發(fā)布時間：2018-03-18 23:40

  本文選題：同倫算法　切入點(diǎn)：雙參數(shù)指數(shù)同倫算子　出處：《四川師范大學(xué)》2017年碩士論文　論文類型：學(xué)位論文

【摘要】：在求解非線性方程組的數(shù)值方法中,同倫算法是一種具有大范圍收斂的算法.盡管在同倫算法中初值的取值范圍得到了進(jìn)一步擴(kuò)大,但是它的收斂范圍卻受到同倫算子構(gòu)造的影響而發(fā)生變化,同時在延拓過程中很難克服Jacobi奇異性.因此,用同倫算法求解某些復(fù)雜非線性方程組時,仍常常發(fā)散.為此,通過構(gòu)造一種新的雙參數(shù)指數(shù)同倫算子,給出了兩種新的同倫算法——雙參數(shù)數(shù)值延拓法和雙參數(shù)微分法.首先,分析了非線性問題在科學(xué)計算中的地位,以及同倫算法在求解非線性問題中的作用;其次,回顧了同倫算法的發(fā)展過程,并討論了其對初值的依賴性和不易克服Jacobi奇異性的問題;再次,介紹了同倫算子構(gòu)造的基本思想,并在此基礎(chǔ)上構(gòu)造了一種新的雙參數(shù)指數(shù)同倫算子;最后,基于數(shù)值延拓法和參數(shù)微分法,分別給出了雙參數(shù)數(shù)值延拓法和雙參數(shù)微分法,并討論了這兩種算法的收斂性.數(shù)值實(shí)驗驗證了雙參數(shù)數(shù)值延拓法和雙參數(shù)微分法的可行性和有效性.相比數(shù)值延拓法、參數(shù)微分法和Newton法,雙參數(shù)數(shù)值延拓法和雙參數(shù)微分法通過改變可控參數(shù)的值來調(diào)節(jié)同倫算子,從而擴(kuò)大它們的收斂范圍,所以這兩種算法不僅解決了數(shù)值延拓法和參數(shù)微分法對初值的依賴性,而且克服了 Jacobi奇異性.此外,由于雙參數(shù)數(shù)值延拓法和雙參數(shù)微分法的收斂范圍隨著可控參數(shù)的改變而改變,所以上述兩種算法為求非線性方程組的所有解提供了一種新途徑.
[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.
【學(xué)位授予單位】：四川師范大學(xué)
【學(xué)位級別】：碩士
【學(xué)位授予年份】：2017
【分類號】：O241.7

【參考文獻(xiàn)】

相關(guān)期刊論文 前5條

1 夏林林;吳開騰;;大范圍求解非線性方程組的指數(shù)同倫法[J];計算數(shù)學(xué);2014年02期

2 陳傳淼;胡宏伶;雷蕾;曾星星;;非線性方程組的Newton流線法[J];計算數(shù)學(xué);2012年03期

3 馬軍星,張華,李險峰,何正嘉;非線性方程組解法對非線性有限元分析精度的影響[J];西北建筑工程學(xué)院學(xué)報(自然科學(xué)版);2001年02期

4 王則柯,高堂安;PL同倫方法進(jìn)展[J];高校應(yīng)用數(shù)學(xué)學(xué)報A輯(中文版);1990年01期

5 李受百;函數(shù)因子法——非線性方程組求解中處理奇異問題的一種新方法[J];計算數(shù)學(xué);1983年02期

相關(guān)碩士學(xué)位論文 前1條

1 夏林林;求解非線性方程組的一種新方法—指數(shù)同倫法[D];四川師范大學(xué);2014年

，

本文編號：1631882