Аналіз обчислювальної стійкості алгоритмів розв’язання систем лінійних алгебричних рівнянь з матрицями

Abstract

Запропоновано новий підхід до обчислювальної стійкості алгоритмів розв’язання систем лінійних алгебричних рівнянь з матрицями. Розглянуто деякі результати з теорії похибок для розв’язання систем лінійних алгебричних рівнянь з матрицями. Проведено аналіз обчислювальної стійкості засобами зворотного аналізу похибок. Показано реалізацію методу розрізання для систем лінійних алгебричних рівнянь з матрицями. Проаналізовано похибки схеми розрізання для розв’язання систем з матрицями.

The linear algebraic equations solution is one of the calculative mathematics actual tasks. Investigating certain processes with the mathematics methods and Electronic Calculative Machine usage firstly the mathematical model of the investigated object was built. Then the built mathematical model is transformed into such appearance that the solution is being found as a numerical result with the help of arithmetical and logical operations. Such transformation is carried out by the numerical methods usage. Naturally at the given stage a number of problems dealing with their calculative stableness arise. That is why the new approach to the calculative stableness system of linear algebraic equations with matrix algorithm solution is proposed in the article. During the linear algebraic equations solution there arise the errors caused by the initial data unpreciseness or the error approximation. Besides, during calculation the errors almost always occur within the task itself (because of the arithmetical operations inaccurate completion). The mistakes of this type (so-called calculative) in many cases being aggregated equal the exact solution of the same task, but with the changed initial data. Even provided that the previous measurements and calculations were carried out with a high preciseness and a stable calculative method was chosen for the solution of the task, the initial data error, though very small would occur. These errors influence, to some extent, the solution of the system solution. Some results from the error theory for the linear algebraic equations with matrix are analysed in the article. Some results from the theory of error for the linear algebraic equations with matrix system solution are studied. The calculative stableness is analyzed by means of the error inverse analysis. Realization of the incision method for the system of the linear algebraic equations with matrix is revealed. The incision scheme errors for the system with matrix solution are analyzed. Theoretical and methodological bases of the investigation are composed by the methods of optimization and mathematical modeling.