The purpose of this detailed introduction to the fundamental ideas of numerical linear algebra is to clarify how to perform computations using numerical methods efficiently and accurately. It is aimed at anyone (advanced undergraduate, graduate student, or scientist) working in disciplines in which numerical methods are used. The prerequisites are an introductory course in linear algebra and the knowledge of a high-level programming language.

The book contains more than enough material for a two-quarter course. It is organized so that, for any major topic, algorithms are introduced early, sometimes by postponing the discussion of theoretical aspects. Furthermore, readers are encouraged to develop their own programs to implement the algorithms and to solve the exercises that form an integral part of the book.

Chapter 1 discusses Gaussian elimination and its variants. In particular, it covers systems of linear equations, triangular systems, Cholesky decomposition for positive definite systems, Gaussian elimination and LU decomposition,  Gaussian  elimination with pivoting, and banded systems. The chapter terminates with a discussion of how the computer system architecture influences the choice of algorithms.

Chapter 2 is devoted to the problems of sensitivity of linear systems (that is, how small changes to the coefficients affect the solution) and accuracy of the computation (how round-off errors made during the computation influence the results). Chapter 3 discusses the least-squares problem and some powerful tools for its solution, including orthogonal matrices, rotators, and reflectors. Furthermore, Watkins presents the Gram-Schmidt method and the geometric approach for the solution of the least-squares problem.

Chapters 4 and 5 are devoted to eigenvalues and eigenvectors. The topics covered in chapter 4 include basic properties, the power method, similarity transformations, reduction to Hessenberg and tridiagonal forms, and the QR algorithm and its implementations. Chapter 5 introduces the  notion  of invariant subspaces (a generalization of eigenspaces) and shows the relations among subspace iteration, simultaneous iteration, and the QR algorithm. The uniqueness theorems for the Hessenberg form are then given, and the related  Lanczos  algorithm is presented. The last part of the chapter gives other algorithms of QR type and discusses the sensitivity of eigenvalues and eigenvectors.

Chapter 6 introduces additional methods for the symmetric eigenvalue problem, which has a wide variety of solutions. The chapter presents the Jacobi method, the slicing method, and Cuppen’s divide and conquer method.

Finally, chapter 7 introduces the concept of singular value decomposition (SVD) of a matrix. Watkins then shows how to compute the SVD and presents some of its basic applications and its relationship to the least-squares problem.

The book contains two appendices. Appendix A covers fast rotators; Appendix B gives some information on available software packages for matrix manipulation. The references are divided into two parts: the first part consists of major text and reference books, while the second part lists other books and papers cited in the text.