The BiCG Algorithm for Solving the Minimal Frobenius Norm Solution of Generalized Sylvester Tensor Equation over the Quaternions
Abstract
:1. Introduction
2. Preliminaries
3. An Iterative Algorithm for Solving the Problems 1.1 and 1.2
Algorithm 1 BiCG-BTF method for solving Equation (2). |
Input: , . Output: The norm and the solutions , . Initialization: ;
|
4. Numerical Examples
5. Conclusions
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
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IT | The Number of Iterations |
---|---|
CPU time | The CPU time elapsed, measured in seconds |
Res | is the residual at kth iteration |
The third-order tensor with pseudo-random values | |
sampled from a uniform distribution over the unit interval | |
The upper triangular part of Hilbert matrix | |
The upper triangular part of matrix with all 1 | |
Identity matrix with size | |
Zero matrix with size | |
The tridiagonal matrix with |
n | IT | CPU Time | Res |
---|---|---|---|
20 | 82 | 8.3272 | |
40 | 156 | 24.8187 | |
60 | 288 | 91.4387 |
l = 10 | l = 25 | l = 30 | |
---|---|---|---|
Algorithm 1 | 25.6884(320) | 140.8233(1270) | 202.7709(1580) |
CGLS [43] | 15.5685(219) | 144.3166(1266) | 230.7340(1832) |
l = 10 | l = 15 | l = 20 | |
---|---|---|---|
Algorithm 1 | 45.4157(576) | 104.4370(1095) | 163.6110(1700) |
CGLS [43] | 44.0701(579) | 118.2419(1296) | 230.7978(2298) |
Algorithm 1 | (248) | (188) | (248) |
CGLS [43] | (106) | (178) | (167) |
Algorithm([r,s]) | Algorithm 1(PSNR/RE) | CGLS(PSNR/RE) |
---|---|---|
[3,3] | 38.6181(0.0235) | 13.8404(0.3769) |
[6,6] | 37.3721(0.0300) | 14.0529(0.3694) |
[8,8] | 33.8073(0.0338) | 14.7958(0.3551) |
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Xie, M.; Wang, Q.-W.; Zhang, Y. The BiCG Algorithm for Solving the Minimal Frobenius Norm Solution of Generalized Sylvester Tensor Equation over the Quaternions. Symmetry 2024, 16, 1167. https://doi.org/10.3390/sym16091167
Xie M, Wang Q-W, Zhang Y. The BiCG Algorithm for Solving the Minimal Frobenius Norm Solution of Generalized Sylvester Tensor Equation over the Quaternions. Symmetry. 2024; 16(9):1167. https://doi.org/10.3390/sym16091167
Chicago/Turabian StyleXie, Mengyan, Qing-Wen Wang, and Yang Zhang. 2024. "The BiCG Algorithm for Solving the Minimal Frobenius Norm Solution of Generalized Sylvester Tensor Equation over the Quaternions" Symmetry 16, no. 9: 1167. https://doi.org/10.3390/sym16091167