×

Introduction to experimental mathematics. (English) Zbl 1372.68001

Cambridge Mathematical Textbooks. Cambridge: Cambridge University Press (ISBN 978-1-107-15613-5/hbk). xv, 303 p. (2017).
This book presents an in-depth analysis of the field of experimental mathematics, namely the study of mathematical theorems and concepts using computational tools. The book begins with a gentle introduction to the field via the presentation of a number of mathematical problems that can be studied with computers, such as the Tower of Hanoi, the Fermat primes, the Collatz conjecture and Pythagorean triples. The second chapter presents an introduction to the Maple programming language, which is used throughout the book for the computational experimentation. The key concepts of the language and sample programs are presented, aimed at a user without any familiarity with the language. The next chapter considers the areas of iteration and recursion in computer programs and many code samples, based on mathematical problems, are provided to expose the power of the Maple programming language.
The fourth chapter considers the visualization of the results and the creation of charts, along with simple data analysis tasks, including problems such as data transformation, linear and nonlinear fitting and the display of probability distributions. This is followed by a lengthy chapter considering the idea of symbolic inversion, looking, among others, at how one represents mathematical notions programmatically. Here, many types of problems are examined, such as continuous fractions, sequences of integers such as Fibonacci and Bernoulli numbers, graph puzzles, paradoxes due to floating-point arithmetic, generating functions and Catalan numbers.
The sixth chapter studies pseudorandomness, including the theory behind the generation of pseudo-random numbers, the Mersenne twister algorithm, Monte Carlo methods, and their use in algorithms for approximate primality testing. The last two chapters consider the important aspect of computational time and memory use as well as techniques to reduce the algorithmic complexity via efficient coding and data structures, and applications of experimental mathematics in graph theory and linear algebra, including the four color theorem.
Each chapter in this very interesting and innovative book concludes with a large list of exercises for the interested student, longer exercises for group work and notes for further reading.

MSC:

68-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to computer science
00A05 Mathematics in general
65Y15 Packaged methods for numerical algorithms
68N15 Theory of programming languages
68W30 Symbolic computation and algebraic computation

Software:

Maple
PDFBibTeX XMLCite