Abstract
Among all integration rules with n points, it is well-known that n-point Gauss–Legendre quadrature rule ∫ −1 1f(x) dx≃∑ i=1 nw if(x i) has the highest possible precision degree and is analytically exact for polynomials of degree at most 2 n−1, where nodes x i are zeros of Legendre polynomial P n ( x), and w i 's are corresponding weights. In this paper we are going to estimate numerical values of nodes x i and weights w i so that the absolute error of introduced quadrature rule is less than a preassigned tolerance ϵ 0, say ϵ 0=10 −8, for monomial functions f(x)=x j, j=0,1,…,2n+1. (Two monomials more than precision degree of Gauss–Legendre quadrature rules.) We also consider some conditions under which the new rules act, numerically, more accurate than the corresponding Gauss–Legendre rules. Some examples are given to show the numerical superiority of presented rules.
Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
