The improved preconditioned iterative integration-exponential method for linear ill-conditioned problems

The preconditioned iterative integration-exponential method is an innovative iterative regularization technique designed to solve ill-conditioned linear problems. However, the preconditioned iterative integration-exponential method has been primarily applied to symmetric positive definite problems, and a notable limitation is its inability to adaptively determine the optimal number of iterations. To overcome this limitation, the present study demonstrates that the preconditioned iterative integration-exponential method can also be effectively applied to nonsymmetric positive definite linear systems. Furthermore, an improved preconditioned iterative integration-exponential method is proposed by combining the iterative refinement algorithm with the original approach. Addressing the challenge of adaptively determining the optimal number of iterations and Krylov subspace can solve the problem of low computational efficiency of the improved preconditioned iterative integration-exponential method in dealing with large-scale and sparse problems. Numerical results show that the newly proposed method is more robust than the original one.