Quadrature Formula of the Highest Algebraic Degree of Accuracy Containing Predefined Nods

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Home Editorial board Publication ethics Peer review guidelines Latest issue All issues Rules for authors Online submission system's guidelines Submit manuscript Contacts Address: Vatutina st. 53, Vladikavkaz, 362025, RNO-A, Russia Phone: (8672)23-00-54 E-mail: rio@smath.ru	DOI: 10.46698/z4764-9590-5591-k Quadrature Formula of the Highest Algebraic Degree of Accuracy Containing Predefined Nods Khubezhty, Sh. S. Vladikavkaz Mathematical Journal 2023. Vol. 25. Issue 1. Abstract: Approximate methods for calculating definite integrals are relevant to this day. Among them, the quadrature methods are the most popular as they enables one to calculate approximately the integral using a finite number of values of the integrable function. In addition, in many cases, less computational labor is required compared to other methods. Using Chebyshev polynomials of the first, second, third, and fourth kind corresponding to the weight functions \(p(x)=\frac{1}{\sqrt{1-x^2}}\), \(p(x)=\sqrt{1- x^2}\), \(p(x)=\sqrt{\frac{1+x}{1-x}}\), \(p(x)=\sqrt{\frac{1-x}{1+x }}\), on the segment \([-1,1]\), quadrature formulas are constructed with predefined nodes \(a_1=-1\), \(a_2=1\), and estimates of the remainder terms with degrees of accuracy \(2n+1\). In this case, a special place is occupied by the construction of orthogonal polynomials with respect to the weight \(p(x)(x^2-1)\) and finding their roots. This problem turned out to be laborious and was solved by methods of computational mathematics. Keywords: weight functions, orthogonal polynomials, quadrature formulas, predetermined nodes, remainder terms, degrees of accuracy Language: Russian Download the full text For citation: Khubezhty, Sh. S. Quadrature Formula of the Highest Algebraic Degree of Accuracy Containing Predefined Nods, Vladikavkaz Math. J., 2023, vol. 25, no. 1, pp.131-140 (in Russian). DOI 10.46698/z4764-9590-5591-k + References 1. Krylov, V. I. Approximate Calculation of Integrals, New York, Macmillan Co, 1962. 2. Bakhvalov, N. S., Zhidkov, N. P. and Kobelkov, G. M. Chislennye metody [Numerical Methods], 2nd ed., Moscow, Fizmatlit, 2002, 598 p. (in Russian). 3. Il'in V. P. Chislennyy analiz. Ch. 1 [Numerical Analysis. Part 1], Novosibirsk, 2004, 334 p. (in Russian). 4. Khubezhty, Sh. S. Kvadraturnye formuly dlja singuljarnyh integralov i nekotorye ih primenenija [Quadrature Formulas for Singular Integrals and Some of their Applications], Vladikavkaz, SMI VSC RAS, 2011 (in Russian). 5. Mardanov, A. A. Vychislenie integralov s osobennostjami i reshenie singuljarnyh integral'nyh uravnenij [Calculation of Integrals with Singularities and Solution of Singular Integral Equations], St. Petersburg, Publ. House, 2017 (in Russian). 6. Pashkovsky, S. Vychislitel'nye primeneniya mnogochlenov i ryadov Chebysheva [Computational Applications of Polynomials and Chebyshev Series], Moscow, Nauka, 1983, 382 p. (in Russian). 7. Khubezhty, Sh. S. and Nartikoev, N. B. Quadrature Formulas with Predetermined Nodes with the Weight Function \(p(x)=\frac{1}{\sqrt{1-x^2}}\), Proceedings of the XIV Intern. Sci.-Tech. Conf., Penza, December 1-4, 2020, 2020, pp. 22-25 (in Russian). 8. Khubezhty, Sh. S. and Nartikoev, N. B. Quadrature Formulas Containing Predetermined Nodes with the Weight Function \(p(x)=\sqrt{1-x^2}\), Proceedings of the XIV Intern. Sci.-Tech. Conf., Penza, December 1-4, 2020, 2020, pp. 98-102 (in Russian). The sole copyright holder of the published work is the Founder of the Vladikavkaz Mathematical Journal. The terms of use of this work are governed by an open license (Creative Commons Attribution-NonCommercial 4.0 International). The use of metadata of the scientific article, including the title, abstract, author information, references, identifiers, and other bibliographic description elements for subsequent unrestricted use, is carried out under the terms of the CC BY or CC0 open licenses. ← Contents of issue


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