Philadelphia, PA: SIAM, Society for Industrial and Applied Mathematics. xii, 361 p. $ 34.50 (1997).
This is a text for an one-semester course for graduate students interested in applying methods from numerical linear algebra to scientific computational problems from natural and engineering sciences. The emphasis is on understanding the algorithms involved, the authors are eager to explain why things happen in a certain way, and often use well-chosen small numerical examples. The text is organised into 40 lectures in a pedagogically interesting order, each giving a new aspect, but still selfcontained enough to be readable separately. Topics covered include (in this order): Vectors and matrices: orthogonality, norms, singular value decomposition (SVD) QR factorization, Gram Schmidt, Householder, least squares. Conditioning, floating point arithmetic, stability. Systems of equations: Gauss elimination, pivoting, stability, Cholesky. Eigenvalues: reduction to Hessenberg form, QR algorithm, algorithm for SVD. Iterative algorithms: Arnoldi, GMRES, Lanczos, conjugate gradients.