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Theory and Computation of Tensors: Multi-Dimensional Arrays [Pehme köide]

4.00/5 (4 hinnangut Goodreads-ist)
(Ph.D student, School of Mathematical Sciences, Fudan University, Shanghai, P.R. of China), (Professor,School of Mathematical Sciences, Fudan University, Shanghai, China)
  • Formaat: Paperback / softback, 148 pages, kõrgus x laius: 229x152 mm, kaal: 180 g
  • Ilmumisaeg: 15-Aug-2016
  • Kirjastus: Academic Press Inc
  • ISBN-10: 0128039531
  • ISBN-13: 9780128039533
Teised raamatud teemal:
Theory and Computation of Tensors: Multi-Dimensional ArraysSuurem pilt
  • Formaat: Paperback / softback, 148 pages, kõrgus x laius: 229x152 mm, kaal: 180 g
  • Ilmumisaeg: 15-Aug-2016
  • Kirjastus: Academic Press Inc
  • ISBN-10: 0128039531
  • ISBN-13: 9780128039533
Teised raamatud teemal:
Püsilink: https://www.kriso.ee/db/9780128039533.html
Märksõnad:

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 tensors
  • Investigates theories and computations of tensors to broaden perspectives on matrices
  • Discusses how to extend numerical linear algebra to numerical multi-linear algebra
  • Offers examples of how researchers and students can engage in research and the applications of tensors and multi-dimensional arrays

Arvustused

"The book should be useful as a reference for research workers in linear algebra, operator theory, mathematical physics and numerical analysis." --MathSciNet

"Roughly half the book is devoted to defining and developing properties of tensors, and the other half to algorithms." --MAA Reviews

Muu info

This timely book introduces tensors, or multi-dimensional arrays, and investigates theories and computations of tensors to broaden perspectives on matrices, including how to extend numerical linear algebra to numerical multi-linear algebra
Preface v
I General Theory
1(36)
1 Introduction and Preliminaries
3(8)
1.1 What Are Tensors?
3(3)
1.2 Basic Operations
6(2)
1.3 Tensor Decompositions
8(2)
1.4 Tensor Eigenvalue Problems
10(1)
2 Generalized Tensor Eigenvalue Problems
11(26)
2.1 A Unified Framework
11(2)
2.2 Basic Definitions
13(1)
2.3 Several Basic Properties
14(6)
2.3.1 Number of Eigenvalues
14(1)
2.3.2 Spectral Radius
15(1)
2.3.3 Diagonalizable Tensor Pairs
15(1)
2.3.4 Gershgorin Circle Theorem
16(3)
2.3.5 Backward Error Analysis
19(1)
2.4 Real Tensor Pairs
20(6)
2.4.1 The Crawford Number
21(1)
2.4.2 Symmetric-Definite Tensor Pairs
22(4)
2.5 Sign-Complex Spectral Radius
26(8)
2.5.1 Definitions
26(1)
2.5.2 Collatz-Wielandt Formula
27(2)
2.5.3 Properties for Single Tensors
29(2)
2.5.4 The Componentwise Distance to Singularity
31(2)
2.5.5 Bauer-Fike Theorem
33(1)
2.6 An Illustrative Example
34(3)
II Hankel Tensors
37(42)
3 Fast Tensor-Vector Products
39(20)
3.1 Hankel Tensors
39(1)
3.2 Exponential Data Fitting
40(5)
3.2.1 The One-Dimensional Case
40(2)
3.2.2 The Multidimensional Case
42(3)
3.3 Anti-Circulant Tensors
45(5)
3.3.1 Diagonalization
46(1)
3.3.2 Singular Values
47(2)
3.3.3 Block Tensors
49(1)
3.4 Fast Hankel Tensor-Vector Product
50(3)
3.5 Numerical Examples
53(6)
4 Inheritance Properties
59(20)
4.1 Inheritance Properties
59(2)
4.2 The First Inheritance Property of Hankel Tensors
61(5)
4.2.1 A Convolution Formula
61(2)
4.2.2 Lower-Order Implies Higher-Order
63(2)
4.2.3 SOS Decomposition of Strong Hankel Tensors
65(1)
4.3 The Second Inheritance Property of Hankel Tensors
66(11)
4.3.1 Strong Hankel Tensors
66(2)
4.3.2 A General Vandermonde Decomposition of Hankel Matrices
68(3)
4.3.3 An Augmented Vandermonde Decomposition of Hankel Tensors
71(4)
4.3.4 The Second Inheritance Property of Hankel Tensors
75(2)
4.4 The Third Inheritance Property of Hankel Tensors
77(2)
III M-Tensors
79(46)
5 Definitions and Basic Properties
81(16)
5.1 Preliminaries
81(2)
5.1.1 Nonnegative Tensor
81(1)
5.1.2 From M-Matrix to M-Tensor
82(1)
5.2 Spectral Properties of M-Tensors
83(1)
5.3 Semi-Positivity
84(6)
5.3.1 Definitions
84(1)
5.3.2 Semi-Positive Z-Tensors
85(2)
5.3.3 Proof of Theorem 5.7
87(2)
5.3.4 General M-Tensors
89(1)
5.4 Monotonicity
90(3)
5.4.1 Definitions
90(1)
5.4.2 Properties
90(3)
5.4.3 A Counter Example
93(1)
5.4.4 A Nontrivial Monotone Z-Tensor
93(1)
5.5 An Extension of M-Tensors
93(2)
5.6 Summation
95(2)
6 Multilinear Systems with M-Tensors
97(28)
6.1 Motivations
97(2)
6.2 Triangular Equations
99(3)
6.3 M-Equations and Beyond
102(6)
6.3.1 M-Equations
102(2)
6.3.2 Nonpositive Right-Hand Side
104(1)
6.3.3 Nonhomogeneous Left-Hand Side
105(1)
6.3.4 Absolute M-Equations
106(1)
6.3.5 Banded M-Equation
107(1)
6.4 Iterative Methods for M-Equations
108(6)
6.4.1 The Classical Iterations
109(2)
6.4.2 The Newton Method for Symmetric M-Equations
111(1)
6.4.3 Numerical Tests
112(2)
6.5 Perturbation Analysis of M-Equations
114(4)
6.5.1 Backward Errors of Triangular M-Equations
115(1)
6.5.2 Condition Numbers
116(2)
6.6 Inverse Iteration
118(7)
Bibliography 125(10)
Subject Index 135
Yimin Wei is a Professor at the School of Mathematical Sciences, Fudan University, Shanghai, P.R. of China. He has has published three English books and over 100 research papers in international journals. His studies on tensors are supported by the National Natural Science Foundation of China. Weiyang Ding is a Ph.D student under the supervision of Professor Wei, at the School of Mathematical Sciences, Fudan University, Shanghai, P.R. of China.