Study On Solving Real Nonlinear Algebra Systems By Hybrid Method

Nonlinear polynomial equations are a fundamental and important subject in the study of nonlinear real algebra systems.In this thesis,we provide a hybrid method,which is based on the resultant computation and slope-based Hansen-Sengupta operator,to solve nonlinear polynomial equations.We prove the correctness and termination of the resulting algorithm and implement it.We not only show the performance of this algorithm by some examples coming from literatures and random generation,but also apply it to solve some problems in real life.In real algebra geometry,many problems can be translated into proving algebraic inequalities.In particular,constructive geometric theorems are always translated into proving the radical inequalities.In general,proving inequalities is more difficult than proving equalities.The main reason is that real algebra and real geometry are often involved in the relative algorithms,which increase the complexity of the algorithms.We set up a semi-mechanization method for proving an inequality conjecture in combinatorial geometry.To prove the radical inequalities,we provide a hybrid method which has both numeric and symbolic part.The main original results of this thesis are as follows:1.Based on the root bounds of univariate polynomials,we improve and implement the numerical subdivision method for solving nonlinear polynomial equations.We also uses the root bounds to show the relationship between the significant digits of isolating interval and the width of the isolating interval;2.We provide a hybrid method(named as HybridMethod) by taking the advantages of resultant computation and slope-based Hansen-Sengupta operator.We prove the correctness and claim the termination of the algorithm.We implement the algorithm in Maple and Visual C++ and illustrate its performance through some comparative experiments;3.A study of sample points in connected region of algebra decomposition is a fundamental topic.It has numerous applications such as solving or proving polynomial equations and inequalities.Moreover it can describe the solutions of real algebra functions.Combining the critical point method and the hybrid method Hybrid-Method in an algorithmic way yields an algorithm(named as ISP) for finding at least one sample point in each connected region of algebra decomposition.We implement the algorithm and compare with PCAD(partial cylindrical algebra decomposition) method through some examples.The algorithm ISP can shrink a lot of redundant sample points and improve the efficiency;4.We study an optimization problem about the Heilbronn's triangle.Due to the complexity, it is difficult to prove the problem directly.Our main idea is to first analyze the features of the optimal configuration to reduce the free variables and redundant inequalities,and then use mechanization method to solve corresponding nonlinear optimization problem.By this method,we prove the validity of conjecture when n = 8.Furthermore,we study the case n = 9.5.We present an algorithm for verifying radical inequalities.In real algebra geometry, many theorems can be translated into proving radical inequalities.The general methods are quantifier elimination(QE) method and dimension-decreasing algorithm. In this thesis,we provide a hybrid method(named as Decide),which consists of both numeric part and symbolic part,for proving radical inequalities.The numeric part,based on Hansen-Sengupta operator,first finds finitely critical points and then checks whether the inequalities hold on these sample points by interval arithmetic.If the numeric method is fail,then switch to the symbolic method.In the last section,we conclude our works in this thesis and speak of some future work.