Preface

Numerical analysis is not a catalogue of formulas. It is the discipline that explains how mathematical problems are transformed into reliable computational procedures. A numerical method must be derived, its assumptions must be visible, its error must be estimated, its stability must be understood, and its computational behavior must be tested on meaningful examples.

This book was written with that philosophy. It presents numerical analysis as a structured bridge between rigorous mathematics and scientific computation. Each chapter begins with the mathematical problem, derives the numerical method, analyzes its accuracy and stability, gives algorithmic formulations, and then places the method in the broader computational landscape. The goal is not only to teach how a method is used, but also why it works, when it fails, and how it should be diagnosed in practice.

The text is intended for advanced undergraduate students, graduate students, researchers, and instructors who need a serious and self-contained treatment of modern numerical methods. It can be used as a textbook for a two-semester sequence in numerical analysis, as a reference for graduate courses in computational mathematics, or as a technical guide for researchers working with differential equations, optimization, spectral methods, and fractional models.

A distinctive feature of the book is its integration of classical and modern topics. The early chapters cover floating-point arithmetic, nonlinear equations, interpolation, linear systems, least squares, numerical differentiation, numerical integration, and ordinary differential equations. The later chapters move toward partial differential equations, spectral and pseudospectral methods, optimization, nonlinear systems, and numerical methods for fractional differential equations. The final chapter is deliberately extensive because fractional models require a careful combination of analysis, quadrature, memory evaluation, and residual diagnostics.

The book emphasizes worked examples. A method is easier to understand when the reader sees it applied to a concrete problem, step by step. For this reason, examples are used to illustrate derivations, matrix assembly, error behavior, stability restrictions, quadrature rules, spectral differentiation, nonlinear iterations, and fractional memory evaluation. The examples are not decorative; they are part of the mathematical development.

The exercises are also an essential part of the text. They are organized from basic computations to advanced and research-level questions. Some exercises ask the reader to verify formulas. Others ask for proofs, implementation, comparison of algorithms, diagnostic experiments, or extensions of the methods to nonstandard models. This structure allows the book to serve both as a teaching text and as a starting point for research projects.

The computational point of view is practical. Algorithms are written in a form that can be implemented directly. Whenever possible, the text explains how to test a method: residuals, convergence tables, stability checks, condition numbers, error profiles, and comparison with exact or manufactured solutions. The reader is encouraged to treat computation not as a black box, but as a mathematical experiment that must be designed and interpreted carefully.

The final chapter, on fractional differential equations, receives special attention. It reviews the principal definitions of fractional derivatives, with emphasis on Caputo and Caputo–Katugampola operators. It derives Volterra integral formulations, power-function formulas, fractional clocks, classical schemes such as L1, Alikhanov-type formulas, Grunwald–Letnikov methods, Lubich convolution quadrature, PECE methods, fast convolution, and high-order Chebyshev–Lobatto panel formulas. It also explains how singular memory kernels can be evaluated by beta-function weights and how fractional residuals can be minimized in parameterized models.

The intended message of the book is simple: a numerical method is a mathematical object, an algorithmic object, and an experimental object at the same time. A good numerical analyst must understand all three aspects.

Álvaro H. Salas S.
Universidad Nacional de Colombia, Sede Manizales
FIZMAKO Research Group