于妍,董波,于波.求解混合三角多项式方程组的直接GBQ方法[J].数学研究及应用,2017,37(2):127~136
求解混合三角多项式方程组的直接GBQ方法
Direct GBQ Algorithm for Solving Mixed Trigonometric Polynomial Systems
投稿时间:2016-01-05  最后修改时间:2016-05-06
DOI:10.3770/j.issn:2095-2651.2017.02.001
中文关键词:  混合三角多项式方程组  多项式方程组  同伦方法  GBQ方法  解个数上界
英文关键词:mixed trigonometric polynomial system  polynomial system  homotopy method  GBQ algorithm  upper bound
基金项目:国家自然科学基金(Grant Nos.11101067;11171051); 中央高校基本科研业务费专项资金(Grant No.DUT16LK04).
作者单位
于妍 大连理工大学数学科学学院, 辽宁 大连 116024; 沈阳农业大学理学院, 辽宁 沈阳 110866 
董波 大连理工大学数学科学学院, 辽宁 大连 116024 
于波 大连理工大学数学科学学院, 辽宁 大连 116024 
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中文摘要:
      许多科学与工程领域,我们经常需要求混合三角多项式方程组的全部解.一般来说,混合三角多项式方程组可以通过变量替换及增加二次多项式转化为多项式方程组,进而利用数值方法进行求解,但这种转化会增大问题的规模从而增加计算量.在本文中,我们不将问题转化,考虑利用直接同伦方法求解,并给出基于GBQ方法构造的初始方程组及同伦定理的证明.数值实验结果表明我们构造的直接同伦方法较已有的直接同伦方法更加有效.
英文摘要:
      In many fields of science and engineering, it is needed to find all solutions of mixed trigonometric polynomial systems. Commonly, mixed trigonometric polynomial systems are transformed into polynomial systems by variable substitution and adding some quadratic equations, and then solved by some numerical methods. However, transformation of a mixed trigonometric polynomial system into a polynomial system will increase the dimension of the system and hence induces extra computational work. In this paper, we consider to solve the mixed trigonometric polynomial systems by homotopy method directly. Homotopy with the start system constructed by GBQ-algorithm is presented and homotopy theorems are proved. Preliminary numerical results show that our constructed direct homotopy method is more efficient than the existent direct homotopy methods.
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