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

• Applicant: Professor Dr. Gerhard Opfer
• Subject Area: Mathematics
• Term: from 2010 to 2016
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 188687996

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.

• DFG Programme: Research Grants
• International Connection: Czech Republic
• Participating Person: Professorin Dr. Drahoslava Janovská