This text aims to help the reader develop a working knowledge of computational matrix analysis, which is sometimes simply called numerical linear algebra. The intended audience includes graduate students in mathematics, computer science, engineering and the sciences. Prerequisites include calculus and enough exposure to linear algebra to include some familiarity with singular value decomposition and pseudoinverses. Matrix Analysis for Scientists and Engineers, by the same author, would provide an adequate background as would at least the first part of Stewart’s Introduction to Matrix Computations or Strang’s Linear Algebra and Its Applications.
After a short chapter on notation and preliminary material, the author addresses fundamental concepts common to nearly all numerical computation in the next three chapters. Perhaps the most important of these deals with the nature and limitations of finite precision arithmetic. After that the author discusses some closely related topics: conditioning (sensitivity of a problem to its data), numerical stability (robustness of the algorithm used to solve the problem), and rounding analysis.
The rest of the book is devoted to topics specific to numerical linear algebra. Most of these are standard, but the treatment here is clear, to the point, and shows an experienced practitioner’s awareness of what can go wrong. These topics include Gaussian elimination, other techniques for solving linear systems, least squares problems, the computation of eigenvalues and eigenvectors, and some advanced topics in matrix factorization. A final chapter describes some applications; these include computation of matrix functions (such as the matrix exponential) and solution of some matrix equations of particular interest in control theory (the Riccati and Sylvester equations).
This is a book for people intending to do numerical linear algebra who also want to understand the theoretical basis of their computations. It is intermediate in sophistication between more introductory accounts and Golub and Van Loan’s canonical Matrix Computations. The current book takes on fewer topics than the latter but is more accessible and has plenty of practical advice as well.
Bill Satzer (email@example.com) is a senior intellectual property scientist at 3M Company, having previously been a lab manager at 3M for composites and electromagnetic materials. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.