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Proceedings of the Computational Methods in Systems and Software

CoMeSySo 2017: Cybernetics Approaches in Intelligent Systems pp 62–75Cite as

Geometric Algebra, Extended Cross-Product and Laplace Transform for Multidimensional Dynamical Systems

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Part of the book series: Advances in Intelligent Systems and Computing ((AISC,volume 661))

Abstract

This contribution describes a new approach for solving linear system of algebraic equations and differential equations using Laplace transform by the extended-cross product. It will be shown that a solution of a linear system of equations Ax = 0 or Ax = b is equivalent to the extended cross-product if the projective extension of the Euclidean system and the principle of duality are used. Using the Laplace transform differential equations are transformed to a system of linear algebraic equations, which can be solved using the extended cross-product (outer product). The presented approach enables to avoid division operation and extents numerical precision as well. It also offers applications of matrix-vector and vector-vector operations in symbolic manipulation, which can leads to new algorithms and/or new formula. The proposed approach can be applied also for stability evaluation of dynamical systems. In the case of numerical computation, it supports vector operation and SSE instructions or GPU can be used efficiently.

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Acknowledgment

The author would like to thank to colleagues at the University of West Bohemia in Plzen for fruitful discussions and to anonymous reviewers for their comments and hints, which helped to improve the manuscript significantly.

The Czech Science Foundation GACR, No. GA17-05534S, supported this project.

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Authors and Affiliations

  1. Department of Computer Science and Engineering, Faculty of Applied Sciences, University of West Bohemia, Plzen, Czech Republic

    Vaclav Skala

Authors
  1. Vaclav Skala
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Correspondence to Vaclav Skala .

Editor information

Editors and Affiliations

  1. Faculty of Applied Informatics, Tomas Bata University in Zlín, Zlín, Czech Republic

    Radek Silhavy

  2. Faculty of Applied Informatics, Tomas Bata University in Zlín, Zlín, Czech Republic

    Petr Silhavy

  3. Faculty of Applied Informatics, Tomas Bata University in Zlín, Zlín, Czech Republic

    Zdenka Prokopova

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Skala, V. (2018). Geometric Algebra, Extended Cross-Product and Laplace Transform for Multidimensional Dynamical Systems. In: Silhavy, R., Silhavy, P., Prokopova, Z. (eds) Cybernetics Approaches in Intelligent Systems. CoMeSySo 2017. Advances in Intelligent Systems and Computing, vol 661. Springer, Cham. https://doi.org/10.1007/978-3-319-67618-0_7

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  • DOI: https://doi.org/10.1007/978-3-319-67618-0_7

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-67617-3

  • Online ISBN: 978-3-319-67618-0

  • eBook Packages: EngineeringEngineering (R0)

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