Forming a мetodology for transforming a model as the basis for expanding its informativeness

Abstract

A problem on building the methodology for transforming the implicit form of a model has been stated and solved, which improves the efficiency of replacing complex nonlinear forms of mathematical models to reducing them to a recurrent sequence in the form of analytical expressions that allow quick express calculations. New explicit forms of the models have been proposed that make it possible to use recurrent sequences for representing a solution to the problem and forming an error estimation expression, as well as and additional information. Given the fact that for many attributes the solution and an error estimation are critical, the analyticity of expressions reveals new properties and possibilities. Based on such factors as authenticity, accuracy, depth, materiality and completeness, the adequacy of the model is represented by a single analytical expression that would make it possible in future to simplify the process of comparison through the use of quantitative methods. Representation of transformations, according to which the connection between the error of two consecutive approximations and dependence on the approximation number was established, is predetermined by the necessity to analyze convergence dynamics based on the iteration number. Another variant, not less important, which can characterize the dynamics of convergence is the connection between an error of the first approximation and the arbitrary approximation. Based on the overall expansion of the implicit form of the model and the mean value theorem, the relationship between the two consecutive errors or norms has been established. It is demonstrated that if the error or a norm of error are assigned, the estimates for the first and second derivatives would make it possible to determine the boundary number of the iteration starting from which the error is less than the assigned one. An example has been given for deriving an estimation for the general model of the magnitude of a maximally possible error, the boundary number of the iteration, starting from which the error acquires a value less than the assigned one. A comprehensive analytical assessment of adequacy based on a single expression has also been derived. Representation of informational attributes in the quantitative form is predetermined by new opportunities that would emerge due to the obtained tools for quantitative analysis. Numerical modelling has been performed; the character of dynamics of new informational indicators and attributes has been examined. The data given for nine iterations demonstrate efficiency and completeness of information to perform a quick analysis and draw a conclusion. Based on the dynamics of quantitative attributes for a relative error, as well as new ones, proposed based on the results from implementing the model transformation methodology, it has been shown that the possibilities emerge to run a quick analysis and draw a conclusion. It was demonstrated that the introduced attributes expand the informativeness of the methodology implementation for the further representation of a nonlinear model in the form of a recurrent sequence