Abstract
We present an efficient algorithm to multiply two arbitrary biquaternions. The schoolbook multiplication of two biquaternions requires 64 real multiplications and 56 real additions. More effective solutions still do not exist. We show how to compute a product of the biquaternions with 24 real multiplications and 56 real additions. During synthesis of the discussed algorithm we use the fact that product of two biquaternions may be represented as a matrix–vector product. The matrix multiplicand that participates in the product calculating has unique structural properties that allow performing its advantageous factorization. Namely this factorization leads to significant reducing of the computational complexity of biquaternion multiplication.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
Using the notation \((\cdot {)}'\) here and subsequently we shall denote a modified version of the matrix inscribed inside the parentheses.
References
Kantor, I., Solodovnikov, A.: Hypercomplex numbers: an elementary introduction to algebras. Springer, New York (2011). Softcover reprint of the original 1st ed. 1989 edition
Alfsmann, D., Göckler, H.G., Sangwine, S.J., Ell, T.A.: Hypercomplex algebras in digital signal processing: benefits and drawbacks (tutorial). In: Proceedings of the EURASIP 15th European Signal Processing Conference, pp. 1322–1326. Poznań (2007)
Moxey, C.E., Sangwine, S.J., Ell, T.A.: Hypercomplex correlation techniques for vector images. IEEE Trans. Signal Process. 51, 1941–1953 (2003)
Calderbank, R., Das, S., Al Dhahir, N., Diggavi, S.: Construction and analysis of a new quaternionic space-time code for 4 transmit antennas. Commun. Inf. Syst. 5, 97–122 (2005)
Malekian, E., Zakerolhosseini, A., Mashatan, A.: QTRU: quaternionic version of the NTRU public-key cryptosystems. Int. J. Inf. Secur. 3, 29–42 (2011)
Sangwine, S.J., Ell, T.A., Le Bihan, N.: Fundamental representations and algebraic properties of biquaternions or complexified quaternions. Appl. Clifford Algebras 21(3), 607–636 (2010)
Makarov, O.M.: An algorithm for the multiplication of two quaternions. Zh. Vychisl. Mat. Mat. Fiz. 17(6), 1574–1575 (1977)
Cariow, A., Cariowa, G.: Algorithm for multiplying two octonions. Radioelectronics and Communications Systems, pp. 464–473. Allerton Press, Inc., New York (2012)
Cariow, A., Cariowa, G.: An algorithm for fast multiplication of sedenions. Inf. Process. Lett. 113, 324–331 (2013)
Tariov, A.: Algorytmiczne aspekty racjonalizacji obliczeń w cyfrowym przetwarzaniu sygnałów, Wydawnictwo Uczelniane ZUT (2011)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2015 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Cariow, A., Cariowa, G. (2015). On the Multiplication of Biquaternions. In: Wiliński, A., Fray, I., Pejaś, J. (eds) Soft Computing in Computer and Information Science. Advances in Intelligent Systems and Computing, vol 342. Springer, Cham. https://doi.org/10.1007/978-3-319-15147-2_35
Download citation
DOI: https://doi.org/10.1007/978-3-319-15147-2_35
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-15146-5
Online ISBN: 978-3-319-15147-2
eBook Packages: EngineeringEngineering (R0)