Matrix iterative algorithms for least-squares problem in quaternionic quantum theory

Abstract

Quaternionic least squares (QLS) is an efficient method for solving approximate problems in quaternionic quantum theory. Based on Paige's algorithms LSQR and residual-reducing version of LSQR proposed in Paige and Saunders [LSQR: An algorithm for sparse linear equations and sparse least squares, ACM Trans. Math. Softw. 8(1) (1982), pp. 43–71], we provide two matrix iterative algorithms for finding solution with the least norm to the QLS problem by making use of structure of real representation matrices. Numerical experiments are presented to illustrate the efficiency of our algorithms.

Keywords:

• quaternion matrix
• quaternionic least squares
• LSQR
• RRLSQR
• iterative algorithm

2000 AMS Subject Classifications:

• 65F10
• 65F30

Acknowledgements

We thank the anonymous referees and the editor for their helpful comments and suggestions. The first author is supported by the Fundamental Research Funds for the Central Universities under grant 2012QNB22 and the second author is supported by the National Natural Science Foundation of China under grant 11201193.