The goal of the research is to construct practicable numerical algorithms for stiff systems of ordinary differential equations which let you increase the accuracy of the approximate solution without decreasing the length of the time interval. To achieve this goal, we have constructed a family of new iterative analytic processes generalising the Picard process. For a basic representative of this family, we demon-strate its better convergence properties on a scalar linear problem in comparison with the classical Picard process. For the general form of such iterative processes, we discuss their connection with existing methods for operator equations and propose a method for choosing their parameters. The efficiency of this parameter determination method is justified with a numerical experiment. In conclusion we propose a general approach to the construction of numerical algorithms which is based on the discretisation of the constructed iterative analytic processes.