Optimal quadrature formulas with derivatives in Sobolev space

2014/10/30 by Kh. M. Shadimetov, Shadimetov, Kh. M., A.R. Hayotov +3
Computer Science · Mathematics · #65D32 #FOS: Mathematics #Iterative Methods for Nonlinear Equations #Mathematical functions and polynomials #Matrix Theory and Algorithms #Numerical Analysis (math.NA)

paper · pdf · doi:10.48550/arxiv.1410.8423

openalex publication_date 2014/10/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In the present paper the problem of construction of optimal quadrature formulas in the sense of Sard in the space L₂^(m)(0,1)is considered. Here the quadrature sum consists of values of the integrand at nodes and values of the first and the third derivatives of the integrand at the end points of the integration interval. The coefficients of optimal quadrature formulas are found and the norm of the optimal error functional is calculated for arbitrary natural number N and for any m≥ 4 using S.L. Sobolev method which is based on discrete analogue of the differential operator d^2m/dx^2m. In particular, for m=4, 5 optimality of the classical Euler-Maclaurin quadrature formula is obtained. Starting from m=6 new optimal quadrature formulas are obtained.

Citations

• The Discrete Analogue of the Operator d^2m/dx^2m and its Properties
• Optimal quadrature formulas of closed type in the space L₂^(m)(0,1)

Related

• On an optimal quadrature formula in Sobolev space L₂^(m) (0,1)
• Optimal Quadrature Formulas in the Sense of Sard in W₂^(m,m-1) Space
• Optimal Quadrature Formulas with Positive Coefficients in L₂^(m)(0,1) Space
• Optimal quadrature formulas of closed type in the space L₂^(m)(0,1)
• Construction of optimal quadrature formulas exact for exponentional-trigonometric functions by Sobolev's method