Sherman–Morrison–Woodbury Formula‐Based Efficient Preconditioners for Sequences of Shifted Linear Systems

ABSTRACT This study explores iterative methods for solving sequences of shifted linear systems expressed as . To address these systems efficiently, we utilize the preconditioned generalized minimum residual (PGMRES) method, enhanced by a tailored preconditioning strategy. Our approach combines the Sherman‐Morrison‐Woodbury (SMW) formula with selective dropping techniques to optimize performance. The preconditioners are constructed through an incomplete factorization of , leveraging a biconjugate gradient algorithm. Our results show that the eigenvalues of the preconditioned matrices exhibit tighter clustering around unity compared to those generated by conventional preconditioners. We also provide a rigorous convergence analysis for the iterative process. Finally, numerical experiments validate the effectiveness of our proposed method.