A block upper triangular preconditioner for block three-by-three linear systems arising form the large indefinite least squares problem

Abstract In this paper, to solve the large indefinite least squares problem, we first rewrite the normal equation of it as the sparse block three-by-three linear systems with non-singular diagonal blocks, then GMRES method is used to solve the linear systems. In order to make GMRES method converge quickly, a block upper triangular preconditioner is deduced. Theoretically, the iteration method under the preconditioner is unconditional convergent. Furthermore, the all eigenvalues of the preconditioned matrix are real number and located in a positive interval. Numerical results are provided to support the validity of the theoretical results and demonstrate the effectiveness of the studied precondition. Mathematics Subject Classification (2010) 65F10 · 65F20