Advertisement

A New Algorithm for Finding Positive Eigenvectors for a Class of Nonlinear Operators Associated with M-matrices

  • Yuriy V. Shlapak
Part of the Operator Theory: Advances and Applications book series (OT, volume 199)

Abstract

In this paper we state the sufficient conditions for the existence and uniqueness of positive eigenvectors for a class of nonlinear operators associated with M-matrices. We also show how to construct a convergent iterative process for finding these eigenvectors. The details of numerical implementation of this algorithm for some spectral methods of discretization of elliptic partial differential equations are also discussed. Some results of numerical experiments for the Gross-Pitaevskii Equation with non-separable potentials in a rectangular domain are given in the end of the paper.

Mathematics Subject Classification (2000)

47J10 

Keywords

Positive eigenvectors Nonlinear operators M-matrices Monotone fixed point theorem Gross-Pitaevskii equation 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Choi Y.S., Koltracht I., McKenna P.J. A generalization of the Perron-Frobenius theorem for non-linear perturbations of Stiltjes Matrices. Contemporary Mathematics, Volume 281, 2001, pp. 325–330MathSciNetGoogle Scholar
  2. [2]
    Choi Y.S., Javanainen J., Koltracht I., Kostrun M., McKenna P. J., Savytska N., A fast algorithm for the solution of the time-independent Gross-Pitaevskii equation. Journal of Computational Physics, 190 (2003), pp. 1–21zbMATHCrossRefMathSciNetGoogle Scholar
  3. [3]
    Kantorovich L.V., Vulikh B.Z., Pinsker A.G., Functional Analysis in Semi-ordered Spaces. (in Russian Language). Moscow, GosIzdat Technico-Teoreticheskoi Literatury, 1950.Google Scholar
  4. [4]
    Gottlieb D., Orszag S.A., Numerical Analysis of Spectral Methods: Theory and Applications. Philadelphia, Society for Industrial and Applied Mathematics, 1977.zbMATHGoogle Scholar

Copyright information

© Birkhäuser Verlag Basel/Switzerland 2010

Authors and Affiliations

  • Yuriy V. Shlapak
    • 1
  1. 1.Department of MathematicsUniversity of ConnecticutStorrsUSA

Personalised recommendations