ABSTRACT
In solving numerous problems in mathematics, mechanics, and physics, one is faced with the necessity of calculating different singular integrals. Many hundreds of works have been devoted to approximate methods for the calculation of singular integrals since the 1950s and the flow of publications increases. In this connection, it is necessary to develop criteria for which these methods can be compared. One of such tests is optimality of algorithms. The optimal methods of calculating the singular and hypersingular integrals are investigated. The chapter is devoted to algorithms which are optimal with respect to accuracy for the calculation of singular integrals with fixed singularity, Cauchy and Hilbert kernels, polysingular, many-dimensional singular and hypersingular integrals. It utilizes the definitions of optimal with respect to accuracy and complexity algorithms from. Several authors have studied the calculation of singular integrals on various classes of analytical functions.
