Spectral properties of a boundary value problem for the discrete beam equation

This paper deals with developing four efficient algorithms (including the conjugate gradient least-squares, least-squares with QR factorization, least-squares minimal residual and Paige algorithms) to numerically find the (least-squares) solutions of the following (in-) consistent quaternion matrix equation

$$\begin{aligned} {A_1}X + {\left( {{A_1}X} \right) ^{\eta H}} + {B_1}YB_1^{\eta H} + {C_1}ZC_1^{\eta H} = {D_1}, \end{aligned}$$

in which the coefficient matrices are large and sparse. More precisely, we construct four efficient iterative algorithms for determining triple least-squares solutions (X, Y, Z) such that X may have a special assumed structure, Y and Z can be either \(\eta \)-Hermitian or \(\eta \)-anti-Hermitian matrices. In order to speed up the convergence of the offered algorithms for the case that the coefficient matrices are possibly ill-conditioned, a preconditioned technique is employed. Some numerical test problems are examined to illustrate the effectiveness and feasibility of presented algorithms.