Convergence of Iterations for Linear Equations - Softcover

Inhaltsangabe

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 Green's 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.

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