# Zeroing Dynamics, Gradient Dynamics, and Newton Iterations

DOI link for Zeroing Dynamics, Gradient Dynamics, and Newton Iterations

Zeroing Dynamics, Gradient Dynamics, and Newton Iterations book

# Zeroing Dynamics, Gradient Dynamics, and Newton Iterations

DOI link for Zeroing Dynamics, Gradient Dynamics, and Newton Iterations

Zeroing Dynamics, Gradient Dynamics, and Newton Iterations book

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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.

*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 *4*th Root Finding

Introduction

Problem Formulation and ZD Models

GD Model

Illustrative Examples

Time-Varying *5*th 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 *p*th 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 *X*_{0}

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.