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Numerical Algorithms

, Volume 71, Issue 4, pp 907–913 | Cite as

Modify Levinson algorithm for symmetric positive definite Toeplitz system

  • Nasser Akhoundi
  • Iman Alimirzaei
Original Paper
  • 122 Downloads

Abstract

This paper describes a new \(O(N^{\frac {3}{2}}\log (N))\) solver for the symmetric positive definite Toeplitz system T N x N = b N . The method is based on the block QR decomposition of T N accompanied with Levinson algorithm and its generalized version for solving Schur complements S m of size m. In our algorithm we use a formula for displacement rank representation of the S m in terms of generating vectors of the matrix T N , and we assume that N = l m with \(l, m\in \mathbb {N}\). The new algorithm is faster than the classical O(N 2)-algorithm for N > 29. Numerical experiments confirm the good computational properties of the new method.

Keywords

Toeplitz matrices Gohberg-Semencul formula Schur complement Displacement rank representation Recursive-based method Block QR decomposition 

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Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.School of Mathematics and Computer SciencesDamghan UniversityDamghanIran

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