基于区间分析的定积分可靠计算

Reliable Computing of Definite Integrals Based on Interval Analysis

研究了定积分的可靠计算问题,即是根据定积分的区间包含定理,借助专业的区间运算软件,构造算法程序严格计算定积分值的包含区间首先,利用泰勒公式、区间函数的包含单调性和区间四则运算给出了被积函数在积分区间上的包含区间多项式.然后,运用区间积分理论构建了计算定积分值包含区间的数学公式,该公式由两个积分区间相同的多项式积分组成,能计算出宽度任意小的包含区间最后,根据此计算公式,借助执行向外舍入规则的区间运算软件设计编写了用于实际计算的算法程序。理论分析和数值实验均表明了文章所提算法的性能.

This paper mainly studies the reliable computing of definite integral,that is,based on interval inclusion theorem,with the help of professional interval arithmetic software to construct algorithm programs to rigorously compute interval enclosures of the definite integral values.Firstly,the interval polynomial enclosure of the integrand on the integration interval is given by using Taylor's formula,inclusion isotonicity of interval function and interval arithmetic operators;Then,the mathematical formula for computing the inclusion interval of definite integral value is constructed by using the interval integral theory,which is composed of two polynomial integrals with the same integral interval,and can calculate the interval enclosure of arbitrarily narrow width;Finally,according to the formula,an algorithm program for practical computation is designed by using the interval operation software that performs outward rounding.Theoretical analysis and numerical experiments illustrate the performance of the proposed algorithm.