A Block Conjugate Gradient Method for Quaternion Linear

Year 2023, Volume: 28 Issue: 2, 394 - 403, 31.08.2023

Sinem Şimşek , Ayça Körükçü

https://doi.org/10.53433/yyufbed.1168844

Cited By: 1

Abstract

This study aims at the simultaneous solution of several quaternion linear systems with the same Hermitian and positive definite coefficient matrix by employing the conjugate gradient method. We consider the setting when the quaternion Hermitian positive definite coefficient matrix at hand is very large so that direct methods are not applicable. In the study, we first transform linear quaternion systems into real linear systems. The transformed real linear systems have special structure due to the fact that they are real representations of quaternion systems. Benefitting from the special structure, we further reduce the size of these linear systems. Then a block conjugate gradient method is applied to the resulting reduced real linear systems. The solution obtained after applying the conjugate gradient method is a real representation of the solution of the original quaternion problem. Thus, a conversion of this real solution to the quaternion setting is performed in the end.

Keywords

Block conjugate gradient method, Krylov subspace, Linear quaternion systems

Kuaterniyon Lineer Sistemler için Blok Eşlenik Gradyan Metot

Abstract

Bu çalışmada katsayılar matrisi aynı ve Hermitian, pozitif tanımlı olan bir takım lineer kuaterniyon sistemlerinin eşlenik gradyan metodu kullanılarak eş zamanlı çözümü amaçlanmıştır. Kuaterniyon Hermitian pozitif tanımlı katsayılar matrisinin boyutunun çok büyük olması durumunda, lineer sistemlerin çözümü için direk metotlar kullanıma uygun değildir. Çalışmada, öncelikle lineer kuaterniyon sistemlerini reel lineer sistemlere dönüştürdük. Dönüştürülen reel lineer sistemler, kuaterniyon sistemlerin reel temsilleri olmalarından ötürü özel yapıya sahiplerdir. Bu özel yapıyı kullanarak reel lineer sistemlerin boyutlarını indirgedik. Daha sonra elde edilen indirgenmiş reel sistemlere yinelemeli bir yöntem olan blok eşlenik gradyan metodunu uyguladık. Blok eşlenik gradyan metodu uygulandıktan sonra elde edilen çözümler, orijinal kuaterniyon lineer sistemlerinin çözümlerinin bir reel temsilidir. Son olarak bu reel çözümleri original sistemin kuaterniyon çözümlerine dönüştürdük.

Keywords

Blok eşlenik gradyan metot, Krylov altuzay, Lineer kuaterniyon sistemler

References

• Adler, S. L. (1995). Quaternionic Quantum Mechanics and Quantum Fields. New York, USA: Oxford University Press.
• Arena, P., Fortuna, L., Muscato, G., & Xibilia, M. G. (1998). Neural Networks in Multidimensional Domains. London, UK: Springer.
• Caccavale, F., Natale, C., Siciliano, B., & Villani, L. (1999). Six-DOF impedance control based on angle/axis representations. IEEE Transactions on Robotics and Automation, 15(2), 289-300. doi:10.1109/70.760350
• Farenick, D. R., & Pidkowich, B. A. F. (2003). The spectral theorem in quaternions. Linear Algebra and its Applications, 371, 75-102. doi:10.1016/S0024-3795(03)00420-8
• Feng, Y. T., Owen, D. R. J., & Peric, D. (1995). A block conjugate gradient method applied to linear systems with multiple right-hand sides. Computer Methods in Applied Mechanics and Engineering, 127, 203-215. doi:10.1016/0045-7825(95)00832-2
• He, Z. H., Wang, X. X., & Zhao, Y. F. (2023). Eigenvalues of quaternion tensors with applications to color video processing. Journal of Scientific Computing, 94, 1-15. doi:10.1007/s10915-022-02058-5
• Ji, H., & Li, Y. (2017) Block conjugate gradient algorithms for least squares problems. Journal of Computational and Applied Mathematics, 317, 203-217. doi:10.1016/j.cam.2016.11.031
• Jia, Z., & Ng, M. K. (2021). Structure preserving quaternion generalized minimal residual method. Journal on Matrix Analysis and Applications, 42(2), 616-634. doi:10.1137/20M133751X
• O’Leary, D. P. (1980). The block conjugate gradient algorithm and related methods. Linear Algebra and Its Applications, 29, 293-322. doi:10.1016/0024-3795(80)90247-5
• Opfer, G. (2005). The conjugate gradient algorithm applied to quaternion valued matrices. Journal of Applied Mathematics and Mechanics, 85(9), 660-672. doi:10.1002/zamm.200410191
• Rodman, L. (2014). Topics in Quaternion Linear Algebra. New Jersey, USA: Princeton University Press.
• Sangwine, S. J. (1996). Fourier transforms of colour images using quaternion or hypercomplex, numbers. Electronics Letters, 32(21), 1979-1980. doi:10.1049/el:19961331
• Wei, M., Li, Y., Zhang, F., & Zhao, J. (2018). Quaternion Matrix computations. New York, USA: Nova Science Publishers.
• Wendland, H. (2018). Numerical Linear Algebra: An Introduction. Cambridge, UK: Cambridge University Press.