LSQR algorithm with structured preconditioner for the least squares problem in quaternionic quantum theory

The solution of a linear quaternionic least squares (QLS) problem can be transformed into that of a linear least squares (LS) problem with JRS-symmetric real coefficient matrix, which is suitable to be solved by developing structured iterative methods when the coefficient matrix is large and sparse. The main aim of this work is to construct a structured preconditioner to accelerate the LSQR convergence. The preconditioner is based on structure-preserving tridiagonalization to the real counterpart of the coefficient matrix of the normal equation, and the incomplete inverse upper-lower factorization related to only one symmetric positive definite tridiagonal matrix rather than four, so it is reliable and has low storage requirements. The performances of the LSQR algorithm with structured preconditioner are demonstrated by numerical experiments.