求解混合三角多项式方程组的同伦方法
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摘要
非线性方程组的数值计算是科学与工程计算中的重要问题,而关于求方程组全部解的研究是其难点。同伦方法是求多项式方程组全部解的一种有效的数值方法。本文主要研究利用同伦方法求解混合三角多项式方程组及由混合三角多项式方程组转化来的多项式方程组。考虑以下问题:
1、不进行变元替换,直接求解混合三角多项式方程组。
2、利用混合三角多项式方程组转化过来的多项式方程组的特殊结构,构造更加有效的同伦进行求解。
第一章首先对同伦方法特别是求解多项式方程组的同伦方法做了简要的综述。然后给出混合三角多项式方程组的一般模型,例举了一些它在工程和科学领域中的应用,并阐述了混合三角多项式方程组与多项式方程组之间的相互转化关系。
第二章给出了一些求解混合三角多项式方程组的直接同伦方法,即不将其化为多项式组而直接构造同伦方法。这样可避免增加问题的维数,使路径跟踪过程效率更高。我们首先给出了求解一般混合三角多项式方程组的标准同伦方法,进一步的,针对实际应用中经常出现的亏欠混合三角多项式方程组,我们给出两种行之有效的随机线性乘积同伦:多重齐次同伦以及基于广义Bezout数构造的同伦,并且给出了一种新的变元分组方法。我们从理论上证明了所提出的方法的可用性,并将算法利用Matlab语言编程实现。通过数值试验验证了它们的实际有效性。
第三章给出两种求解由混合三角多项式方程组转化而来的多项式方程组的高效率同伦方法。利用这类问题的特殊结构,我们提出了混合同伦方法,不仅同伦的形式是混合的,而且求解方法也是符号计算方法和数值方法的结合。进一步利用这类方程组的部分对称性,我们给出了一种更加有效的方法:对称混合同伦方法。我们建立了所提出方法的理论基础并将其利用C++语言实现,通过数值试验验证了它们的有效性。
第四章是进一步的数值试验及实际应用。首先利用直接同伦方法和混合同伦方法两种方法分别求解不同类型的混合三角多项式方程组,给出了数值实验结果,说明两种方法各自适合求解的混合三角多项式方程组的类型;其后,我们着重讨论一个具有挑战性的实际工程问题一声纳和雷达信号处理问题。该问题用已有的方法很难求解,而当维数较大时,甚至不能求解。利用本文提出的混合同伦方法并结合系数参数同伦方法,我们很好地解决了这个实际问题,实现了快速求解。
Solving nonlinear systems is a major task of computational mathematics. Finding all solutions to a nonlinear system is a challenging problem and has practical applications in many fields of science and engineering. Homotopy method is an efficient numerical method for finding all isolated solutions to some special kinds of nonlinear systems, e.g., polynomial systems. In this dissertation, we consider to solve mixed trigonometric polynomial systems and polynomial systems transformed from them more efficiently. We present two kinds of methods for such problems:
1. Direct homotoy methods to solve mixed trigonometric polynomial systems;
2. For the polynomial systems transformed from the mixed trigonometric polynomial systems, we utilize its special structure to construct more efficient homotopies.
In Chapter 1, we give an introduction of the homotopy method and its applications in the field of science and engineering, especially homotopy methods for solving polynomial systems. Also we formulate the general form of mixed trigonometric polynomial systemsas as well as transformations between a mixed trigonometric polynomial system and a polynomial system. Some practical examples are also listed.
In Chapter 2, we present some direct homotopy methods for mixed trigonometric polynomial systems, that is, we construct homotopy directly for mixed trigonometric polynomial systems and do not transform them into polynomial systems. By doing like this, no additional variables is introduced and hence can solve the problem more efficiently. For general mixed trigonometric polynomial systems, we present standard homotopies. Furthermore, because mixed trigonometric polynomial systems arising in practice are mainly deficient, we present two efficient random linear product homotopies: multi-homogeneous homotopy and product homotopy based on the generalized Bézout number to solve this class of systems, and in the latter method, a new and more efficient variable partition method is presented. We give some theoretical results, implement the methods by applying Matlab programming language, and make a comparison between our direct homotopy methods and the existing methods to show their effectiveness.
In Chapter 3, efficient methods for solving polynomial systems transformed from mixed trigonometric polynomial systems are given. This class of polynomial systems have a special structure. Applying this special structure, an efficient hybrid method is presented. It combines the homotopy method, in which the homotopy is a combination of coefficient parameter homotopy and the random product homotopy, with symbolic computation methods, such as decomposition, variable substitution and reduction techniques. Furthermore, based on the symmetric structure of the lower part of the target system, a symmetric homotopy and hybrid method are presented. This method can keep the symmetric structure of the target system, which can save computational work greatly. We prove some theoretical results, implement our method by applying C++ programming language and make a comparison between our methods and the existing methods to show their effectiveness.
In Chapter 4, by further numerical experiments, we discuss the advantages and disadvantages of direct homotopy methods and hybrid methods in solving different classes of mixed trigonometric polynomial systems. Then, we turn to a challenging practical problem, which arises in signal processing of sonar and radar and is hard to be solved by existing solving methods. We give a fast solving method which is the combination of the symmetric hybrid method and the coefficient-parameter homotopy method.
1、不进行变元替换,直接求解混合三角多项式方程组。
2、利用混合三角多项式方程组转化过来的多项式方程组的特殊结构,构造更加有效的同伦进行求解。
第一章首先对同伦方法特别是求解多项式方程组的同伦方法做了简要的综述。然后给出混合三角多项式方程组的一般模型,例举了一些它在工程和科学领域中的应用,并阐述了混合三角多项式方程组与多项式方程组之间的相互转化关系。
第二章给出了一些求解混合三角多项式方程组的直接同伦方法,即不将其化为多项式组而直接构造同伦方法。这样可避免增加问题的维数,使路径跟踪过程效率更高。我们首先给出了求解一般混合三角多项式方程组的标准同伦方法,进一步的,针对实际应用中经常出现的亏欠混合三角多项式方程组,我们给出两种行之有效的随机线性乘积同伦:多重齐次同伦以及基于广义Bezout数构造的同伦,并且给出了一种新的变元分组方法。我们从理论上证明了所提出的方法的可用性,并将算法利用Matlab语言编程实现。通过数值试验验证了它们的实际有效性。
第三章给出两种求解由混合三角多项式方程组转化而来的多项式方程组的高效率同伦方法。利用这类问题的特殊结构,我们提出了混合同伦方法,不仅同伦的形式是混合的,而且求解方法也是符号计算方法和数值方法的结合。进一步利用这类方程组的部分对称性,我们给出了一种更加有效的方法:对称混合同伦方法。我们建立了所提出方法的理论基础并将其利用C++语言实现,通过数值试验验证了它们的有效性。
第四章是进一步的数值试验及实际应用。首先利用直接同伦方法和混合同伦方法两种方法分别求解不同类型的混合三角多项式方程组,给出了数值实验结果,说明两种方法各自适合求解的混合三角多项式方程组的类型;其后,我们着重讨论一个具有挑战性的实际工程问题一声纳和雷达信号处理问题。该问题用已有的方法很难求解,而当维数较大时,甚至不能求解。利用本文提出的混合同伦方法并结合系数参数同伦方法,我们很好地解决了这个实际问题,实现了快速求解。
Solving nonlinear systems is a major task of computational mathematics. Finding all solutions to a nonlinear system is a challenging problem and has practical applications in many fields of science and engineering. Homotopy method is an efficient numerical method for finding all isolated solutions to some special kinds of nonlinear systems, e.g., polynomial systems. In this dissertation, we consider to solve mixed trigonometric polynomial systems and polynomial systems transformed from them more efficiently. We present two kinds of methods for such problems:
1. Direct homotoy methods to solve mixed trigonometric polynomial systems;
2. For the polynomial systems transformed from the mixed trigonometric polynomial systems, we utilize its special structure to construct more efficient homotopies.
In Chapter 1, we give an introduction of the homotopy method and its applications in the field of science and engineering, especially homotopy methods for solving polynomial systems. Also we formulate the general form of mixed trigonometric polynomial systemsas as well as transformations between a mixed trigonometric polynomial system and a polynomial system. Some practical examples are also listed.
In Chapter 2, we present some direct homotopy methods for mixed trigonometric polynomial systems, that is, we construct homotopy directly for mixed trigonometric polynomial systems and do not transform them into polynomial systems. By doing like this, no additional variables is introduced and hence can solve the problem more efficiently. For general mixed trigonometric polynomial systems, we present standard homotopies. Furthermore, because mixed trigonometric polynomial systems arising in practice are mainly deficient, we present two efficient random linear product homotopies: multi-homogeneous homotopy and product homotopy based on the generalized Bézout number to solve this class of systems, and in the latter method, a new and more efficient variable partition method is presented. We give some theoretical results, implement the methods by applying Matlab programming language, and make a comparison between our direct homotopy methods and the existing methods to show their effectiveness.
In Chapter 3, efficient methods for solving polynomial systems transformed from mixed trigonometric polynomial systems are given. This class of polynomial systems have a special structure. Applying this special structure, an efficient hybrid method is presented. It combines the homotopy method, in which the homotopy is a combination of coefficient parameter homotopy and the random product homotopy, with symbolic computation methods, such as decomposition, variable substitution and reduction techniques. Furthermore, based on the symmetric structure of the lower part of the target system, a symmetric homotopy and hybrid method are presented. This method can keep the symmetric structure of the target system, which can save computational work greatly. We prove some theoretical results, implement our method by applying C++ programming language and make a comparison between our methods and the existing methods to show their effectiveness.
In Chapter 4, by further numerical experiments, we discuss the advantages and disadvantages of direct homotopy methods and hybrid methods in solving different classes of mixed trigonometric polynomial systems. Then, we turn to a challenging practical problem, which arises in signal processing of sonar and radar and is hard to be solved by existing solving methods. We give a fast solving method which is the combination of the symmetric hybrid method and the coefficient-parameter homotopy method.
引文
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[3]Kellogg R B,Li T Y,Yorke J A.A constructive proof of the Brouwer fixed point theorem and computational results.SIAM J.Numer.Anal.,1976,4:473-483.
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[6]Allgower E,Georg K.Simplicial and continuation methods for approximating fixed points and solutions to systems of eqnations.SIAM Review,1980,22:28-85.
[7]Allgower E,Georg K.Numerical continuation methods.Springer Verlag,New York,1990.
[8]Garcia C B,Zangwill W I.Pathways to solutions,Fixed Points and Equilibria.Prentice-Hall,New Jersey,1981.
[9]Watson L T,Billups S C,Morgan A P.HOMPACK:A Suite of codes for globally convergent homotopy algorithms.ACM Trans.Math.Software,1987,13,281-310.
[10]Scart H E.The approximation of fixed points of a continuous mapping.SIAM J.Appl.Math.,1967,15:1328-1341.
[11]Morgan A.Solving polynomial systems using continuation for engineering and scientific problems.Prentice-Hall,Englewood Cliffs,N.J.,1987.
[63]Wu W J.Basic principles of mechanical theorem proving in elementary geometries.J.Sys.Sci.and Math.Scis.,1984,4(3):207-235.
[13]Wu W J.On the decision problem and the mechanization of theorem-proving in elementary geometry,in Automated Theorem Proving:After 25 years,edited by W.Bledsoe and D.Loveland,Contemporary Mathematics 29,American Mathematical Society,Providence,Rhode Island,213-234.
[14]Buchberger B.Groebner bases:an algorithmic method in polynomial ideal theory.Multidimeusion Systems Theory,Reidel,Dordrecht,1985,184-232.
[15]Lazard D.Resolution des systems dequations algebriques.Theor.Comp.Sci.,1981,16:77-110.
[16]Cox D,Little J,O'shea D.Using Algebraic Geometry.Berlin:Springer,1998.
[17]Auzinger W,Stetter H J.An elimination algorithm for the computation of all zeros of a system of multivariate polynomial equations.Numerical mathematics,Singapore 1988,11-30,Internat.Schriftenreihe Numer.Math.,86,Birkh?user,Basel,1988.
[18]Starter H J.Multivariate polynomial equations as matrix eigenproblems.Contributions in numerical mathematics,355-371,World Sci.Ser.Appl.Anal.,2,World Sci.Publ.,River Edge,NJ,1993.
[19]Feng G C,Wu W D,Zhang S G.The eigenvalue problem equivalent to multivariate polynomial system.Numer.Math.J.Chinese Univ.(English Ser.) 2(1993),no.2,234-241.
[20]Feng G C,Wu W D,Huang K.Second equivalence theorem of multivariate polynomial system and eigenproblem.Adv.in Math.(China) 22(1993),no.5,476-477.
[21]Feng G C,Zhang S G,Liu Y.The structure of solutions to algebraic system and matrices in eiganvalue method.Northeast.Math.J.11(1995),no.4,383-386.
[22]Garcia C B,Zangwill W I.Finding all solutions to polynomial systems and other systems of equations.Math.Programming,1979,16(2):159-176.
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