{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,18]],"date-time":"2026-03-18T20:23:11Z","timestamp":1773865391529,"version":"3.50.1"},"reference-count":37,"publisher":"Wiley","issue":"5","license":[{"start":{"date-parts":[[2012,6,20]],"date-time":"2012-06-20T00:00:00Z","timestamp":1340150400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/2.zoppoz.workers.dev:443\/http\/onlinelibrary.wiley.com\/termsAndConditions#vor"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Numerical Linear Algebra App"],"published-print":{"date-parts":[[2013,10]]},"abstract":"<jats:title>SUMMARY<\/jats:title><jats:p>Amongst recent contributions to preconditioning methods for saddle point systems, standard iterative methods in nonstandard inner products have been usefully employed. Krzy\u017canowski (<jats:italic>Numerical Linear Algebra with Applications<\/jats:italic>2011;<jats:bold>18<\/jats:bold>:123\u2013140) identified a two\u2010parameter family of preconditioners in this context and Stoll and Wathen (<jats:italic>SIAM Journal on Matrix Analysis and Applications<\/jats:italic>2008;<jats:bold>30<\/jats:bold>:582\u2013608) introduced combination preconditioning, where two preconditioners, self\u2010adjoint with respect to different inner products, can lead to further preconditioners and associated bilinear forms or inner products. Preconditioners that render the preconditioned saddle point matrix nonsymmetric but self\u2010adjoint with respect to a nonstandard inner product always allow a MINRES\u2010type method (<jats:inline-graphic xmlns:xlink=\"https:\/\/2.zoppoz.workers.dev:443\/http\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/nla1843-math-0001.gif\" xlink:title=\"urn:x-wiley:10705325:media:nla1843:nla1843-math-0001\"\/>\u2010PMINRES) to be applied in the relevant inner product. If the preconditioned matrix is also positive definite with respect to the inner product, a more efficient CG\u2010like method (<jats:inline-graphic xmlns:xlink=\"https:\/\/2.zoppoz.workers.dev:443\/http\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/nla1843-math-0002.gif\" xlink:title=\"urn:x-wiley:10705325:media:nla1843:nla1843-math-0002\"\/>\u2010PCG) can be reliably used. We establish eigenvalue expressions for Krzy\u017canowski preconditioners and show that for a specific choice of parameters, although the Krzy\u017canowski preconditioned saddle point matrix is self\u2010adjoint with respect to an inner product, it is never positive definite. We provide explicit expressions for the combination of certain preconditioners and prove the rather counterintuitive result that the combination of two specific preconditioners for which only<jats:inline-graphic xmlns:xlink=\"https:\/\/2.zoppoz.workers.dev:443\/http\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/nla1843-math-0003.gif\" xlink:title=\"urn:x-wiley:10705325:media:nla1843:nla1843-math-0003\"\/>\u2010PMINRES can be reliably used leads to a preconditioner for which, for certain parameter choices,<jats:inline-graphic xmlns:xlink=\"https:\/\/2.zoppoz.workers.dev:443\/http\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/nla1843-math-0004.gif\" xlink:title=\"urn:x-wiley:10705325:media:nla1843:nla1843-math-0004\"\/>\u2010PCG is reliably applicable. That is, combining two indefinite preconditioners can lead to a positive definite preconditioner. This combination preconditioner outperforms either of the two preconditioners from which it is formed for a number of test problems. Copyright \u00a9 2012 John Wiley &amp; Sons, Ltd.<\/jats:p>","DOI":"10.1002\/nla.1843","type":"journal-article","created":{"date-parts":[[2012,6,20]],"date-time":"2012-06-20T12:33:45Z","timestamp":1340195625000},"page":"785-808","source":"Crossref","is-referenced-by-count":16,"title":["Combination preconditioning of saddle point systems for positive definiteness"],"prefix":"10.1002","volume":"20","author":[{"given":"J.","family":"Pestana","sequence":"first","affiliation":[{"name":"Mathematical Institute University of Oxford 24\u201329 St Giles' Oxford OX1 3LB U.K."}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"A. 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