An Optimal Quadrature Formula Exact to the Exponential Function
DOI:
https://doi.org/10.71310/pcam.5_69.2025.06Keywords:
Hilbert space, phi-function method, optimal quadrature formula, the errorAbstract
Numerical integration of definite integrals plays a crucial role in both fundamental and applied sciences. The precision of approximate integral calculations depends on the initial data and specific conditions, which impose various requirements on the resulting computations. Traditional methods for numerical analysis of definite integrals, such as the quadrature formulas developed by Gregory, Newton-Cotes, Euler, Gauss, Markov, and others, are well-established. Since the mid-20th century, the theory of creating optimal formulas for numerical integration through variational methods has evolved. It is important to mention that optimal quadrature formulas exist in the sense defined by Nikolsky and Sard. This paper focuses on the challenge of constructing an optimal quadrature formula according to Sard’s approach. In this process, the method of phi-functions is applied. The error of the formula is estimated by integrating the square of the phi function from a specific Hilbert space. Afterward, the appropriate function is chosen such that the integral of its square within this interval is minimized. Finally, the coefficients of the optimal quadrature formula are determined using the resulting phi function.
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