Summary: Several applied problems are characterized by the need to numerically solve equations with an operator function (matrix function). In particular, in the last decade, mathematical models with a fractional power of an elliptic operator and numerical methods for their study have been actively discussed. Computational algorithms for such non-standard problems are based on approximations by the operator function. In this case, an approximate solution is determined by solving auxiliary standard problems. This paper discusses the main directions for constructing acceptable approximations of operator functions when solving equations.
The most widespread are the approaches using various options for rational approximation. Also, we note the methods that relate to approximation by exponential sums. We propose to use a new approach, which is based on the application of approximation by exponential products. The solution of an equation with an operator function is based on the transition to standard stationary or evolutionary problems. Estimates of the accuracy of the approximate solution of the operator equation at known absolute or relative errors of approximation functions are obtained. The influence of the error in the solution of auxiliary problems is investigated separately.
General approaches are illustrated by a problem with a fractional power of the operator. The first class of methods is based on the integral representation of the operator function under rational approximation, approximation by exponential sums, and approximation by exponential products. The second class of methods is associated with solving an auxiliary Cauchy problem for some evolutionary equation. In this case, we can distinguish a method based on approximation by exponential products.