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The Ritz vectors obtained by oldi’s method may not be good approximations and even may not converge even if the corresponding Ritz values do. In order to improve the quality of Ritz vectors and enhance the efficiency of oldi type algorithms, we propose a strategy that uses Ritz values obtained from an m-dimensional Krylov subspace but chooses modified approximate eigenvectors in an (m+1)-dimensional Krylov subspace. Residual norm of each new approximate eigenpair is minimal over the span of the Ritz vector and the (m+1)th basis vector,which is available when the m-step oldi process is run. The resulting modi-fied m-step oldi method is better than the standard m-step one in theory and cheaper than the standard (m+1)-step one. Based on this strategy, we present a modified m-step restarted oldi algorithm. Numerical examples show that the modified m-step restarted algorithm and its version with Chebyshev acceleration are often considerably more efficient than the standard (m + 1)-step restarted ones.