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

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.