On Rayleigh Quotient Iteration for the Dual Quaternion Hermitian Eigenvalue Problem
Abstract
1. Introduction
2. Dual Quaternions and Dual Quaternion Matrices
2.1. Dual Numbers, Quaternions and Dual Quaternions
2.2. Dual Quaternion Matrices
- (1)
- ;
- (2)
3. Dual Representation of the Dual Quaternion Matrix
- (1)
- .
- (2)
- , .
- (3)
- .
- (1)
- .
- (2)
- .
- (3)
- .
4. Rayleigh Quotient Iteration for Computing the Appreciable Eigenvalue of a Dual Quaternion Hermitian Matrix
4.1. A Structure-Preserving Method for Solving the Linear System
Algorithm 1 The Rayleigh quotient iteration (RQI) |
Input: Given a normalized initial guess with , the maximal iteration number and the stopping tolerance . Output: Eigenvalue and eigenvector .
|
4.2. Convergence Analysis
4.3. Computing All Appreciable Eigenvalues of a Dual Quaternion Hermitian Matrix
Algorithm 2 Computing all appreciable eigenvalues |
Input: Given a dual quaternion Hermitian matrix , the tolerance . Output: Eigenvalues and eigenvectors .
|
5. Numerical Experiments
6. Conclusions
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
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Method | Residual | Relative Error | IT | CPU (s) |
---|---|---|---|---|
RQI | 6.425678537436753 | 7.6342876438642964 | 3 | 0.0064 |
PM | 3.163485367353436 | 1.7456863573577357 | 41 | 0.0288 |
Method | n | 10 | 20 | 50 | 100 | 200 | 400 |
---|---|---|---|---|---|---|---|
IT | 5 | 5 | 4 | 4 | 5 | 5 | |
RQI | CPU (s) | 0.0103 | 0.0159 | 0.0148 | 0.0783 | 0.1734 | 1.1012 |
RSE | 1.0436 | 1.2212 | 3.3790 | 8.2080 | 1.1310 | 6.3420 | |
IT | 219 | 873 | 4856 | — | — | — | |
PM | CPU (s) | 0.1347 | 0.5200 | 2.9406 | — | — | — |
RSE | 9.0942 | 9.8542 | 9.9711 | — | — | — |
Method | n | 10 | 20 | 50 | 100 | 200 | 400 |
---|---|---|---|---|---|---|---|
IT | 5 | 3 | 7 | 5 | 5 | 6 | |
RQI | CPU (s) | 0.0113 | 0.0849 | 0.0288 | 0.0963 | 0.2547 | 2.7683 |
RSE | 6.7524 | 5.9735 | 3.3790 | 7.7547 | 3.5472 | 8.7542 | |
IT | 533 | 1055 | 4666 | — | — | — | |
PM | CPU (s) | 0.1634 | 0.9198 | 3.0257 | — | — | — |
RSE | 1.7548 | 9.9296 | 9.9883 | — | — | — |
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Duan, S.-Q.; Wang, Q.-W.; Duan, X.-F. On Rayleigh Quotient Iteration for the Dual Quaternion Hermitian Eigenvalue Problem. Mathematics 2024, 12, 4006. https://doi.org/10.3390/math12244006
Duan S-Q, Wang Q-W, Duan X-F. On Rayleigh Quotient Iteration for the Dual Quaternion Hermitian Eigenvalue Problem. Mathematics. 2024; 12(24):4006. https://doi.org/10.3390/math12244006
Chicago/Turabian StyleDuan, Shan-Qi, Qing-Wen Wang, and Xue-Feng Duan. 2024. "On Rayleigh Quotient Iteration for the Dual Quaternion Hermitian Eigenvalue Problem" Mathematics 12, no. 24: 4006. https://doi.org/10.3390/math12244006
APA StyleDuan, S.-Q., Wang, Q.-W., & Duan, X.-F. (2024). On Rayleigh Quotient Iteration for the Dual Quaternion Hermitian Eigenvalue Problem. Mathematics, 12(24), 4006. https://doi.org/10.3390/math12244006