Foundations of Applied Mathematics Volume 2: Algorithms, Approximation, Optimization

This book is the second volume in a planned 4-volume set of textbooks for an undergraduate program in applied mathematics.  This curriculum has been implemented at Brigham Young University as the Applied & Computational Mathematics Emphasis (ACME) program.  ACME is a tightly integrated set of four year-long courses that are taken by juniors and seniors.  Each course is 4 credit hours per semester including computer labs.  The volume reviewed here is intended for use in a two-semester junior-level course sequence on algorithms, approximation, and optimization.

 

The first section of the book contains material that would typically be taught in a sophomore-level course on data structures and algorithms.  This is material that undergraduates in mathematics don't typically encounter unless they double major or minor in computer science. It provides important background that is used extensively in later sections of the book.

 

The second section of the book covers floating-point arithmetic, polynomial approximation, Fourier series, the discrete Fourier transform, and wavelets that might normally be found in an introductory undergraduate course on numerical analysis.  However, there is no treatment of methods the numerical solution of ODE's or PDE's.  These topics will presumably appear in volume 4 of the series, Modeling with Dynamics and Control.  Similarly, there is no discussion of numerical linear algebra in this volume, but some numerical linear algebra material does appear in the first volume of the series, Mathematical Analysis.

 

The final section of the book covers material that would typically be found in an introductory course on optimization at the advanced undergraduate or graduate level.  The selection of topics here is somewhat old fashioned, with little on recent developments in first-order methods for convex optimization and applications of optimization to machine learning.  However, applications to machine learning will presumably appear in the third volume of the series, Modeling with Uncertainty and Data.

 

The authors have been very selective in the material that they include, so none of the three parts are really as thorough as a typical textbook intended for a semester long 3 credit-hour course.  Even so, the book is close 800 pages long and there is more than enough material for a two semester course sequence.

 

With this selection of topics, the authors have made a strong statement that undergraduate programs in applied mathematics should include more computer science, numerical analysis, and optimization. The material presented in this textbook is essential to modern computational applied mathematics.  It is presented in a very clear and appropriately rigorous way.  However, it would be difficult to use this volume as a textbook for any conventional course outside of a sequence of courses similar to ACME.  Faculty may find this book and the entire series most useful as an inspiration and starting point for discussions about what should be included in a modern applied and computational mathematics curriculum.

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Brian Borchers is a professor mathematics at New Mexico Tech and the editor of MAA Reviews.