INTERPRETATIONS METHOD AND GAUSSIAN QUADRATURES

Abstract

Any mathematical model is the interpretation of natural, technological, mental process in mathematical language. In scientific researches one faces interpretations method at every step. It is sufficient to mention the graph theory, analytic geometry, differential equations, Laplace transformation, Fourier transformation, encoding theory etc. As a rule in the interpretations method the problem of one branch of mathematics is interpreted in other branch, where it is either simplified or better responds to our intuition or allows usage of other approaches etc. We paid our attention to Gaussian quadratures not only because they are used in modern standard programs of integration. We made sure that there is certain didactic potential in Gaussian quadratures, which can be useful for those who study and teach mathematical modelling. We have selected problems in which nodes of Gaussian quadrature appear unexpectedly as a result of received solution. Traditionally the search of nodes and weighting factors of Gaussian quadrature involves making and solving the system of (non-linear!) algebraic equations, while simple mathematical folklore requires more ‘trivial’ proves which are good for simple understanding. Simple quadrature formula of Gauss (two nodes of integration) has been reviewed in the work. Examples of problems which contain latent connection to Gaussian quadrature are given. These problems are peculiar combination of simplicity and non-triviality in which a reader can find something interesting to his/her taste. It is natural that every problem is formulated on two ‘canonic’ intervals: [−1, 1] and [0, 1], to cover two versions of quadrature: Gauss-Legendre and Gauss-Bernoulli ones. Reviewed examples give new subjects for reflections and observations. It is worth noting that approaches suggested in the work has been successfully tested for ‘clearness + briefness + convenience’ among students of higher education institutions. We agree with the point of view of Lithuanian mathematician R. Kashuba who thinks that interpolations method contributes to spread of democracy because it improves the ability to change point of view.