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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.
Directed Interval Arithmetic in Mathematica: Implementation and Applications
, 1996
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Only intervals preserve the invertibility of arithmetic operations, Reliable Computing
, 1997
"... In standard arithmetic, if we, e.g., accidentally added a wrong number y to the preliminary result x, we can undo this operation by subtracting y from the result x+ y. A similar possibility to invert (undo) addition holds for intervals. In this paper, we show that if we add a single noninterval set ..."
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Cited by 4 (2 self)
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In standard arithmetic, if we, e.g., accidentally added a wrong number y to the preliminary result x, we can undo this operation by subtracting y from the result x+ y. A similar possibility to invert (undo) addition holds for intervals. In this paper, we show that if we add a single noninterval set, we lose invertibility. Thus, invertibility requirement leads to a new characterization of the class of all intervals. 1 Formulation of the Problem Why Invertible? Many computer operations, including addition x → x + y, are invertible in the sense that if we accidentally added the wrong number y, we can always reconstruct the original value x by simply subtracting y from the result of the addition: (x+ y) − y = x. Case of Partial Knowledge In many real life situations, our knowledge about the actual values x and y is
Computer graphics, linear interpolation, and nonstandard intervals, Manuscript
, 2008
"... Abstract. This document is an assessment of the value of optimal linear interpolation enclosures and of nonstandard intervals, especially with respect to applications in computer graphics, and of the extent a future IEEE interval standard should support these. It turns out that essentially all prese ..."
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Abstract. This document is an assessment of the value of optimal linear interpolation enclosures and of nonstandard intervals, especially with respect to applications in computer graphics, and of the extent a future IEEE interval standard should support these. It turns out that essentially all present applications of nonstandard intervals to practical problems can be matched by similarly efficient approaches based on standard intervals only. On the other hand, a number of applications were inspired by the use of nonstandard arithmetic. This suggests the requirement of a minimal support for nonstandard intervals, allowing implementations of nonstandard interval arithmetic to be compatible with the standard, while a full support by making one of the conflicting variants required seems not appropriate.
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 PASCAL–XSC 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. Расширенная интервальная арифметика в среде стандарта IEEE для чисел с плавающей точкой Е. Д. Попова Описывается реализация расширения интервальной арифметики, которое включает в себя: (1) дополнение множества обычных интервалов несобственными интервалами; (2) дополнение множества арифметических операций нестандартными операциями; (3) введение бесконечных интервалов. Представлена схема реализации такой универсальной интервальной
Tawards Credible Implementation of Inner Interval Operations
 15th IMACS World Congress on Scientific Computation, Modelling and Applied Mathematics. Volume 2 Numerical Mathematics
, 1997
"... This paper briefly outlines interval arithmetic, extended by four supplementary interval operations, discuss a source of numerical errors at the implementation of floatingpoint inner interval operations and shows different ways for their suppression. The goal is to make computations involving these ..."
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This paper briefly outlines interval arithmetic, extended by four supplementary interval operations, discuss a source of numerical errors at the implementation of floatingpoint inner interval operations and shows different ways for their suppression. The goal is to make computations involving these operations more accurate and credible.
The Biography of A. A. Markov
 In [62
"... We consider interpolation of families of functions depending on a parameter by families of interpolation polynomials. Inner and outer inclusions for the interpolating families are constructed in terms of interval and extended interval arithmetic. We achieve tighter inclusions under certain monotonic ..."
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We consider interpolation of families of functions depending on a parameter by families of interpolation polynomials. Inner and outer inclusions for the interpolating families are constructed in terms of interval and extended interval arithmetic. We achieve tighter inclusions under certain monotonicity assumptions with respect to the parameters involved. Some interpolation polynomials involving directed intervals are studied. Некоторые интерполяционные задачи, использующие интервальные данные С. М. Марков Рассматривается интерполяция семейств функций, зависящих от некоторого параметра, семействами интерполяционных многочленов. Внутреннее и внешнее включения для этих интерполяционных семейств строятся с использованием интервальной и расширенной интервальной арифметик. При некоторых предположениях о монотонности входящих в задачу параметров достигаются более тесные включения. Исследуются некоторые интерполяционные многочлены, в которые входят направленные интервалы.
Fuzzy Numbers are the Only Fuzzy Sets That Keep Invertible Operations Invertible
"... In standard arithmetic, if we, e.g., accidentally add a wrong number y to the preliminary result x, we can undo this operation by subtracting y from the result x+ y. In this paper, we prove the following two results: First, a similar possibility to invert (undo) addition holds for fuzzy numbers (al ..."
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In standard arithmetic, if we, e.g., accidentally add a wrong number y to the preliminary result x, we can undo this operation by subtracting y from the result x+ y. In this paper, we prove the following two results: First, a similar possibility to invert (undo) addition holds for fuzzy numbers (although in case of fuzzy numbers, we cannot simply undo addition by subtracting y from the sum). Second, if we add a single fuzzy set that is not a fuzzy number, we lose invertibility. Thus, invertibility requirement leads to a new characterization of the class of all fuzzy numbers.
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"... 1 Introduction Conventional interval arithmetic [1], [30] has been extended in the following three main directions: * Extension of the set of normal (proper) intervals by improper intervals, which involves an extension of the definitions of the intervalarithmetic operations for the set of proper an ..."
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1 Introduction Conventional interval arithmetic [1], [30] has been extended in the following three main directions: * Extension of the set of normal (proper) intervals by improper intervals, which involves an extension of the definitions of the intervalarithmetic operations for the set of proper and improper intervals. The corresponding extended interval arithmetic structure K has been studied by E. Kaucher [14][16], H.J. Ortolf [31], E. Gardenes [11], [12] and others. The conditional distributivity in K has been recently formulated [8].