Author | Nevanlinna, Olavi. author |
---|---|
Title | Convergence of Iterations for Linear Equations [electronic resource] / by Olavi Nevanlinna |
Imprint | Basel : Birkhรคuser Basel : Imprint: Birkhรคuser, 1993 |
Connect to | http://dx.doi.org/10.1007/978-3-0348-8547-8 |
Descript | VIII, 180 p. 15 illus. online resource |
1. Motivation, problem and notation -- 1.1 Motivation -- 1.2 Problem formulation -- 1.3 Usual tools -- 1.4 Notation for polynomial acceleration -- 1.5 Minimal error and minimal residual -- 1.6 Approximation of the solution operator -- 1.7 Location of zeros -- 1.8 Heuristics -- Comments to Chapter 1 -- 2. Spectrum, resolvent and power boundedness -- 2.1 The spectrum -- 2.2 The resolvent -- 2.3 The spectral mapping theorem -- 2.4 Continuity of the spectrum -- 2.5 Equivalent norms -- 2.6 The Yosida approximation -- 2.7 Power bounded operators -- 2.8 Minimal polynomials and algebraic operators -- 2.9 Quasialgebraic operators -- 2.10 Polynomial numerical hull -- Comments to Chapter 2 -- 3. Linear convergence -- 3.1 Preliminaries -- 3.2 Generating functions and asymptotic convergence factors -- 3.3 Optimal reduction factor -- 3.4 Greenโs function for G? -- 3.5 Optimal polynomials for -- 3.6 Simply connected G?(L) -- 3.7 Stationary recursions -- 3.8 Simple examples -- Comments to Chapter 3 -- 4. Sublinear convergence -- 4.1 Introduction -- 4.2 Convergence of Lk(L?1) -- 4.3 Splitting into invariant subspaces -- 4.4 Uniform convergence -- 4.5 Nonisolated singularity and successive approximation -- 4.6 Nonisolated singularity and polynomial acceleration -- 4.7 Fractional powers of operators -- 4.8 Convergence of iterates -- 4.9 Convergence with speed -- Comments to Chapter 4 -- 5. Superlinear convergence -- 5.1 What is superlinear -- 5.2 Introductory examples -- 5.3 Order and type -- 5.4 Finite termination -- 5.5 Lower and upper bounds for optimal polynomials -- 5.6 Infinite products -- 5.7 Almost algebraic operators -- 5.8 Estimates using singular values -- 5.9 Multiple clusters -- 5.10 Approximation with algebraic operators -- 5.11 Locally superlinear implies superlinear -- Comments to Chapter 5 -- References -- Definitions