This unique volume covers two fundamental elements of computational methods — numbers and functions. It provides an in-depth discussion of the behaviors of numbers, including both real and complex numbers. The discussion leads to the important closure properties of numbers, ensuring solution consistence and existence, and also possible failure in numerical computations in science and engineering.This text then introduces types of functions that take numbers as independent variables and produce a single number. Approaches for constructing inverse functions are also provided. Frequently used basis functions are introduced, with detailed studies on their properties and behaviors. Techniques are presented for constructing basis functions and their use in function approximation in computational methods.
Taking advantage of the fast development of computer hardware and software, our discussions will largely be accompanied by computer codes, so that the concepts, theories, and formulations can be immediately demonstrated using examples with results plotted. We will use primarily Python in the first handful of volumes on more fundamental topics and possibly other suitable languages for more advanced topics.
This book is the first volume of the book series. It covers the two most fundamental concepts: numbers and functions. A number is the basic building block for all objects used in computational methods, including functions, vectors, matrices, tensors. A function is the most essential device that converts numbers to numbers, and more generally, objects to objects.
Our discussion will be applications-focused without too much on theoretic proofs and overly lengthy derivations. Thus, much effort has been placed on coding to demonstrate the concepts, theory, and techniques. Python codes are used because they are currently, and in the foreseeable time, the most popular and easy-to-use tool. The idea is to break down all these difficult topics into simpler ones and conquere one by one immediately using the codes provided. As Python codes are provided, readers can easily see how the theory is formulated and how the solutions are obtained in terms of formulas, numbers, and graphs. Readers may also deepen their understanding via playing with codes and even further develop his/her own codes for solving other related problems. The book is written in Jupyter Notebook format, so that description of theory, formulation, derivation, and coding can all be done in a unified document.
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