Abstract
We consider the problem of finding interval enclosures of all zeros of a nonlinear system of polynomial equations. We present a method which combines the method of Gröbner bases (used as a preprocessing step), some techniques from interval analysis, and a special version of the algorithm of E. Hansen for solving nonlinear equations in one variable. The latter is applied to a triangular form of the system of equations, which is generated by the preprocessing step. Our method is able to check if the given system has a finite number of zeros and to compute verfied enclosures for all these zeros. Several test results demonstrate that our method is much faster than the application of Hansen’s multidimensional algorithm (or similar methods) to the original nonlinear systems of polynomial equations.
Abstract
Рассматрнвается залача нахожлення интервальных оболочек всех корней нелннейной снстемы полиномиальных уравнений. Прелстаилена процелура, обьелиняюшая метод базисов Грёбнера (нснользуемый на нрелварнтельном этаце вычцслений), некоторые методнки интервального аналнза н особую разновидность алгоритма Е. Хансена для решення нелинейных уравнений с одной иеременной. Послелний прнменяется к треугольному уредставлению, снстемы уравнений, сформированному на предварнтельном этапе. Онисываемый метод снособен проверять, нмеет, ли данная снстема конечное чнсло корней, и вычнслять вернфицированные ободочкн для всех корней. Несколько чнсленных прнмеров ноказывают, что наш метод является намного более быстрым, чем многомерный алгорнтм Нансена (илп аналогичные методы) в нрименении к исходным нелииейным снстемам полнномнальных уравнений.
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Jäger, C., Ratz, D., йегер, К. et al. A combined method for enclosing all solutions of nonlinear systems of polynomial equations. Reliable Comput 1, 41–64 (1995). https://doi.org/10.1007/BF02390521
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DOI: https://doi.org/10.1007/BF02390521