This book, a translation from the German, is an introduction to numerical linear algebra at the junior/senior undergraduate level. Although the book itself does not specify the prerequisites for reading it, it would seem that, at a minimum, a prospective reader should have some familiarity with basic linear algebra and with programming.

The book is quite short, only about a hundred pages of actual text (plus some appendices). Its European origins are very apparent in the exposition, which is terse and efficient. The author says what needs to be said, but does not engage in a lot of handholding. This is not a chatty or conversational book; it belongs to the “Just the facts, Ma’am” school of exposition.

Computer programs illustrating the basic ideas appear throughout the text in MATLAB, and these programs also appear, in an Appendix, in the language Julia. The exclusive use of these two specific languages rather than pseudocode may limit the potential audience for the book, but the author clearly believes that working in at least one of these languages is important, and teaching that would appear to be one of the goals of the book. Brief introduction to both of these languages appear as appendices.

There are five chapters. The first is an introductory one that introduces MATLAB, discusses certain types of important matrices (triangular and unitary), and also has a section on the execution time of an algorithm. The second chapter discusses direct methods for solving a system of linear equations by factoring the coefficient matrix: specific factorizations discussed here include the triangular (LU) decomposition, the Cholesky factorization and the QR decomposition. Householder reflection matrices, often introduced in connection with the QR decomposition, are not discussed in this section, but are discussed in an appendix at the end of the book.

This is followed by a chapter on error analysis, including discussions of condition numbers.

Chapter IV, which is all of five pages long, gives a very quick introduction to the least squares problem. The normal equations are derived by analysis, and then the QR factorization is employed to give another algorithm. Finally, chapter V looks at eigenvalues and their calculation. Topics touched upon here include power iteration, inverse iteration and the QR algorithm.

The book has an interesting format. In addition to the traditional division of chapters into sections, each section here is further subdivided into separately-numbered portions, many of them only a paragraph or two long. Each of these portions deals with a basic idea or concept: so, for example, section 5, on triangular matrices, begins with a paragraph numbered 5.1 in which lower and upper triangular matrices are defined, then proceeds to paragraph 5.2 where triangularity is expressed in terms of how the matrix treats an ordered basis, etc. Dividing the text into short chunks like this may have some pedagogical advantage in helping the students organize their thoughts, but some people may miss the idea of a smoothly-flowing narrative.

I should also note that the book I received to review was an actual paperback, not an e-book. This may have some significance. Certain terms and phrases throughout the book are printed in blue or red; the preface explains that the blue phrases are (in electronic copies of the book) links to other sections of the book and red phrases are links to outside sources. Out of curiosity I downloaded an e-book from the SpringerLink website (like many universities, Iowa State has a contract with Springer that allows free downloading of most new books published by them); neither the red nor blue links worked on this PDF copy, however. This can cause problems: on one occasion, for example, the authors point out that because eigenvalues are the roots of the characteristic polynomial of a matrix, the Abel-Ruffini theorem implies that there is no formula for determining the eigenvalues of a matrix that has size \(5 \times 5\) or higher. The phrase “Abel-Ruffini theorem” appears in red. Those readers who know that the Abel-Ruffini theorem is the one that says that a polynomial of degree \(5\) or greater cannot be solved in radicals will immediately understand what this statement means, but those who don’t will not, and if they own a physical copy of the book or an e-copy that doesn’t provide links, they will be out of luck. (This problem potentially exists, of course, whenever a phrase appears in red; elsewhere in the book, for example, the authors use the phrase “singular value decomposition”, linked in red, without explanation as to what that term means.)

As a mentioned earlier, this book is very short; it looks almost like a thick pamphlet. In a book of this size, it is inevitable that a number of topics that one might expect to find in a text on this subject are omitted. Iterative methods for solving systems of linear equations, such as the Jacobi, Gauss-Seidel and SOR methods, are not discussed. Neither is the conjugate gradient method. As far as eigenvalue determination goes, the Gershgorin disc theorem appears only in the exercises, and even here we only get a statement of the main result; variations on the theme (such as the fact that if a collection of *p* Gershgorin discs constitute a connected region that is disjoint from the remaining discs, then these *p* discs between them contain exactly *p* eigenvalues) are not mentioned at all. A prospective adopter of this book will want to carefully check to see if all the topics that he or she plans to cover are included.

The inclusion of this book in the *Springer Undergraduate Mathematics Series* is a little puzzling. It is, for one thing, much shorter than most or all of the other books in the SUMS series. Also, the books in that series are, according to Springer’s advertising, “supported by a wealth of examples, problems and fully-worked solutions”. There is certainly not a “wealth” of examples here. There are also relatively few homework problems (62 appear in the main body of the text, and 36 more appear in an appendix), and none of them are accompanied by solutions. Perhaps a better home for this book would have been the *Compact Textbooks in Mathematics* series.

There is a reasonable textbook literature on the subject of numerical linear algebra, including at least two books that were published within months of this one: *Numerical Linear Algebra: Theory and Applications* by Beilina et al. (another Springer book) and Wendland’s *Numerical Linear Algebra*. Both of these books have greater topic coverage than does the book now under review, and my guess is that students would also find either of these two books to be somewhat more reader-friendly.

Mark Hunacek ([email protected]) teaches mathematics at Iowa State University.

## Comments

## Update to Review

It now appears that the problems that I originally noted with malfunctioning hyperlinks no longer exist. After being contacted by a Springer editor, I downloaded another copy of the book from springerlink.com and found that the hyperlinks now work fine.