519.685.7 This article examines some topics of optimization of computations, which have been discussed at 25 seminar-schools and symposia organized by the V. M. Glushkov Institute of Cybernetics of the Ukrainian Academy of Sciences since 1969. We describe the main directions in the development of computational mathematics and present some of our own results that reflect a certain design conception of speed-optimal and accuracy-optimal (or nearly optimal) algorithms for various classes of problems, as well as a certain approach to optimization of computer computations.The vigorous development of computer science in the 1950s-1960s has left its imprint on computational mathematics, which studies the methods of generating solutions of various mathematical problems in numerical form (whether approximate or exact). The traditional divisions of computational mathematics mainly include [ 1] function evaluation, computational methods of linear algebra, numerical solution of algebraic and transcendental equations, numerical differentiation and integration, numerical solution of differential and integral equations, and numerical minimization methods for functions and functionals. New directions are developing in parallel with this traditional body of knowledge, including theory and numerical methods of solution of ill-posed problems, linear and nonlinear programming, theory of data processing systems, fitting of functions and functionals, optimization methods, etc. Each of these disciplines is characterized by its own numerical methods, which are based on different ideas. Alongside the iterative methods, which have rapidly grown and are efficiently used today for many classes of problems, we are witnessing particularly wide development of various approximation (discretization) methods that replace the original (exact) problem with an approximate (in some sense) problem for which an exact or an approximate solution is obtained. These include projection methods, finite-difference methods, projection-iteration methods, and others. Approximating equations are constructed by these methods in such a way that their solution normally reduces to solving a finite system of scalar equations. We know that the quality and the efficiency of specific numerical methods is determined by the set of their characteristics, some of which are applicable only to numerical methods of a certain class (such as initial approximation and rate of convergence that are applicable for iterative methods) while others apply to numerical methods of all classes. Such general characteristics include various errors of the numerical method, running time on a computer, and the computer memory requirements.Many mathematicians are engaged in the theoretical analysis of numerical methods with the object of estimating their various characteristics. These studies have focused on discretization of continuous problems, effectively testable conditions of applicability of various methods, rate of convergence bounds, prior and posterior error estimates, choice of init...
