Low-rank approximation (LRA) is an abstract framework for approximately solving many highly complex problems in science and engineering within acceptable time limits. Concrete implementations of LRA-based algorithms for specific problems are therefore highly relevant in many applications, and students and practitioners in such areas will benefit from a good textbook on this topic.

Markovsky’s book is marketed by the publisher as the second edition of a book with the same title, written by the same author, published in 2012, and favorably reviewed in Computing Reviews [1]. A cursory inspection of the two editions’ tables of contents reveals, however, that quite a lot of the text has been more or less completely rewritten, thus this second edition is essentially a new book.

The book starts with an introductory chapter that explains the basic ideas and concepts--in particular, the behavioral paradigm of data modeling that is fundamental for most of the content and provides a large set of potential applications for LRA techniques.

The core of the material is then organized in two parts. The first part is dedicated to the discussion of linear modeling problems, and the second part deals with applications and more general (that is, nonlinear) problems. Overall, the book provides a very thorough and complete overview of the topic; it is thus accessible to readers with relatively few prerequisites. Many ideas and principles that need to be exploited in order to implement LRA algorithms for various application scenarios are well described, and the readers receive good advice on choosing a particular method for their problem of interest in a sensible way.

Each chapter ends with a section containing additional useful notes. Even more material is promised on an accompanying website; however, as of writing this review, the website does not seem to exist.

The style of writing is strongly tied to the principle of literate programming, that is, the text is interwoven with code fragments written in MATLAB. Readers may need some time to get accustomed to this form of presentation, which sometimes interrupts the flow of reading, but in many instances this also helps them understand the underlying ideas and concepts. Moreover, this approach is also quite helpful when one effectively wants to implement such algorithms.

Markovsky’s book is certainly well suited for graduate students and more experienced readers, and should also be useful to people who need to apply LRA methods in their daily work.