  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. The most widespread are the approaches
using various options for rational approximation. Also, we note the methods
that relate to approximation by exponential sums. In this paper, the
possibility of using approximation by exponential products is noted. The
solution of an equation with an operator function is based on the transition to
standard stationary or evolutionary problems. 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.
