Numerical Algorithms

, Volume 44, Issue 2, pp 159–172 | Cite as

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 


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  1. 1.
    Berlinet, A., Roland, Ch.: Acceleration schemes with application to the EM algorithm. Comput. Stat. Data Anal. 51, 3689–3702 (2007)CrossRefGoogle Scholar
  2. 2.
    Brezinski, C., Redivo Zaglia, M.: Extrapolation Methods Theory and Practice. North-Holland, Amsterdam (1991)MATHGoogle Scholar
  3. 3.
    Brezinski, C.: Vector sequence transformations: methodology and applications to linear systems. J. Comput. Appl. Math. 98, 149–175 (1998)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Brezinski, C.: A classification of quasi-Newton methods. Numer. Algorithms 33, 123–145 (2003)MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Dempster, A.P., Laird, N.M., Rubin, D.B.: Maximum likelihood from incomplete data via the EM algorithm (with discussion). J.R. Stat. Soc. B 39, 1–38 (1977)MATHMathSciNetGoogle Scholar
  6. 6.
    Jbilou, K., Sadok, H.: Vector extrapolation methods. Applications and numerical comparison. J. Comput. Appl. Math. 122, 149–165 (2000)MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Lemaréchal, C.: Une méthode de résolution de certains systèmes non linéaires bien posés. C. R. Acad. Sci. Paris Ser. A 272, 747–756 (1971)Google Scholar
  8. 8.
    Marder, H., Weitzner, B.: A bifurcation problem in E-layer equilibria. Plasma Phys. 12, 435–445 (1970)CrossRefGoogle Scholar
  9. 9.
    R Development Core Team: R: a language and environment for statistical computing. Version 2.3.1., (2006)
  10. 10.
    Raydan, M., Svaiter, B.F.: Relaxed steepest descent and Cauchy–Barzilai–Borwein method. Comput. Optim. Appl. 21, 155–167 (2002)MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Roland, Ch., Varadhan, R.: New iterative schemes for nonlinear fixed point problems, with applications to problems with bifurcations and incomplete-data problems. Appl. Numer. Math. 55, 215–226 (2005)MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Sidi, A.: Efficient implementation of minimal polynomial and reduced rank extrapolation methods. J. Comput. Appl. Math. 36, 305–337 (1991)MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Varadhan, R., Roland, Ch.: Squared extrapolation methods (SQUAREM): 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)
  14. 14.
    Vardi, Y., Shepp, L.A., Kaufman, L.: A statistical model for positron emission tomography. J. Amer. Stat. Assoc. 80, 8–37 (1985)MATHCrossRefMathSciNetGoogle Scholar

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

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