Nonlinear Algebra in an ACORN

The following sections are included:

• Dedication
• Foreword
• Preface
• Acknowledgments
• Contents

The following sections are included:

• Two Simple Optimization Schemes
• The Levenberg-Marquardt Optimization Algorithm
• General Characteristics of Optimization Algorithms
• Optimization Algorithms and Dynamical Systems
• Attractors and Basins of Attraction
• Basins of Attraction for Optimization Algorithms

The following sections are included:

• Transform Nonlinear Equations into a Matrix Equation
• Lee’s Iteration Scheme for Solving Nonlinear Equation Set
• Fine-Tuning the Path to Convergence
• From Equation Solver to Optimizer

In this chapter we will discuss how the ACORN optimization algorithm distinguishes itself from the rest of the pack available commercially by simple examples.

In this and later chapters, we will discuss examples of practical applications of the ACORN algorithm.

In high energy physics experiments, measuring the high energy particle beam profile inside the accelerator beam-line is the first step to quantify the beam quality achieved by the accelerator. The beam profile is measured by sticking metallic wire inside the vacuum chamber of the beam-line to obtain 1-D or 2-D beam intensity profiles. From the raw data, physicists attempt to build mathematical model to describe the beam intensity profile so that they have a better description and characterization of the beam.

The following sections are included:

• Artificial Neural Networks — A Brief Review
• Deep-learning Neural Networks and Optimization in High-dimensional Systems
• Examples of Neural Network with ACORN Learning Algorithm
• Looking Ahead

Since the first draft of this book, the responses are encouraging. Here are some enthusiastic remarks we received from aficionados of our work.

“Congratulations on publishing the textbook on nonlinear algebra. It is going to transform the way mathematics is taught, from algebra to calculus to topography and multivariate space etc. Great contribution to our knowledge”…

The following sections are included:

• Appendix A: Code for Plotting the Basin of Attraction (BOA) in Octave
• Appendix B: ACORN Code Implementation in Octave
  • B.1 Listing of Driver Code for ACORN
• Appendix C: Details for Optimizing the Objective Function in NN8 (Chapter 5)
• Appendix D: Details for Optimizing the Objective Function in NN481 (Chapter 5)

The following section is included:

• Bibliography