A Block Conjugate Gradient Method for Quaternion Linear
Öz
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.
Anahtar Kelimeler
Block conjugate gradient method, Krylov subspace, Linear quaternion systems
Kaynakça
- 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