System for Finding Optimal Coefficients of Hermite-type Quadrature Formulas with third-Order Derivatives
DOI:
https://doi.org/10.71310/pcam.4_68.2025.07Keywords:
Sobolev space, derivative optimal quadrature formula, error functional, optimal coefficientsAbstract
In the world, special attention is given to developing various optimal computational methods for the approximate calculation of definite integrals. This article focuses on constructing composite optimal quadrature formulas in the space of differentiable functions using the Sobolev method. The quadrature formula consists of a linear combination of function values and its derivatives up to and including the third order at all nodes of the interval [0,1]. The accuracy of quadrature formulas is evaluated by the norm of the error functional of the quadrature formulas: The error of composite quadrature formulas is estimated through the norm of the error functional of the quadrature formula under onsideration and the norm of the function. The norm of the error functional of the quadrature formula will be determined through the extremal function of this quadrature formula. It is known that the norm of the error functional of a composite quadrature formula is expressed in terms of the coefficients of this quadrature formula. By minimizing this norm with respect to the coefficients, systems of linear equations will be obtained for finding the optimal coefficients of composite quadrature formulas.
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