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Geometric Algebra Levenberg-Marquardt

Part of the Lecture Notes in Computer Science book series (LNIP,volume 11542)


This paper introduces a novel and matrix-free implementation of the widely used Levenberg-Marquardt algorithm, in the language of Geometric Algebra. The resulting algorithm is shown to be compact, geometrically intuitive, numerically stable and well suited for efficient GPU implementation. An implementation of the algorithm and the examples in this paper are publicly available.


  • Geometric Algebra
  • Levenberg-Marquardt
  • Automatic differentiation
  • Non-linear estimation

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  • DOI: 10.1007/978-3-030-22514-8_51
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  1. Tingelstad, L., Egeland, O.: Motor parameterization. Adv. Appl. Clifford Algebras 28, 34 (2018)

    Google Scholar 

  2. Lasenby, J., Fitzgerald, W.J., Lasenby, A.N., Doran, C.J.L.: New geometric methods for computer vision: an application to structure and motion estimation. Int. J. Comput. Vis. 26(3), 191–213 (1998)

    CrossRef  Google Scholar 

  3. Guennebaud, G., Jacob, B. et al.: Eigen v3 (2010).

  4. Moré, J.J., Sorensen, D.C., Hillstrom, K.E., Garbow, B.S.: The MINPACK project, in sources and development of mathematical software. In: Cowell, W.J. (ed.) pp. 88–111. Prentice-Hall (1984).

  5. Press, W.H., Teukolsky, S.A., Vetterling, W.T., Flannery, B.P.: Numerical Recipes in C : The Art of Scientific Computing. Cambridge University Press, Cambridge (1992)

    Google Scholar 

  6. Tingelstad, L., Egeland, O.: Automatic multivector differentiation and optimization. Adv. Appl. Clifford Algebras 27, 707 (2017).

    MathSciNet  CrossRef  Google Scholar 

  7. Fletcher, R.: A modified marquardt subroutine for nonlinear least squares. Atomic Energy Research Establishment report R6799, Harwell, England (1971)

    Google Scholar 

  8. Gunn, C.: Geometry, Kinematics, and Rigid Body Mechanics in Cayley-Klein Geometries. Ph.D. thesis, Technical University, Berlin (2011).

  9. Gunn, C.: On the homogeneous model of Euclidean geometry. In: Dorst, L., Lasenby, J. (eds.) A Guide to Geometric Algebra in Practice, chapter 15, pp. 297–327. Springer, London (2011).,

    CrossRef  Google Scholar 

  10. Gunn, C.: Geometric algebras for Euclidean geometry. Adv. Appl. Clifford Algebras 27(1), 185–208 (2017).

    MathSciNet  CrossRef  Google Scholar 

  11. Rall, L.B. (ed.): Automatic Differentiation: Techniques and Applications. LNCS, vol. 120. Springer, Heidelberg (1981).

    CrossRef  MATH  Google Scholar 

  12. De Keninck, S.: Ganja.js: Geometric Algebra - Not Just Algebra (2017).

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The authors would like to thank Charles Gunn for his valuable feedback on, and Hugo Hadfield and Vincent Nozick for their proofreading of an early version of this article.

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Correspondence to Steven De Keninck .

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De Keninck, S., Dorst, L. (2019). Geometric Algebra Levenberg-Marquardt. In: Gavrilova, M., Chang, J., Thalmann, N., Hitzer, E., Ishikawa, H. (eds) Advances in Computer Graphics. CGI 2019. Lecture Notes in Computer Science(), vol 11542. Springer, Cham.

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  • Print ISBN: 978-3-030-22513-1

  • Online ISBN: 978-3-030-22514-8

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