J. Comp. Math., 30 (2012), pp. 24-33. |
Linear Convergence of the LZI Algorithm for Weakly Positive Tensors Liping Zhang ^{1}, Liqun Qi ^{2}, Yi Xu ^{2} 1 Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China2 Department of Applied Mathematics, The Hong Kong Polytechnic University, Hong Kong Received 2011-2-14 Accepted 2011-6-19 Available online 2012-1-09 doi:10.4208/jcm.1110-m11si09 Abstract We define weakly positive tensors and study the relations among essentially positive tensors, weakly positive tensors, and primitive tensors. In particular, an explicit linear convergence rate of the Liu-Zhou-Ibrahim(LZI) algorithm for finding the largest eigenvalue of an irreducible nonnegative tensor, is established for weakly positive tensors. Numerical results are given to demonstrate linear convergence of the LZI algorithm for weakly positive tensors. Key words: Irreducible nonnegative tensor, Weakly positive tensor, Largest eigenvalue, Linear convergence. AMS subject classifications: 74B99, 15A18, 15A69. Email: lzhang@math.tsinghua.edu.cn, maqilq@polyu.edu.hk, yi.xu1983@gmail.com |