A new implementation of LSMR algorithm for the quaternionic least squares problem

Abstract

This paper is endeavored to present a new version of the LSMR algorithm for solving the linear least squares problem in quaternion field, by means of direct quaternion arithmetics rather than the usually used real or complex representation methods. The present new algorithm is based on the classical Golub-Kahan bidiagonalization process, but is instead of using two QR factorizations. It has several advantages as follows: (i) does not make the scale of the problem dilate exponentially, compared to the conventional complex representation or real representation methods, (ii) has monotonic and smooth convergence behavior, compared to the Q-LSQR algorithm, and (iii) the new algorithm is more straightforward, and there is no expensive matrix inversion or decomposition. It may reduce the number of iterations in some cases. The performances of the algorithm are illustrated by some numerical experiments.