In this book, we study theoretical and practical aspects of computing methods for mathematical modelling of nonlinear systems. A number of computing techniques are considered, such as methods of operator approximation with any given accuracy; operator interpolation techniques including a non-Lagrange interpolation; methods of system representation subject to constraints associated with concepts of causality, memory and stationarity; methods of system representation with an accuracy that is the best within a given class of models; methods of covariance matrix estimation;
methods for low-rank matrix approximations; hybrid methods based on a combination of iterative procedures and best operator approximation; and
methods for information compression and filtering under condition that a filter model should satisfy restrictions associated with causality and different types of memory.

As a result, the book represents a blend of new methods in general computational analysis,
and specific, but also generic, techniques for study of systems theory ant its particular
branches, such as optimal filtering and information compression.

- Best operator approximation,
- Non-Lagrange interpolation,
- Generic Karhunen-Loeve transform
- Generalised low-rank matrix approximation
- Optimal data compression
- Optimal nonlinear filtering

• Front Cover 2
• Methods of Nonlinear Analysis 5
• Copyright Page 6
• Preface 9
• Contents 17
• Chapter 1. First- and Second-order Differential Equations 25
  • 1.1. Introduction 25
  • 1.2. The First-order Linear Differential Equation 26
  • 1.3. Fundamental Inequality 27
  • 1.4. Second-order Linear Differential Equations 29
  • 1.5. Inhomogeneous Equation 31
  • 1.6. Lagrange Variation of Parameters 32
  • 1.7. Two-point Boundary Value Problem 34
  • 1.8. Connection with Calculus of Variations 35
  • 1.9. Green’s Functions 36