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This paper proposes an algorithm for calculating approximate values of  roots of algebraic equations with a specified limit of absolute errors. A mathematical basis of the algorithm is an analytical-numerical method of solving nonlinear integral-differential equations with non-stationary coefficients. The analytical-numerical method belongs to the class of one-step continuous methods of variable order with an adaptive procedure for choosing a calculation step, a formalized estimate of the error of the performed calculations at each step and the error accumulated during the calculation. The proposed algorithm for calculating the approximate values of the roots of an algebraic equation with specified limit absolute errors consists of two stages. The results of the first stage are numerical intervals containing the unknown exact values of the roots of the algebraic equation. At the second stage, the approximate values of these roots with the specified limit absolute errors are calculated. As an example of the use of the proposed algorithm, defining the roots of the fifth-order algebraic equation with three different values of the limiting absolute error is presented. The obtained results allow drawing the following conclusions. The proposed algorithm enables to select numeric intervals that contain unknown exact values of the roots. Knowledge of these intervals facilitates the calculation of the approximate root values under any specified limiting absolute error. The algorithm efficiency, i.e., the guarantee of achieving the goal, does not depend on the choice of initial conditions. The algorithm is not iterative, so the number of calculation steps required for extracting a numerical interval containing an unknown exact value of any root of an algebraic equation is always restricted. The algorithm of determining a certain root of the algebraic equation is computationally completely autonomous.