The GMRES method for solving the large indefinite least squares problem via an accelerated preconditioner

In this research, to solve the large indefinite least squares problem, we firstly transform its normal equation into a sparse block three-by-three linear systems, then use GMRES method with an accelerated preconditioner to solve it. The construction idea of the preconditioner comes from the thought of Luo et.al [Luo, WH., Gu, XM., Carpentieri, B., BIT 62, 1983-2004(2022)], and the advantage of this is that the preconditioner is closer to the coefficient matrix of the block three-by-three linear systems when the parameter approachs zero. Theoretically, the iteration method under the preconditioner satisfies the conditional convergence, and all eigenvalues of the preconditioned matrix are real numbers and gathered at point $(1,0)$ as parameter is close to $0$. In the end, numerical results reflect that the theoretical results is correct and the proposed preconditioner is effective by comparing with serval existing preconditioners.