双参数指数同伦算法及其在求解非线性方程组中的应用

发布时间：2018-03-18 23:40

本文选题：同伦算法　切入点：双参数指数同伦算子　出处：《四川师范大学》2017年硕士论文　论文类型：学位论文

【摘要】：在求解非线性方程组的数值方法中,同伦算法是一种具有大范围收敛的算法.尽管在同伦算法中初值的取值范围得到了进一步扩大,但是它的收敛范围却受到同伦算子构造的影响而发生变化,同时在延拓过程中很难克服Jacobi奇异性.因此,用同伦算法求解某些复杂非线性方程组时,仍常常发散.为此,通过构造一种新的双参数指数同伦算子,给出了两种新的同伦算法——双参数数值延拓法和双参数微分法.首先,分析了非线性问题在科学计算中的地位,以及同伦算法在求解非线性问题中的作用;其次,回顾了同伦算法的发展过程,并讨论了其对初值的依赖性和不易克服Jacobi奇异性的问题;再次,介绍了同伦算子构造的基本思想,并在此基础上构造了一种新的双参数指数同伦算子;最后,基于数值延拓法和参数微分法,分别给出了双参数数值延拓法和双参数微分法,并讨论了这两种算法的收敛性.数值实验验证了双参数数值延拓法和双参数微分法的可行性和有效性.相比数值延拓法、参数微分法和Newton法,双参数数值延拓法和双参数微分法通过改变可控参数的值来调节同伦算子,从而扩大它们的收敛范围,所以这两种算法不仅解决了数值延拓法和参数微分法对初值的依赖性,而且克服了 Jacobi奇异性.此外,由于双参数数值延拓法和双参数微分法的收敛范围随着可控参数的改变而改变,所以上述两种算法为求非线性方程组的所有解提供了一种新途径.
[Abstract]:In the numerical method for solving nonlinear equations, the homotopy algorithm is a kind of algorithm with large range convergence, although the initial value range of the homotopy algorithm has been further expanded. However, its convergence range is influenced by the construction of homotopy operators, and it is difficult to overcome the Jacobi singularity in the continuation process. Therefore, the homotopy algorithm still often diverges when solving some complex nonlinear equations. By constructing a new double parameter exponential homotopy operator, two new homotopy algorithms, the double parameter numerical continuation method and the two parameter differential method, are given. Firstly, the position of nonlinear problems in scientific calculation is analyzed. And the role of homotopy algorithm in solving nonlinear problems. Secondly, the development process of homotopy algorithm is reviewed, and its dependence on initial values and the problem that it is difficult to overcome the singularity of Jacobi are discussed. In this paper, the basic idea of constructing homotopy operator is introduced, and a new double parameter exponential homotopy operator is constructed, finally, based on numerical continuation method and parameter differential method, two parameter numerical continuation method and two parameter differential method are given respectively. The convergence of the two algorithms is also discussed. The feasibility and validity of the two-parameter numerical continuation method and the two-parameter differential method are verified by numerical experiments. Compared with the numerical continuation method, the parametric differential method and the Newton method are compared. The two-parameter numerical continuation method and the two-parameter differential method adjust the homotopy operator by changing the value of controllable parameters, so they can not only solve the dependence of the numerical continuation method and the parameter differential method on the initial value, but also enlarge their convergence range. Moreover, the convergence range of the two-parameter numerical continuation method and the two-parameter differential method change with the change of controllable parameters. Therefore, the above two algorithms provide a new way for finding all the solutions of nonlinear equations.
【学位授予单位】：四川师范大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O241.7

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