Theory and Computation of Tensors: Multi-Dimensional Arrays

Description

Theory and Computation of Tensors: Multi-Dimensional Arrays investigates theories and computations of tensors to broaden perspectives on matrices. Data in the Big Data Era is not only growing larger but also becoming much more complicated. Tensors (multi-dimensional arrays) arise naturally from many engineering or scientific disciplines because they can represent multi-relational data or nonlinear relationships. Provides an introduction of recent results about tensorsInvestigates theories and computations of tensors to broaden perspectives on matricesDiscusses how to extend numerical linear algebra to numerical multi-linear algebraOffers examples of how researchers and students can engage in research and the applications of tensors and multi-dimensional arrays Content: Front Cover Theory and Computation of Tensors: Multi-Dimensional Arrays Copyright Preface Contents Part I: General Theory Chapter 1: Introduction and Preliminaries 1.1 What Are Tensors? 1.2 Basic Operations 1.3 Tensor Decompositions 1.4 Tensor Eigenvalue Problems Chapter 2: Generalized Tensor Eigenvalue Problems 2.1 A Uni ed Framework 2.2 Basic De nitions 2.3 Several Basic Properties 2.4 Real Tensor Pairs 2.5 Sign-Complex Spectral Radius 2.6 An Illustrative Example Part II: Hankel Tensors Chapter 3: Fast Tensor-Vector Products 3.1 Hankel Tensors. 3.2 Exponential Data Fitting3.3 Anti-Circulant Tensors 3.4 Fast Hankel Tensor-Vector Product 3.5 Numerical Examples Chapter 4: Inheritance Properties 4.1 Inheritance Properties 4.2 The First Inheritance Property of Hankel Tensors 4.3 The Second Inheritance Property of Hankel Tensors 4.4 The Third Inheritance Property of Hankel Tensors Part III: M-Tensors Chapter 5: De nitions and Basic Properties 5.1 Preliminaries 5.2 Spectral Properties of M-Tensors 5.3 Semi-Positivity 5.4 Monotonicity 5.5 An Extension of M-Tensors 5.6 Summation. Chapter 6: Multilinear Systems with M-Tensors 6.1 Motivations 6.2 Triangular Equations 6.3 M-Equations and Beyond 6.4 Iterative Methods for M-Equations 6.5 Perturbation Analysis of M-Equations 6.6 Inverse Iteration Appendix Bibliography Subject Index Back Cover.