一个高精度数值积分公式

A HIGHER ORDER ACCURACY NUMERICAL QUADRATURE RULE

摘要: 本文讨论了一个具有高精度的求积公式∫_a^bf(x)dx)=(b-a)(7f(a)+16f((a+b)/(2))+7f(b)+(b-a)(f'(a)-f'(b)))/30+Ef其中Ef=((b-a)⁷/(604800))f⁽⁶⁾(ξ),a<ξ<b及其复合公式∫_a^b(f(x)dx)=(b-a)((7f(a+2ih)+7f(a+2ih)+16???19880410-2???f(a+2i-1h)+(b-a)(f'(a)-f(b))/2n)/30n+E_nf这里E_nf=((b-a)⁷)/(604800n⁶)f⁽⁶⁾(η),a< η< b而h=(b-a)/2n它具有辛普生公式的一切优点,但精确度比辛普公式高2阶。数值试验表明,这是一个非常有效的求积公式。

Abstract: This paper discusses the following quadrature rule with higher degree of accuracy ∫_a^bf(x)dx)=(b-a)(7f(a)+16f((a+b)/(2))+7f(b)+(b-a)(f'(a)-f'(b)))/30+Ef Where Ef=((b-a)⁷/(604800))f⁽⁶⁾(ξ),a<ξ<b andits Compositrule are presented aswell:∫_a^b(f(x)dx)=(b-a)((7f(a+2ih)+7???19880410-2???f(a+2ih)+16f(a+2i-1h)+(b-a)(f'(a)-f(b))/2n)/30n+E_nf Where E_nf=((b-a)⁷)/(604800n⁶)f⁽⁶⁾(η),a< η< b h=(b-a)/2n which possess the all advantages of simpsons rule, but the degree of accuracy isincreased by two order than Simpson's rule.The numerical tests that the quadrature formula of this paper is very efficient.