Abstract. The computer method of piecewise polynomial approximation of functions and of the Cauchy problem solution for ordinary differential equations based on the Newton polynomial is presented. The approximating polynomial on a subinterval is converted to the form with numerical coefficients, the degree of the polynomial and the number of subintervals varies. The uniform convergence of the method at the rate of geometric progression is shown under conditions of double continuous differentiability of the function and of the right-hand side of the system. The approximate solution of the system is continuous, continuously differentiable, and is characterized by low error rate, in particular, when solving stiff problems.

Keywords: piecewise polynomial approximation of functions, solution of ordinary differential equations, error minimization, stiff systems.