Combinatorial Preconditioners

Research output: Chapter in Book/Report/Conference proceedingChapterpeer-review


The Conjugate Gradient (CG) method is an iterative algorithm for solving linear systems of equations Ax = b, where A is symmetric and positive definite. The convergence of the method depends on the spectrum of A; when its eigenvalues are clustered, the method converges rapidly. In partic- ular, CG converges to within a fixed tolerance in O( √ κ) iterations, where κ = κ(A) = λmax(A)/λmin(A) is the spectral condition number of A. This upper bound is tight when eigenvalues are poorly distributed (i.e., they are not clustered). For example, if there are only two distinct eigenvalues, CG converges in two iterations.
Original languageEnglish
Title of host publicationCombinatorial Scientific Computing
EditorsUwe Naumann, Olaf Schenk
PublisherChapman and Hall/CRC
Number of pages26
ISBN (Electronic)9781439827369
ISBN (Print)9781439827369
StatePublished - 2012


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