A generalized nonlinear diagonal dominance

Abstract

The concept of strictly β-diagonally dominant matrices is introduced. This concept is directly related to solving a system of equations. In the linear case, the strictly β -diagonally dominant matrices contain the strictly diagonally dominant matrices. However, for systems of nonlinear equations, it is a challenge to present a better condition to make sure that the iteration algorithm used to solve them is convergent. A new class of nonlinear mappings and some properties are offered; moreover, the corresponding convergence theorems of the nonlinear Gauss–Seidel and Jacobi iterations are presented.

Keywords:

• Diagonally dominant matrix
• Nonlinear diagonal dominance
• Nonlinear mapping

Acknowledgements

The authors would like to thank the referees for their valuable comments and suggestions. This work was supported by NCET of the Chinese Ministry of Education (NCET-04-0893), the scientific and technological key project of the Chinese Ministry of Education (107098), the Sichuan province foundation for applied basic research (05JY029-068-2) and project for academic leader + group of UESTC.