{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,10,11]],"date-time":"2023-10-11T09:20:18Z","timestamp":1697016018362},"reference-count":36,"publisher":"Wiley","issue":"1","license":[{"start":{"date-parts":[[2012,2,13]],"date-time":"2012-02-13T00:00:00Z","timestamp":1329091200000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/onlinelibrary.wiley.com\/termsAndConditions#vor"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Numerical Linear Algebra App"],"published-print":{"date-parts":[[2013,1]]},"abstract":"SUMMARY<\/jats:title>The preconditioned inverse iteration is an efficient method to compute the smallest eigenpair of a symmetric positive definite matrix M<\/jats:italic>. Here we use this method to find the smallest eigenvalues of a hierarchical matrix. The storage complexity of the data\u2010sparse \u2010matrices is almost linear. We use \u2010arithmetic to precondition with an approximate inverse of M<\/jats:italic> or an approximate Cholesky decomposition of M<\/jats:italic>. In general, \u2010arithmetic is of linear\u2010polylogarithmic complexity, so the computation of one eigenvalue is cheap. We extend the ideas to the computation of inner eigenvalues by computing an invariant subspace S<\/jats:italic> of (M<\/jats:italic>\u2009\u2212\u2009\u03bcI<\/jats:italic>)2<\/jats:sup> by subspace preconditioned inverse iteration. The eigenvalues of the generalized matrix Rayleigh quotient \u03bc<\/jats:italic>M<\/jats:italic><\/jats:sub>(S<\/jats:italic>) are the desired inner eigenvalues of M<\/jats:italic>. The idea of using (M<\/jats:italic>\u2009\u2212\u2009\u03bcI<\/jats:italic>)2<\/jats:sup> instead of M<\/jats:italic> is known as the folded spectrum method. As we rely on the positive definiteness of the shifted matrix, we cannot simply apply shifted inverse iteration therefor. Numerical results substantiate the convergence properties and show that the computation of the eigenvalues is superior to existing algorithms for non\u2010sparse matrices.Copyright \u00a9 2012 John Wiley & Sons, Ltd.<\/jats:p>","DOI":"10.1002\/nla.1830","type":"journal-article","created":{"date-parts":[[2012,2,14]],"date-time":"2012-02-14T04:37:37Z","timestamp":1329194257000},"page":"150-166","source":"Crossref","is-referenced-by-count":4,"title":["The preconditioned inverse iteration for hierarchical matrices"],"prefix":"10.1002","volume":"20","author":[{"given":"Peter","family":"Benner","sequence":"first","affiliation":[{"name":"Max Planck Institute for Dynamics of Complex Technical Systems 39106 Magdeburg Germany"},{"name":"Department of Mathematics Chemnitz University of Technology 09107 Chemnitz Germany"}]},{"given":"Thomas","family":"Mach","sequence":"additional","affiliation":[{"name":"Max Planck Institute for Dynamics of Complex Technical Systems 39106 Magdeburg Germany"}]}],"member":"311","published-online":{"date-parts":[[2012,2,13]]},"reference":[{"key":"e_1_2_9_2_1","doi-asserted-by":"publisher","DOI":"10.4171\/OWR\/2009\/37"},{"key":"e_1_2_9_3_1","first-page":"360","article-title":"Direct minimization for calculating invariant subspaces in density functional computations of the electronic structure","volume":"27","author":"Schneider R","year":"2009","journal-title":"Journal of Computational Mathematics"},{"key":"e_1_2_9_4_1","doi-asserted-by":"publisher","DOI":"10.1007\/s00791-005-0001-x"},{"key":"e_1_2_9_5_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.compchemeng.2006.07.012"},{"key":"e_1_2_9_6_1","volume-title":"Matrix Computations","author":"Golub G","year":"1996"},{"key":"e_1_2_9_7_1","doi-asserted-by":"publisher","DOI":"10.1137\/S0036144500378648"},{"key":"e_1_2_9_8_1","doi-asserted-by":"publisher","DOI":"10.1021\/j100059a032"},{"key":"e_1_2_9_9_1","first-page":"104","article-title":"Preconditioned eigensolvers \u2014 an oxymoron?","volume":"7","author":"Knyazev AV","year":"1998","journal-title":"Electronic Transactions on Numerical Analysis"},{"key":"e_1_2_9_10_1","unstructured":"NeymeyrK.A Hierarchy of Preconditioned Eigensolvers for Elliptic Differential Operators. 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