1st Edition

Zeroing Dynamics, Gradient Dynamics, and Newton Iterations



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ISBN 9781498753760
Published December 1, 2015 by CRC Press
310 Pages 294 B/W Illustrations

USD $175.00

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Book Description

Neural networks and neural dynamics are powerful approaches for the online solution of mathematical problems arising in many areas of science, engineering, and business. Compared with conventional gradient neural networks that only deal with static problems of constant coefficient matrices and vectors, the authors’ new method called zeroing dynamics solves time-varying problems.

Zeroing Dynamics, Gradient Dynamics, and Newton Iterations is the first book that shows how to accurately and efficiently solve time-varying problems in real-time or online using continuous- or discrete-time zeroing dynamics. The book brings together research in the developing fields of neural networks, neural dynamics, computer mathematics, numerical algorithms, time-varying computation and optimization, simulation and modeling, analog and digital hardware, and fractals.

The authors provide a comprehensive treatment of the theory of both static and dynamic neural networks. Readers will discover how novel theoretical results have been successfully applied to many practical problems. The authors develop, analyze, model, simulate, and compare zeroing dynamics models for the online solution of numerous time-varying problems, such as root finding, nonlinear equation solving, matrix inversion, matrix square root finding, quadratic optimization, and inequality solving.

Table of Contents

Time-Varying Root Finding
Time-Varying Square Root Finding
Introduction
Problem Formulation and Continuous-Time (CT) Models
S-DTZD Model and Newton Iteration
Illustrative Examples

Time-Varying Cube Root Finding
Introduction
ZD Models for Time-Varying Case
Simplified ZD Models for Constant Case and Newton Iteration
Illustrative Examples

Time-Varying 4th Root Finding
Introduction
Problem Formulation and ZD Models
GD Model
Illustrative Examples

Time-Varying 5th Root Finding
Introduction
ZD Models for Time-Varying Case
Simplified ZD Models for Constant Case and Newton Iteration
Illustrative Examples
Appendix: Extension to Time-Varying pth Root Finding

Nonlinear Equation Solving
Time-Varying Nonlinear Equation Solving
Introduction
Problem Formulation and Solution Models
Convergence Analysis
Illustrative Example

Static Nonlinear Equation Solving
Problem Formulation and Continuous-Time Models
DTZD Models
Comparison between CTZD Model and Newton Iteration
Further Discussion to Avoid Local Minimum

System of Nonlinear Equations Solving
Problem Formulation and CTZD Model
Discrete-Time Models

Matrix Inversion
ZD Models and Newton Iteration
Introduction
ZD Models
Choices of Initial State X0
Choices of Step Size h
Illustrative Examples
New DTZD Models Aided with Line-Search Algorithm

Moore–Penrose Inversion
Introduction
Preliminaries
ZD Models for Moore–Penrose Inverse
Comparison between ZD and GD Models
Simulation and Verification
Application to Robot Arm

Matrix Square Root Finding
ZD Models and Newton Iteration
Introduction
Problem Formulation and ZD Models
Link and Explanation to Newton Iteration
Line-Search Algorithm
Illustrative Examples

ZD Model Using Hyperbolic Sine Activation Functions
Model and Activation Functions
Convergence Analysis
Robustness Analysis
Illustrative Examples

Time-Varying Quadratic Optimization
ZD Models for Quadratic Minimization
Introduction
Problem Formulation and CTZD Model
DTZD Models
GD Models
Illustrative Example

ZD Models for Quadratic Programming
Introduction
CTZD Model
DTZD Models
Illustrative Examples

Simulative and Experimental Application to Robot Arms
Problem Formulation and Reformulation
Solution Models
Computer Simulations
Hardware Experiments

Time-Varying Inequality Solving
Linear Inequality Solving
Introduction
Time-Varying Linear Inequality
Constant Linear Inequality
Illustrative Examples
System of Time-Varying Linear Inequalities
Illustrative Examples

System of Time-Varying Nonlinear Inequalities Solving
Introduction
Problem Formulation
CZD Model and Convergence Analysis
MZD Model and Convergence Analysis
Illustrative Example

Application to Fractal
Fractals Yielded via Static Nonlinear Equation
Introduction
Complex-Valued ZD Models
Illustrative Examples

Fractals Yielded via Time-Varying Nonlinear Equation
Introduction
Complex-Valued ZD Models
Illustrative Examples

A summary appears at the end of each chapter.

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Author(s)

Biography

Yunong Zhang is a professor in the School of Information Science and Technology at Sun Yat-sen University. He is also with the SYSU-CMU Shunde International Joint Research Institute for cooperative research. He has published more than 375 scientific works of various types and has been a winner of the Best Paper Award of ISSCAA and the Best Paper Award of ICAL. He was among the 2014 Highly Cited Scholars of China. His main research interests include neural networks, robotics, computation, and optimization. He earned a PhD from the Chinese University of Hong Kong.

Lin Xiao is a lecturer in the College of Information Science and Engineering at Jishou University. His current research interests include neural networks, intelligent information processing, robotics, and related areas. He earned a PhD from Sun Yat-sen University.

Zhengli Xiao is currently pursuing an MS in the Department of Computer Science in the School of Information Science and Technology at Sun Yat-sen University. He is also with the SYSU-CMU Shunde International Joint Research Institute for cooperative research. His current research interests include neural networks, intelligent information processing, and learning machines. He earned a BS in software engineering from Changchun University of Science and Technology.

Mingzhi Mao is an associate professor in the School of Information Science and Technology at Sun Yat-sen University. His main research interests include intelligence algorithms, software engineering, and management information systems. He earned a PhD from the Department of Computer Science at Sun Yat-sen University.