Newton’s Approach to General Algebraic Equations over Clifford Algebras

  • Drahoslava JanovskáEmail author
  • Gerhard Opfer
Part of the Trends in Mathematics book series (TM)


There is a short section describing how Newton’s method works for algebraic problems over Clifford algebras. There are two applications. Zeros of unilateral polynomials over a Clifford algebra in \(\mathbb {R}^8\) and solutions of a Riccati equation over all eight Clifford algebras in \(\mathbb {R}^4\).


Clifford algebras Newton’s method Algebraic equations over Clifford algebras Riccati equation 

Mathematics Subject Classification (2010)

Primary 15A66; Secondary 12E10 1604 


  1. 1.
    D.J.H. Garling, Clifford Algebras: An Introduction (Cambridge Univerity Press, Cambridge, 2011), 200 ppCrossRefGoogle Scholar
  2. 2.
    K. Gürlebeck, W. Sprössig, Quaternionic and Clifford Calculus for Physicists and Engineers (Wiley, Chichester, 1997), 371 ppzbMATHGoogle Scholar
  3. 3.
    D. Janovská, G. Opfer, The algebraic Riccati equation for quaternions. Dedicated to Ivo Marek on the occasion of his 80th birthday. Adv. Appl. Clifford Algebras 23, 907–918 (2013)Google Scholar
  4. 4.
    D. Janovská, G. Opfer, Zeros and singular points for one-sided, coquaternionic polynomials with an extension to other \(\mathbb {R}^4\) algebras. Electron. Trans. Numer. Anal. 41, 133–158 (2014)Google Scholar
  5. 5.
    R. Lauterbach, G. Opfer, The Jacobi matrix for functions in noncommutative algebras. Adv. Appl. Clifford Algebras 24, 1059–1073 (2014). Erratum: Adv. Appl. Clifford Algebras, 24 (2014), p. 1075Google Scholar
  6. 6.
    A.A. Pogurui, R.M. Rodriguez-Dagnino, Some algebraic and analytic properties of coquaternion algebra. Adv. Appl. Clifford Algebras 20, 79–84 (2010)MathSciNetCrossRefGoogle Scholar
  7. 7.
    B. Schmeikal, Tessarinen, Nektarinen und andere Vierheiten. Beweis einer Beobachtung von Gerhard Opfer. Mitt. Math. Ges. Hamburg 34, 81–108 (2014)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.University of Chemistry and TechnologyPrague 6Czech Republic
  2. 2.University of HamburgFaculty on Mathematics, Informatics, and Natural Sciences [MIN]HamburgGermany

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