Krylov solvers for linear algebraic systems : Krylov solvers The first four chapters of this book give a comprehensive and unified theory of the Krylov methods. Many of these are shown to be particular examples of the block conjugate gradient algorithm and it is this observation that permits the unification of the theory. The two major sub classes of those methods the Lanczos and the Hestenes Stiefel are developed in parallel as natural generalisations of the Orthodir (GCR) and Orthomin algorithms. These are themselves based on Arnoldis algorithm and a generalised Gram Schmidt algorithm and their properties in particular their stability properties are determined by the two matrices that define the block conjugate gradient algorithm. These are the matrix of coefficients and the preconditioning matrix. In Chapter 5 the transpose free algorithms based on the conjugate gradient squared algorithm are presented while Chapter 6 examines the various ways in which the QMR technique has been exploited. Look ahead methods and general block methods are dealt with in Chapters 7 and 8 while Chapter 9 is devoted to error analysis of two basic algorithms. In Chapter 10 the results of numerical testing of the more important algorithms in their basic forms (i.e. without look ahead or preconditioning) are presented and these are related to the structure of the algorithms and the general theory. Graphs illustrating the performances of various algorithm problem combinations are given via a CD ROM. Chapter 11 by far the longest gives a survey of preconditioning techniques. These range from the old idea of polynomial preconditioning via SOR and ILU preconditioning to methods like SpAI AInv and the multigrid methods that were developed specifically for use with parallel computers. Chapter 12 is devoted to dual algorithms like Orthores and the reverse algorithms of Hegedus. Finally certain ancillary matters like reduction to Hessenberg form Chebychev polynomials and the companion matrix are described in a series of appendices. comprehensive and unified approach up to date chapter on preconditioners complete theory of stability includes dual and reverse methods comparison of algorithms on CD ROM objective assessment of algorithms Contents 1. Introduction. 2. The long recurrences. 3. The short recurrences. 4. The Krylov aspects. 5. Transpose free methods. 6. More on QMR. 7. Look ahead methods. 8. General block methods. 10. And in practice 11. Preconditioning. 12. Duality. Appendices. A. Reduction of upper Hessenberg matrix to upper triangular form by plane rotations. B. Schur complements. C. The Jordan form. D. Chebychev polynomials. E. The companion matrix. F. Algorithmic details