推广的双向积分不等式及其应用

对一类开型的最佳两点积分法则:ba∫f(t)dt=b-a2f(3a+b4)+f(a+3b4)]+R(f),通过构造函数p1(t),p3(t),导出了两点积分法则上下误差界的双向积分不等式,并得到了相应的最佳上下误差界;另外,考虑到两点积分法则的一个扰动,这个扰动后的法则优越于原始的积分法则;最后,给出了这些结果在数值积分中的应用。

Extended Double Integral Inequalities and Applications

In this paper,a class of optimal and simple 2-point quadrature rule of open type is discussed,that is,∫baf(t)dt=b-a2f(3a+b4)+f(a+3b4)]+R(f).Accordingly,upper and lower error bounds for this rule are derived by constructing two functions p1(t),p3(t).Double integral inequalities are obtained,which give upper and lower bounds for this quadrature rule.Furthermore,it is shown that these error bounds are sharp.We also consider a perturbation of the quadrature rule(1.1),Which has some advantages over the original rule(1.1).Finally,applications of the above mentioned results in numerical integration are given.