Hermitian tridiagonal solution with the least norm to quaternionic least squares problem

Quaternionic least squares (QLS) is an efficient method for solving approximate problems in quaternionic quantum theory. In view of the extensive applications of Hermitian tridiagonal matrices in physics, in this paper we list some properties of basis matrices and subvectors related to tridiagonal matrices, and give an iterative algorithm for finding Hermitian tridiagonal solution with the least norm to the quaternionic least squares problem by making the best use of structure of real representation matrices, we also propose a preconditioning strategy for the Algorithm LSQR-Q in Wang, Wei and Feng (2008) [14] and our algorithm. Numerical experiments are provided to verify the effectiveness of our method.