You are here

Difference Equations: From Rabbits to Chaos

Paul Cull, Mary Flahive, and Robby Robson
Springer Verlag
Publication Date: 
Number of Pages: 
Undergaduate Texts in Mathematics
[Reviewed by
Henry Ricardo
, on

Recognizing the increasing importance of difference equation models in such areas as mathematics, computer science, engineering, and biology and decrying the unevenness of treatments found in discrete mathematics texts, the authors aimed to write a book that provides a solid foundation in the field and is accessible to undergraduates.  The entire (short) first chapter of this book uses Fibonacci’s rabbit problem to introduce basic concepts of difference equations.  Subsequent chapters cover homogeneous linear recurrence relations, finite difference equations, generating functions, nonnegative difference equations, Leslie’s population matrix model, matrix difference equations, modular recurrences, computational complexity, and some nonlinear recurrences.  Four appendices cover (additional) worked examples of kth-order linear recurrences, complex numbers, basic linear algebra, and an algorithmic treatment of Marden’s method for calculating the number of roots of a polynomial within the unit circle.

Intended for junior-senior level students (in contradiction of the back cover blurb) in one of the mathematical sciences, this text is written in an informal style and with an “algorithmic spirit.”  Many explicit algorithms written in pseudocode are given, although the authors admit that MATLAB, Maple, and Mathematica have good packages that solve difference equations and recurrence relations.   There are many worked examples and the exercise sets are good, but there are no answers at the back of the book.  There is apparently no Instructor’s Manual available.

This text, although requiring some computational and/or theoretical maturity, seems more suitable to an undergraduate course than such recent books as Difference Equations: An Introduction with Applications (Second Edition) by W. G. Kelley and A. C. Peterson (San Diego: Academic Press, 2000) or An Introduction to Difference Equations (Third Edition) by S. Elaydi (New York: Springer, 2005).   The text under review does not go as deeply into the subject as these other books, but it provides an accessible introduction to the material and a firm foundation for applications in various scientific fields.

Henry Ricardo ( is Professor of Mathematics at Medgar Evers College of The City University of New York and Secretary of the Metropolitan NY Section of the MAA. His book, A Modern Introduction to Differential Equations, was published by Houghton Mifflin in January, 2002; and he is currently writing a linear algebra text.

Preface * Fibonacci Numbers * Homogeneous Linear Recurrence Relations * Finite Difference Equations * Generating Functions * Nonnegative Difference Equations * Leslie's Population Matrix Model * Matrix Difference Equations * Modular Recurrences * Computational Complexity * Some Nonlinear Recurrences * Appendix A: Worked Examples * Appendix B: Complex Numbers * Appendix C: Highlights of Linear Algebra * Appendix D: Roots in the Unit Circle * References * Index