On computing minimal H-eigenvalue of sign-structured tensors  

作　　者：Haibin CHEN Yiju WANG 

机构地区：[1]School of Management Science, Qufu Normal University, Rizhao 276826, China

出　　处：《Frontiers of Mathematics in China》2017年第6期1289-1302,共14页中国高等学校学术文摘·数学（英文）

基　　金：This work was done during the first authors＇ postdoctoral period in Qufu Normal University. This work was supported in part by the National Natural Science Foundation of China （Grant Nos. 11601261, 11671228） and the Natural Science Foundation of Shandong Province （No. ZR2016AQ12）.

摘　　要：Finding the minimal H-eigenvalue of tensors is an important topic in tensor computation and numerical multilinear algebra. This paper is devoted to a sum-of-squares （SOS） algorithm for computing the minimal H-eigenvalues of tensors with some sign structures called extended essentially nonnegative tensors （EEN-tensors）, which includes nonnegative tensors as a subclass. In the even-order symmetric case, we first discuss the positive semi-definiteness of EEN-tensors, and show that a positive semi-definite EEN-tensor is a non- negative tensor or an M-tensor or the sum of a nonnegative tensor and an M-tensor, then we establish a checkable sufficient condition for the SOS decomposition of EEN-tensors. Finally, we present an efficient algorithm to compute the minimal H-eigenvalues of even-order symmetric EEN-tensors based on the SOS decomposition. Numerical experiments are given to show the efficiency of the proposed algorithm.Finding the minimal H-eigenvalue of tensors is an important topic in tensor computation and numerical multilinear algebra. This paper is devoted to a sum-of-squares （SOS） algorithm for computing the minimal H-eigenvalues of tensors with some sign structures called extended essentially nonnegative tensors （EEN-tensors）, which includes nonnegative tensors as a subclass. In the even-order symmetric case, we first discuss the positive semi-definiteness of EEN-tensors, and show that a positive semi-definite EEN-tensor is a non- negative tensor or an M-tensor or the sum of a nonnegative tensor and an M-tensor, then we establish a checkable sufficient condition for the SOS decomposition of EEN-tensors. Finally, we present an efficient algorithm to compute the minimal H-eigenvalues of even-order symmetric EEN-tensors based on the SOS decomposition. Numerical experiments are given to show the efficiency of the proposed algorithm.

关 键 词：Extended essentially nonnegative tensor （EEN-tensor） positive semi-definiteness H-eigenvalue sum-of-squares （SOS） polynomial 

分 类 号：T0[一般工业技术]