investigated.
An Inclusion Algorithm for Global Optimization in a Portable PASCALXSC Implementation
, 1992
"... An algorithm for computing inclusions for all global minimizers of a function f : IR n ! IR with f 2 C 2 (IR n ) in a compact interval vector and its implementation in PASCALXSC are presented. The algorithm is based on the method of E. Hansen using branchandbound techniques and interval a ..."
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An algorithm for computing inclusions for all global minimizers of a function f : IR n ! IR with f 2 C 2 (IR n ) in a compact interval vector and its implementation in PASCALXSC are presented. The algorithm is based on the method of E. Hansen using branchandbound techniques and interval arithmetic. First, some basic concepts, the essential parts of the algorithm, and some modifications are described. Subsequently, the comfortable programming of the algorithm in PASCALXSC and the easy use of the optimization program is demonstrated. Due to the Cbased implementation of PASCALXSC the compiler as well as the optimization algorithm is portable. Thus, numerical results and performance tests of different computer types are presented for some optimization problems. 1 Introduction We present an implementation of an algorithm for computing guaranteed bounds for all solutions of the global unconstrained optimization problem minf(x) subject to x 2 X; (1) where X ` IR n is a com...
Extended Interval Arithmetic in IEEE FloatingPoint Environment
 Interval Computations
, 1994
"... This paper describes an implementation of a general interval arithmetic extension, which comprises the following extensions of the conventional interval arithmetic: (1) extension of the set of normal intervals by improper intervals; (2) extension of the set of arithmetic operations for normal interv ..."
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This paper describes an implementation of a general interval arithmetic extension, which comprises the following extensions of the conventional interval arithmetic: (1) extension of the set of normal intervals by improper intervals; (2) extension of the set of arithmetic operations for normal intervals by nonstandard operations; (3) extension by infinite intervals. We give a possible realization scheme of such an universal interval arithmetic in any programming environment supporting IEEE floatingpoint arithmetic. A PASCALXSC module is reported which allows easy programming of numerical algorithms formulated in terms of conventional interval arithmetic or of any of the enlisted extended interval spaces, and provides a common base for comparison of such numerical algorithms. 1
All about Generalized Interval Distributive Relations. I. Complete . . .
, 2000
"... The arithmetic on an extended set of proper and improper intervals presents algebraic completion of the conventional interval arithmetic allowing thus e cient solution of interval algebraic problems. This paper generalizes the distributive relations, known by now, on multiplication and addition of p ..."
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The arithmetic on an extended set of proper and improper intervals presents algebraic completion of the conventional interval arithmetic allowing thus e cient solution of interval algebraic problems. This paper generalizes the distributive relations, known by now, on multiplication and addition of proper and improper intervals. A complete proof of the main results is presented, demonstrating an original technique based on functional notations and transition formulae between different interval structures. A variety of equivalent forms and different representations are discussed together with some examples. This paper is an extraction from [19] and will be updated permanently to include current improvements, generalizations and applications of the conditionally distributive relations. The second part of the paper is scheduled for the end of 2000 and will include several directions for the application of the generalized distributive relations. Keywords: