Numerical Algorithms

, Volume 44, Issue 2, pp 159–172

Squared polynomial extrapolation methods with cycling: an application to the positron emission tomography problem

Original Paper


Roland and Varadhan (Appl. Numer. Math., 55:215–226, 2005) presented a new idea called “squaring” to improve the convergence of Lemaréchal’s scheme for solving nonlinear fixed-point problems. Varadhan and Roland (Squared extrapolation methods: A new class of simple and efficient numerical schemes for accelerating the convergence of the EM algorithm, Department of Biostatistics Working Paper. Johns Hopkins University,, 2004) noted that Lemaréchal’s scheme can be viewed as a member of the class of polynomial extrapolation methods with cycling that uses two fixed-point iterations per cycle. Here we combine these two ideas, cycled extrapolation and squaring, and construct a new class of methods, called squared polynomial methods (SQUAREM), for accelerating the convergence of fixed-point iterations. Our main goal is to evaluate whether the squaring device is effective in improving the rate of convergence of cycled extrapolation methods that use more than two fixed-point iterations per cycle. We study the behavior of the new schemes on an image reconstruction problem for positron emission tomography (PET) using simulated data. Our numerical experiments show the effectiveness of first- and higher-order squared polynomial extrapolation methods in accelerating image reconstruction, and also their relative superiority compared to the classical, “unsquared” vector polynomial methods.


Nonlinear systems Fixed-point methods Polynomial extrapolation methods Squaring Linear systems 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Copyright information

© Springer Science+Business Media LLC 2007

Authors and Affiliations

  1. 1.Laboratoire Paul Painlevé, UFR de Mathématiques Pures et AppliquéesUniversité des Sciences et Technologies de LilleVilleneuve d’Ascq cedexFrance
  2. 2.The Center on Aging and HealthJohns Hopkins UniversityBaltimoreUSA
  3. 3.Department of BiostatisticsJohns Hopkins UniversityBaltimoreUSA

Personalised recommendations