An optimal quadrature formula exact to the exponential function by the phi function method

The numerical integration of definite integrals is essential in fundamental and applied sciences. The accuracy of approximate integral calculations is contingent upon the initial data and specific requirements, leading to the imposition of diverse conditions on the resultant computations. Classical methods for the numerical analysis of definite integrals are known, such as the quadrature formulas of Gregory, Newton-Cotes, Euler, Gauss, Markov, etc. Since the middle of the last century, the theory of constructing optimal formulas for numerical integration based on variational methods began to develop. It should be noted that there are optimal quadrature formulas in the sense of Nikolsky and Sard. In this paper, we study the problem of constructing an optimal quadrature formula in the sense of Sard. When constructing a quadrature formula, the method of ϕ-functions is used. The error of the formula is estimated from above using the integral of the square of the function ϕ from a specific Hilbert space. Next, such a ϕ function is selected, and the integral of the square in this interval takes the smallest value. The coefficients of the optimal quadrature formula are calculated using the resulting ϕ function. The optimal quadrature formula in this work is exact on the functions eσx and e−σx, where σ is a nonzero real parameter. Mathematics Subject Classification (2010): 65D30, 65D32. Keywords: Hilbert space, phi-function method, optimal quadrature formula, error quadrature formula.