This article explores the accuracy of numerical solutions, and suggests methods for analyzing accuracy depending on the properties of the problem. Numerical studies of complex expensive objects of technology and physics require that the computational results be obtained with guaranteed accuracy. It also depends on the fact that in the work of technical objects there are large intervals of operation time not observed experimentally. Therefore, there is a need to describe the location of the observed and calculated values, as well as the accuracy with which they are calculated. The effect of strong growth in estimates of error bounds is manifested for a large number of methods used to estimate the error of a numerical solution. This means the lack of correctness of algorithms for evaluating the accuracy of numerical solutions due to the failure of the stability conditions with respect to perturbations of the right-hand side. For many problems, among all the algorithms, the backward analysis of errors turned out to be the most effective method for assessing the accuracy of numerical solutions. The backward analysis of errors consists in the fact that when assessing the accuracy (error) of a numerical solution, the numerical solution is considered as an exact solution to a problem close to the original problem. A backward error analysis was proposed and developed in the algorithms of J Wilkinson in the context of the numerical solution of problems of linear algebra, and in the algorithms of V V Voevodin, who widely distributed it to many areas of numerical analysis. In the framework of the backward error analysis, the regularization of the algorithm for estimating the error of a numerical solution is reasonably applied. This article explores methods for the backward analysis of errors of numerical solutions.

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In this article we discuss methods of computing the guaranteed values of the states of a technical system in order to estimate the safety of a technical system The danger of a system functioning is a threat, possibility, probability of damage, system catastrophe, that is, potential damage to a technical system in certain conditions and situations. An analysis of the boundaries of the safety areas of technical systems helps to obtain quantitative estimates of the possibility of dangerous situations. The presence of such estimates may allow a more reasonable search and development of a set of measures to eliminate or mitigate the consequences of such situations. The article discusses the new results of computing the guaranteed boundaries of solution sets and the results of their application for assessing the boundaries of security areas and studying practical stability. Methods are used based on the approximation of the shift operator along the trajectory, and taking into account the influence of constantly acting perturbations on the solutions.

The reliability of a technical system is the ability to perform the required functions, being within the specified boundaries of all parameters and conditions of application of the technical system.The concept of safety of a technical system is based on damage. Damage is the amount of deterioration in the quality of the system.The amount of damage determines two properties: the hazard of the system is a characteristic of the ability to suffer or cause damage;safety of system is a characteristic that prevents damage from occurring or reducing its magnitude to an acceptable value.The article deals with the analysis of safe and dangerous states of a technical system based on the study of the properties of their mathematical models, especially the survival of their trajectories.The relationship between the security areas and the practical stability of the technical system as the limitation of all its trajectories is determined.