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

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以一种经济的方式来描述。因此，涉及许多这样的运动的计算机游戏是四元数代数的优选应用领域。工业生产机器人的建造也是如此。人们可以通过使用互联网找到更多的应用程序。然而，所有这些应用基本上是基于在正交变换的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研究的。申请人及其合作伙伴仅在今年（2010年）发现了一个寻找某类四元数多项式的所有零点的良好工作程序（SIAM Journal on Numerical Analysis，48（2010），244-256）在来自巴西、意大利、葡萄牙的小组的许多其他出版物之后，所有工作都在Niven（1941）的一个想法上，这与我们的程序无关，这是一个非常成功的合作的例子。许多其他情况需要调查：（1）特征值问题，（2）特别是幂方法（v.米塞斯迭代），（3）牛顿方法的进一步调查，（4）一般多项式方程（e。G.矩阵变量和系数），（5）多个变量的多个多项式，（6）矩阵分解，（7）解线性方程组。

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.