Dankowicz and Schilder state that the objective of this book “is to present the mathematical methodology known as parameter continuation in a context of a treatment that lends equal importance to the theoretical rigor, algorithm development, and software engineering.” These are the three fundamental pillars of computational science and engineering. It is unfortunate that nowadays one finds papers concerned with just one of these three pillars.

What are continuation methods? Rheinboldt discusses numerical continuation methods that cover embedding methods, homotopy methods, parameter variation methods, and incremental methods, among others [1]. This book concentrates on parameter continuation, finding solutions to equations depending on a parameter or parameters.

The book is divided into six parts. Except for the last part, the epilogue, all other parts consist of several chapters. Each chapter ends with a set of exercises, making the book very useful as a textbook.

The first part, “Design Fundamentals,” starts with a chapter formulating the problem of continuation with an example of minimizing a functional. The authors discuss analytical and numerical solutions to this problem. The second chapter is on encapsulation, the binding of monitor functions, zero functions, continuation variables, and continuation parameters into a coherent package. The third chapter discusses the construction of a restricted continuation problem. Several examples are given, including how to use the software for the construction. The last two chapters of Part 1 develop a toolbox for the software.

Part 2 presents toolbox templates. The first chapter of this part discusses the solution of an ordinary differential equation depending on a set of parameters. The authors describe the collocation method to discretize the problem and the use of vectorization. For each step, the Computational Continuation Core (COCO) code is given. The second chapter gives the collocation continuation problem with several examples and their COCO codes. The third chapter discusses a single-segment continuation problem. Here, the authors give boundary value problems and periodic orbit examples. The fourth chapter deals with multisegment continuation problems. The authors revisit the periodic orbit problem in this context. The second part closes with the variational collocation problem. Each chapter is rich with examples and the appropriate software code for the solution, and each concludes with related exercises.

Part 3 discusses atlas algorithms included in the COCO software. According to the authors, these algorithms are “implementations of a finite-state machine for generating a collection of charts that covers a portion of the solution manifold of the continuation problem.”

Part 4, “Event Handling,” discusses the detection of special points associated with critical properties of the solution.

Part 5 includes three chapters. The first of the three chapters in this section discusses four distinct methods of pointwise adaptation. The next chapter describes the implementation of mesh-preserving discretization, an adaptive change to the discretization error without a change in the number of continuation variables. The last chapter of this part discusses moving meshes. As in all other parts of the book, there are plenty of examples.

The book closes with an epilogue suggesting areas for additional work, such as the parallelization of multidimensional atlas algorithms.

This excellent book guides the reader incrementally through the building of a solution, and shows how to use the COCO software to solve continuation problems. Every chapter includes many detailed examples and a variety of exercises to help readers test their understanding of the material covered. It could be used as a textbook for senior undergraduate and graduate students.