This book presents the most comprehensive discussion to date of the use of the Lanczos and CG methods for computing eigenvalues and solving linear systems in both exact and floating point arithmetic. The author synthesizes the research done over the past 30 years, describing and explaining the 'average' behavior of these methods and providing new insight into their properties in finite precision. Many examples are given that show significant results obtained by researchers in the field. The author details the mathematical properties of both algorithms and emphasizes how they can be used efficiently in finite precision arithmetic, regardless of the growth of rounding errors that occurs. Loss of orthogonality involved with using the Lanczos algorithm, ways to improve the maximum attainable accuracy of CG computations, and what modifications need to be made when the CG method is used with a preconditioner are addressed.
Gerard Meurant is Director of Research in the military applications division at Commissariat ... l'Energie Atomique (CEA) in Bruyeres le Chatel, France. He is the author of Computer Solution of Large Linear Systems (North-Holland, 1999) and serves on the editorial boards of the International Journal of High Speed Computing and Numerical Algorithms. In 1988 Meurant was awarded the Prix CEA and in 1995 the Palmes Academiques, an honor presented each year by the French Ministry of Education.
Preface; 1. The Lanczos algorithm in exact arithmetic; 2. The CG algorithm in exact arithmetic; 3. A historical perspective on the Lanczos algorithm in finite precision; 4. The Lanczos algorithm in finite precision; 5. The CG algorithm in finite precision; 6. The maximum attainable accuracy; 7. Estimates of norms of the error in finite precision; 8. The preconditioned CG algorithm; 9. Miscellaneous; Appendix; Bibliography; Index.