Iterative solution methods. (English) Zbl 0795.65014

Axelsson, Owe

Cambridge: Cambridge University Press. xiii, 654 p. (1994).

The first seven chapters of the book and the first appendix have the character of a textbook for numerical linear algebra including direct solution methods, matrix eigenvalues, the Perron-Frobenius theory for nonnegative matrices, the basic iterative methods and their improvements by the Chebyshev and the splitting methods up to the SOR method and the incomplete factorization preconditioning methods. Both the theory and the algorithms are discussed and, in particular, the convergence rates of the iterative methods. Many examples and exercises with hints are presented, these exercises include generalized inverses, singular values, orthogonalization and QR methods, optimization problems and logarithmic norms.
The last six chapters deal with recent results in the iterative solution of linear systems, mainly using preconditioned conjugate gradient methods. Some further keywords are: approximate matrix inverses, block diagonal preconditionings, Lanczos-type methods, condition numbers and once more convergence rates. The last two appendices concern Chebyshev polynomials and matrix inequalities. Every chapter and appendix is provided with references, which sometimes overlap.
The well written book, which also contains the research of the author and scattered historical remarks, enables the reader not only to use the presented algorithms, but also to derive and to analyse new algorithms for the problems of linear algebra.

Reviewer: L.Berg (Rostock)

MSC:

65F10	Iterative numerical methods for linear systems
65-01	Introductory exposition (textbooks, tutorial papers, etc.) pertaining to numerical analysis
65Fxx	Numerical linear algebra
15-01	Introductory exposition (textbooks, tutorial papers, etc.) pertaining to linear algebra

Keywords:

Chebyshev method; textbook; numerical linear algebra; direct solution methods; matrix eigenvalues; Perron-Frobenius theory; nonnegative matrices; iterative methods; splitting methods; SOR method; incomplete factorization preconditioning methods; algorithms; convergence; exercises; generalized inverses; singular values; orthogonalization; QR methods; optimization; logarithmic norms; conjugate gradient methods; matrix inverses; block diagonal preconditionings; Lanczos-type methods; condition numbers; matrix inequalities