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

四元数是描述刚体运动的非常有用的工具。如果一把椅子被抛向空中，那么它的运动可以用经济上的四元数来描述。因此，涉及许多这样的运动的计算机游戏是四元数代数应用的首选领域。工业生产的机器人的建造也是如此。人们可以通过利用互联网找到更多的应用。然而，所有这些应用基本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，如由四元数定义的矩阵或多项式，主要是从理论的角度进行研究--如果有的话。直到今年(2010年)，申请人和他的合作伙伴(SIAM数值分析杂志，48(2010)，244-256)才发现了寻找某类四元数多项式的所有零点的良好工作程序。在来自巴西、意大利、葡萄牙的许多其他团体发表了许多其他出版物后，所有人都致力于Niven(1941)的一个想法，这与我们的程序无关，这是非常成功的合作的一个例子。还有许多其他情况需要研究：(1)本征值问题，(2)特别是幂方法(v·米塞斯迭代)，(3)牛顿方法的进一步研究，(4)一般多项式方程(如矩阵变量和系数)，(5)多个变量的多项式，(6)矩阵分解，(7)解线性方程。

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.