Interval Computing and Information Technology ∗
Interval Computing and Information Technology ∗
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区间计算与信息技术＊
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Chenyi Hu
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作者：
Chenyi Hu
区间计算已成为应用研究中的有力工具。本文简要介绍了区间计算、现有的软件以及最近的应用，特别是在信息技术中的应用。这是一份涉及技术细节的调查。1区间计算R. E.摩尔在20世纪50年代末引入了区间计算[23，24]。从此，它成为科学计算的一个活跃的研究分支。在本节中，我们简要回顾一下什么是区间计算。1.1数学和机器区间一个非空的数学区间[a，B]是集合{x ∈ <|a ≤ x ≤ B}其中a ≤ B。为了在计算机上执行区间计算，数学区间应该由机器区间表示，其端点是机器可表示的数字。我们说[a <$，B <$]是[a，B]的机器表示，如果[a <$，B <$]<$[a，B]，即a <$≤ a且B ≤ B。我们说机器区间[a B]是数学区间[a，B]的紧表示当且仅当a是小于或等于a的最大机器可表示数，而B是大于或等于B的最小机器可表示数。1.2数学区间上的区间运算定义如下。设a和B是两个数学区间。设op为算术运算+，−，×，之一。则a op B ∈ {a op B：a ∈ a，B ∈ B}，条件是0 6∈ B如果op表示n.这项工作得到了NSF赠款0202042的部分支持
Interval computing has become a powerful tool in applied research. In this paper, we briefly introduce interval computing, available software, and resent applications especially in information technology. It is a survey with points to technical details. 1 Interval computing R. E. Moore introduced interval computing [23, 24] in the late 1950s. Ever since, it has become an active research branch of scientific computation. In this section, we briefly review what is interval computing. 1.1 Mathematical and machine intervals A nonempty mathematical interval [a, b] is the set {x ∈ <|a ≤ x ≤ b} where a ≤ b. To perform interval computing on computers, a mathematical interval should be represented by a machine interval whose endpoints are machine representable numbers. We say that [a∗, b∗] is a machine representation of [a, b] if [a∗, b∗] ⊇ [a, b] i.e. a∗ ≤ a and b ≤ b∗. We say that the machine interval [a∗, b∗] is a tight representation of a mathematical interval [a, b] if and only if a∗ is the greatest machine representable number which is less than or equal to a, and b∗ is the least machine representable number which is greater than or equal to b. 1.2 Interval arithmetic Interval arithmetic on mathematical intervals is defined as follows. Let a and b be two mathematical intervals. Let op be one of the arithmetic operations +,−,×,÷. Then a op b ≡ {a op b : a ∈ a, b ∈ b}, provided that 0 6∈ b if op represents ÷. ∗This work is partially supported by NSF grants 0202042