Iterative Methods for the Solution of Equations.

General Preliminaries: 1. 1 Introduction 1. 2 Basic concepts and notations General Theorems on Iteration Functions: 2. 1 The solution of a fixed-point problem 2. 2 Linear and superlinear convergence 2. 3 The iteration calculus The Mathematics of Difference Relations: 3. 1 Convergence of difference inequalities 3. 2 A theorem on the solutions of certain inhomogeneous difference equations 3. 3 On the roots of certain indicial equations 3. 4 The asymptotic behavior of the solutions of certain difference equations Interpolatory Iteration Functions: 4. 1 Interpolation and the solution of equations 4. 2 The order of interpolatory iteration functions 4. 3 Examples One-Point Iteration Functions: 5. 1 The basic sequence Eₛ 5. 2 Rational approximations to Eₛ 5. 3 A basic sequence of iteration functions generated by direct interpolation 5. 4 The fundamental theorem of one-point iteration functions 5. 5 The coefficients of the error series of Eₛ One-Point Iteration Functions With Memory: 6. 1 Interpolatory iteration functions 6. 2 Derivative-estimated one-point iteration functions with memory 6. 3 Discussion of one-point iteration functions with memory Multiple Roots: 7. 1 Introduction 7. 2 The order of Eₛ 7. 3 The basic sequence Eₛ 7. 4 The coefficients of the error series of Eₛ 7. 5 Iteration functions generated by direct interpolation 7. 6 One-point iteration functions with memory 7. 7 Some general results 7. 8 An iteration function of incommensurate order Multipoint Iteration Functions: 8. 1 The advantages of multipoint iteration functions 8. 2 A new interpolation problem 8. 3 Recursively formed iteration functions 8. 4 Multipoint iteration functions generated by derivative estimation 8. 5 Multipoint iteration functions generated by composition 8. 6 Multipoint iteration functions with memory Multipoint Iteration Functions: Continuation: 9. 1 Introduction 9. 2 Multipoint iteration functions of type 1 9. 3 Multipoint iteration functions of type 2 9. 4 Discussion of criteria for the selection of an iteration function Iteration Functions Which Require No Evaluation of Derivatives: 10. 1 Introduction 10. 2 Interpolatory iteration functions 10. 3 Some additional iteration functions Systems of Equations: 11. 1 Introduction 11. 2 The generation of vector-valued iteration functions by inverse interpolation 11. 3 Error estimates for some vector-valued iteration functions 11. 4 Vector-valued iteration functions which require no derivative evaluations A Compilation of Iteration Functions: 12. 1 Introduction 12. 2 One-point iteration functions 12. 3 One-point iteration functions with memory 12. 4 Multiple roots 12. 5 Multipoint iteration functions 12. 6 Multipoint iteration functions with memory 12. 7 Systems of equations Appendices: A. Interpolation B. On the jth derivative of the inverse function C. Significant figures and computational efficiency D. Acceleration of convergence E. Numerical examples F. Areas for future research Bibliography Index.