The fast bisection Eigenvalue method for Hermitian order one Quasiseparable matrices and computations of norms

Yuli Eidelman, Iulian Haimovici

Research output: Contribution to journalArticlepeer-review

Abstract

Since we can evaluate the characteristic polynomial of an N × N order one quasiseparable Hermitian matrix A in less than 10N arithmetical operations by sharpening results and techniques from Eidelman, Gohberg, and Olshevsky [Linear Algebra Appl., 405 (2005), pp. 1-40], we use the Sturm property with bisection to compute all or selected eigenvalues of A. Moreover, linear complexity algorithms are established for computing norms, in particular, the Frobenius norm ∥A∥F and ∥A∥, ∥A∥1, and other bounds for the initial interval to be bisected. Upper and lower bounds for eigenvalues are given by the Gershgorin Circle Theorem, and we describe an algorithm with linear complexity to compute them for quasiseparable matrices.

Original languageEnglish
Pages (from-to)342-366
Number of pages25
JournalElectronic Transactions on Numerical Analysis
Volume44
StatePublished - 2015

Keywords

  • Bisection
  • Eigenvalues
  • Hermitian
  • Matrix norm
  • Quasiseparable
  • Sturm property

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