Robust numerical methods for nonlinear equations

2006年期间，首席研究员在国际期刊上发表了3篇论文。调查人员不仅在数学上工作，而且还研究了数值方法在解决许多真实的世界问题中的应用。研究人员在赠款的资助下参加了几次国际会议，并在会议上介绍了他们的研究成果。此外，他们还与世界一流的研究人员交流了新的研究成果。主要研究成果如下：1.与新南威尔士大学的Robert Womersley教授一起，将球面t-设计转化为欠定非线性方程组，并证明了球面t-设计的存在性。我们还应用了高斯-牛顿法来寻找近似的球面t-设计，并定义了球面上新的数值积分规则。2.首席研究员与广崎大学的T.Kimura（广崎大学的博士生）一起提出了一种新的桥梁管理有限元曲面拟合方法。采用KITACON公司的方法和数据分析了青森地区桥梁的破坏因素。3.研究组长与南方中心大学的向胜教授一起给出了线性互补问题的新的误差界。新的误差界大大改进了已有的误差界。此外，对于一些特殊的线性互补问题，包括H-矩阵线性互补问题，它们也可以很容易地计算出来。我们收到了各领域专家的非常积极的评价。

During 2006,the head investigator published 3 papers in international journals. The investigators have not only worked on mathematics but also studied applications of numerical methods for solving many real world problems. By the support of the grants, investigators have attended several international conferences and presented their research results in the conferences. Moreover, they exchanged their new research results with world class researchers. Some established results are summarized as the follows.1.The head investigator reformulated the spherical t-design to underdetermined nonlinear equations and proved the existence of spherical t-designs with Prof. Robert Womersley of the University of New South Wales. We also applied the Guass-Newton method to find approximate spherical t-design and defined new numerical integration rules on the sphere.2.The head investigator proposed a new finite element surface fitting method for bridge management with T.Kimura of Hirosaki University ( current Ph. D student in Hirosaki University). We used the method and date from KITACON company to analyze the factors which damage the bridges in Aomori region.3.The head investigator gave new error bounds for linear complementarity problems with Prof. S.Xiang of South Center University. The new error bounds improved existing error bounds greatly. Moreover, they can be computed easily for some special linear complementarity problems including H-matrix linear complementarity problems. We received very positive comments from experts in the areas.