An Iterative Method for Quaternionic Linear Equations

Authors: Bin Yu, Minghui Wang, Juntao Zhang

Abstract:

By the real representation of the quaternionic matrix, an iterative method for quaternionic linear equations Ax = b is proposed. Then the convergence conditions are obtained. At last, a numerical example is given to illustrate the efficiency of this method.

Keywords: Quaternionic linear equations, Real representation, Iterative algorithm.

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1337139

References:

[1] D. Finkelstein , J. Jauch , S. Schiminovich and D. Speiser, Foundations of quaternion quantum mechanics, J. Math. Phys., vol. 3, 1962, pp.207-231.
[2] S. Adler, Quaternionic quantum field theory Commun. Math. Phys., vol. 104, 1986, pp. 611-623.
[3] S. Adler, Quaternionic Quantum Mechanics and Quantum Fields, New York: Oxford University Press, 1995.
[4] J. Jiang, An algorithm for quaternionic linear equations in quaternionic quantum theory, J. Math. Phys., vol. 45, 2004, pp.4218-4228.
[5] J. Jiang, Cramer ruler for quaternionic linear equations in quaternionic quantum theory, Rep. Math. Phys., vol. 57, 2006, pp. 463-467.
[6] J. Jiang, Algebraic algorithms for least squares problem in quaternionic quantum theory, Comput. Phys. Commun., vol. 176, 2007, pp. 481-485.
[7] J. Jiang, Real representiations of quaternion matrices and quaternion matrix equations, Acta Mathematica Scientia, vol. 26A, 2006, pp. 578-584.
[8] S. Yuan, Q. Wang and X. Duan, On solutions of the quaternion matrix equation AX = B and their applications in color image restoration, Applied Mathematics and Computation, vol. 221, 2013, pp.10-20.
[9] M. Wang, M. Wei and Y. Feng, An iterative algorithm for least squares problem in quaternionic quantum theory, Comput. Phys. Commun., vol. 179, 2008, pp. 203-207.
[10] M. Wang, On Positive Definite Solutions of Quaternionic Matrix Equations, World Academy of Science, Engineering and Technology, vol. 37, 2010, pp.535-537.