Numerische Analysis und Numerische Lineare Algebra auf dem Gebiet der Quaternionen, Tschechische Republik

四元数领域的数值分析和数值线性代数，捷克共和国

• 批准号: 188687996
• 负责人: Professor Dr. Gerhard Opfer
• 金额: --
• 依托单位: Bereich Optimierung und Approximation
• 依托单位国家: 德国
• 项目类别: Research Grants
• 财政年份: 2010
• 资助国家: 德国
• 起止时间: 2009-12-31 至 2015-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/188687996?language=en
• 关键词: Numerische; Analysis; und; Lineare; Algebra

Quaternions are a very useful tool for describing motions of rigid bodies. If a chair is thrown into the air, then its motion can be described by the use of quaternions in an economic fashion. Thus, Computer games which involve many such motions, are a preferred field of applications of quaternionic algebra. The same is true for the construction of industrially produced robots. One can find more applications by employing the internet. However, all these applications are essentially based on the capability of the multiplication with a single quaternion in the sense of an orthogonal transformation. More complex (in the sense of complicated) structures like matrices or polynomials defined by quaternions are mainly studied -if at all- from a theoretical point of view. A well working procedure for finding all zeros of a certain class of quaternionic polynomials was found only this year (2010) by the applicant and his cooperation partner (SIAM Journal on Numerical Analysis, 48 (2010), 244-256) after many other publications by groups from Brazil, Italy, Portugal, working all on an idea by Niven (1941), which is not related to our procedure, This is one example for a very successful cooperation. Many other cases need investigations: (1) Eigenvalue Problems, (2) particular the power method (v. Mises Iteration), (3) further investigations of Newton's method, (4) general polynomial equations (e. g. matrix variables and coefficients), (5) several polynomials in several variables, (6) matrix decompositions, (7) solving linear equations.

四元数是描述刚体运动的一个非常有用的工具。如果一把椅子被抛向空中，那么它的运动可以用四元数的经济方式来描述。因此，包含许多此类运动的电脑游戏是四元代数应用的首选领域。工业生产机器人的建造也是如此。人们可以通过互联网找到更多的应用。然而，所有这些应用本质上都是基于在正交变换意义上与单个四元数相乘的能力。更复杂的（在复杂的意义上）结构，如矩阵或由四元数定义的多项式，主要是从理论的角度来研究的-如果有的话。申请人和他的合作伙伴（SIAM Journal on Numerical Analysis, 48(2010), 244-256）在巴西、意大利、葡萄牙的团队发表了许多其他出版物之后，才在今年（2010）发现了一个很好的寻找某一类四元数多项式的所有零的工作程序，这与我们的程序无关，这是一个非常成功的合作的例子。许多其他情况需要研究：(1)特征值问题，(2)特别是幂方法（v. Mises迭代），(3)牛顿方法的进一步研究，(4)一般多项式方程（例如矩阵变量和系数），(5)几个变量中的几个多项式，(6)矩阵分解，(7)求解线性方程。