Numerical linear algebra (NLA) is no longer merely a subtopic of linear algebra or of numerical analysis. In recent years, NLA has become an independent subject for research and teaching and its techniques are important components of scientific computing.
Datta’s book is primarily intended for a first course in NLA for students in mathematics, computer science, and engineering at the advanced undergraduate and beginning graduate levels. It is also described as suitable for self-study and for use as a reference by scientists and engineers.
In terms of content, there is a slightly reworked treatment of the quadratic eigenvalue problem (including the Jacobi-Davidson method) and what Nicholas Higham (Siam Review 41 (1999), p. 608) has called “the best textbook presentation of the generalized eigenvalue problem that I have seen.” The new edition has a separate chapter on iterative methods, emphasizing the development of major Krylov-subspace methods for the solution of linear systems. The Singular Value Decomposition (SVD) is defined early (Chapter 2) and is given a full treatment in Chapter 7. SVD applications to image processing have been added to the interesting applications appearing in the first edition.
The second edition of this book preserves and enhances the pedagogical strengths of the first edition. The text is very readable. There are end-of-chapter summaries, suggestions for further reading, a MATLAB-based toolkit, and a list of key terms. The second edition contains many more pictures and figures; and there are footnotes offering brief biographical sketches of (deceased) prominent practitioners of NLA: Gauss, Hessenberg, Cholesky, Golub, Givens…
This is a second edition that shows care in its preparation. The attractive features of the first edition have been enhanced and minor criticisms have been addressed. Anyone teaching a course in NLA should consider this as both a text and a reference.
Henry Ricardo (firstname.lastname@example.org) has retired from Medgar Evers College (CUNY), but continues to serve as Governor of the Metropolitan NY Section of the MAA. He is the author of A Modern Introduction to Differential Equations (Second Edition). His linear algebra text was published in October 2009 by CRC Press.
Chapter 1: Linear Algebra Problems, Their Importance, and Computational Difficulties;
Chapter 2: A Review of Some Required Concepts from Core Linear Algebra;
Chapter 3: Floating Point Numbers and Errors in Computations;
Chapter 4: Stability of Algorithms and Conditioning of Problems;
Chapter 5: Gaussian Elimination and LU Factorization;
Chapter 6: Numerical Solutions of Linear Systems;
Chapter 7: QR Factorization, Singular Value Decomposition, and Projections;
Chapter 8: Least-Squares Solutions to Linear Systems;
Chapter 9: Numerical Matrix Eigenvalue Problems;
Chapter 10: Numerical Symmetric Eigenvalue Problem and Singular Value Decomposition;
Chapter 11: Generalized and Quadratic Eigenvalue Problems;
Chapter 12: Iterative Methods for Large and Sparse Problems: An Overview;
Chapter 13: Key Terms in Numerical Linear Algebra;