We propose a new iterative method based on two strategies, improvement stabilization by applying two-term recurrences and utilization of associate residual. The former adopts the coupled two-term recurrences proposed by Rutishauser as stabilizing polynomials, and the latter adopts the associate residual vector for computation of the recurrence parameters ζ_k,η_k instead of that of the residual vector. Through numerical experiments, we make clear that the proposed method outperforms other methods from the view points of computational and elapsed time and speed-up on both serial and parallel computers.