IN a wide range of practical and computational problems we have to find the roots of a function which can only be computed approximately for given values of its argument. Examples include the boundary value problem for a nonlinear system of differential equations, solved by specifying missing initial conditions (at one end), or the problem of the experimental selection of the parameters of a process or device in order to satisfy a given condition. Since computation of the function whose root is required often involves a laborious, lengthy or expensive procedure, it seems natural to consider optimization of the method of finding the root. Two problems arise here: first, assuming that the errors in computing the function are known, how to select the points at which it is to be computed; and second, how optimally to distribute the existing resources over the computational stages. By resources we mean what the computational accuracy depends on in a specific case, e.g. computer time, labour or cost of the computations, etc. The present paper states mathematically and solves the problem of an optimal search method for one class of functions. The statement comes in Section 1, while Section 2 defines the optimal search algorithm for the case when the function is computed with known errors (and in particular, accurately). The question of the optimal distribution of the resources over the computations is discussed in Section 3. Some examples are considered, and the results of calculating optimal algorithms are given. In the course of the solution we use ideas of the method of dynamic programming, which were earlier used for devising opt1imal search algorithms in which computational errors were ignored [1].