Positive eigenvector of nonlinear eigenvalue problem with a singular M-matrix and Newton-SOR iterative solution

Cheng-Yi Zhang,Yao-Yan Song,Shuanghua Luo

doi:10.1186/s13660-016-1169-y

Cheng-Yi Zhang, Yao-Yan Song + Show 1 more

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https://doi.org/10.1186/s13660-016-1169-y

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Abstract

Some sufficient conditions are proposed in this paper such that the nonlinear eigenvalue problem with an irreducible singular M-matrix has a unique positive eigenvector. Under these conditions, the Newton-SOR iterative method is proposed for numerically solving such a positive eigenvector and some convergence results on this iterative method are established for the nonlinear eigenvalue problems with an irreducible singular M-matrix, a nonsingular M-matrix, and a general M-matrix, respectively. Finally, a numerical example is given to illustrate that the Newton-SOR iterative method is superior to the Newton iterative method.

Highlights

In research of physics, Bose-Einstein condensation of atoms near absolute zero temperature is modeled by a nonlinear Gross-Pitaevskii equation, see [, ], i.e.,– u + V (x, y, z)u + ku = λu, ( ) ∞∞∞lim u =, u(x, y, z) dx dy dz =, |(x,y,z)|→∞ –∞ –∞ –∞where V is a potential function
In [ – ], some scholars studied the conditions that the nonlinear eigenvalue problem ( ) with an irreducible nonsingular M-matrix has a unique positive eigenvector, applied the Newton iterative method to solve numerically this problem, and established some significant theoretical and numerical results. It is shown in [ – ] that the main contributions were made to the nonlinear eigenvalue problem as follows: (i) any number greater than the smallest positive eigenvalue of the nonsingular M-matrix is an eigenvalue of the nonlinear eigenvalue problems; (ii) the corresponding positive eigenvector is unique, and (iii) the Newton iterative method is convergent for numerically solving the positive eigenvector
Some sufficient conditions will be proposed such that the nonlinear eigenvalue problem with an irreducible singular M-matrix has a unique positive eigenvector

Summary

Introduction

Some sufficient conditions will be proposed such that the nonlinear eigenvalue problem with an irreducible singular M-matrix has a unique positive eigenvector. The Newton-SOR iterative method will be proposed under these conditions for numerically solving such a positive eigenvector, and some convergence results on this iterative method will be established for the nonlinear eigenvalue problems with an irreducible singular M-matrix, a nonsingular M-matrix, and a general M-matrix, respectively.

Results

Conclusion