  In this paper we propose a new iterative method to hierarchically compute a
relatively large number of leftmost eigenpairs of a sparse symmetric positive
matrix under the multiresolution operator compression framework. We exploit the
well-conditioned property of every decomposition components by integrating the
multiresolution framework into the Implicitly restarted Lanczos method. We
achieve this combination by proposing an extension-refinement iterative scheme,
in which the intrinsic idea is to decompose the target spectrum into several
segments such that the corresponding eigenproblem in each segment is
well-conditioned. Theoretical analysis and numerical illustration are also
reported to illustrate the efficiency and effectiveness of this algorithm.
