Abstract
Systems of linear algebraic equations are a mathematical apparatus that is widely used in solving a significant number of problems in the practical application of mathematics and engineering. The analysis of errors in solving linear algebraic equations and the sensitivity function of nonlinear systems using the method of equivalent excitations, which, in turn, makes it possible to make informed decisions on the choice and development of methods for studying the accuracy of computing devices. Methods for constructing estimates «from below» of the distribution functions of the fatal error of the numerical solution of systems of linear algebraic equations are also presented, in particular, a posteriori estimates of the effectiveness of the methods under study are analyzed.
Highlights
At the present stage of the development of analog computing facilities, it is the inaccuracy of the input of the initial data that is the main source of errors in the SLAE modeling, which determines the importance of developing practical methods for assessing the error due to the inaccuracy of the initial data
Often the elements of both matrices and vectors of the right sides that arise in the practice of SLAE (System of linear algebraic equations) are known quantities only with a certain degree of accuracy
We give an analysis of certain stages in the development of guaranteed estimates Cг of the fatal error of the SLAE solution in order to obtain practically more efficient estimates of the accuracy
Summary
At the present stage of the development of analog computing facilities, it is the inaccuracy of the input of the initial data that is the main source of errors in the SLAE modeling, which determines the importance of developing practical methods for assessing the error due to the inaccuracy of the initial data. A practical way of constructing estimates that are «better» in the above sense than guaranteed ones is to determine some estimates «from below» the interested distribution laws, on the basis of which it is possible to obtain confidence estimates Ср of the unremovable error in the solution of the SLAE with a confidence level not less than р In this case, the following relation is valid: Ср ≥ Сор. Most of the guaranteed estimates of the fatal error of the SLAE solution can be used to construct lower estimates of the distribution laws of the corresponding quantities if the excitation elements ∆A and ∆b are random variables with known probabilistic characteristics, which will be assumed in what follows. ‖∆x‖⁄‖x‖, when the calculation does not require knowledge of the quantity ‖x‖
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
