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Abstract
Continuous Seidel method for solving systems of linear and nonlinear algebraic equations is constructed in the article, and the convergence of this method is investigated. According to the method discussed, solving a system of algebraic equations is reduced to solving systems of ordinary differential equations with delay. This allows to use rich arsenal of numerical ODE solution methods while solving systems of algebraic equations. The main advantage of the continuous analogue of the Seidel method compared to the classical one is that it does not require all the elements of the diagonal matrix to be non-zero while solving linear algebraic equations’ systems. The continuous analogue has the similar advantage when solving systems of nonlinear equations.
Highlights
Continuous Seidel method for solving systems of linear and nonlinear algebraic equations is constructed in the article, and the convergence of this method is investigated
According to the method discussed, solving a system of algebraic equations is reduced to solving systems of ordinary dierential equations with delay
The main advantage of the continuous analogue of the Seidel method compared to the classical one is that it does not require all the elements of the diagonal matrix to be non-zero while solving linear algebraic equations' systems
Summary
Äëÿ ðåøåíèÿ ñèñòåì óðàâíåíèé âèäà (1.1) ÷àñòî ïðèìåíÿåòñÿ ìåòîä Çåéäåëÿ [1][4], êîòîðûé çàêëþ÷àåòñÿ â ñëåäóþùåì. Î÷åâèäíî, ÷òî äëÿ ðàçðåøèìîñòè ñèñòåìû (1.3) äèñêðåòíûì ìåòîäîì Çåéäåëÿ íåîáõîäèìî, ÷òîáû aii= 0, i = 1, 2, . Ýòî ÿâëÿåòñÿ ñóùåñòâåííûì íåäîñòàòêîì ìåòîäà, ïîñêîëüêó ïðè ðåøåíèè áîëüøèõ ñèñòåì óðàâíåíèé òðåáóåòñÿ çíà÷èòåëüíîå âðåìÿ (âîçìîæíî, ñîïîñòàâèìîå ñî âðåìåíåì ðåøåíèÿ çàäà÷è) äëÿ ïðèâåäåíèÿ ñèñòåìû ê êàíîíè÷åñêîìó âèäó (â êîòîðîì âñå äèàãîíàëüíûå ýëåìåíòû îòëè÷íû îò íóëÿ). Ìåòîä Çåéäåëÿ äëÿ ðåøåíèÿ ñèñòåì íåëèíåéíûõ óðàâíåíèé çàêëþ÷àåòñÿ â ñëåäóþùåì. Ïðèâåäåì íåîáõîäèìûå è äîñòàòî÷íûå óñëîâèÿ ñõîäèìîñòè ìåòîäà Çåéäåëÿ. N; bij = aij ïðè j ≤ i, i = 1, 2, . Áîëåå óäîáíîå äëÿ ïðîâåðêè äîñòàòî÷íîå óñëîâèå ñõîäèìîñòè ìåòîäà Çåéäåëÿ èìååò ñëåäóþùèé âèä. Åãî ýôôåêòèâíîñòü ïðè ðåøåíèè íåëèíåéíûõ ãèïåðñèíãóëÿðíûõ èíòåãðàëüíûõ óðàâíåíèé ïðîäåìîíñòðèðîâàíà â ðàáîòå [6].
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