Iterative Methods for the Solution of Equations

From the Preface (1964): 'This book presents a general theory of iteration algorithms for the numerical solution of equations and systems of equations. The relationship between the quantity and the quality of information used by an algorithm and the efficiency of the algorithm is investigated. Iteration functions are divided into four classes depending on whether they use new information at one or at several points and whether or not they reuse old information. Known iteration functions are systematized and new classes of computationally effective iteration functions are introduced. Our interest in the efficient use of information is influenced by the widespread use of computing machines...The mathematical foundations of our subject are treated with rigor, but rigor in itself is not the main object. Some of the material is of wider application...Most of the material is new and unpublished. Every attempt has been made to keep the subject in proper historical perspective...'

Table of Contents:
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 $E_s$ 5.2 Rational approximations to $E_s$ 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 $E_s$ 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 $E_s$ 7.3 The basic sequence $\scr{E}_s$ 7.4 The coefficients of the error series of $\scr{E}_s$ 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 $j$th 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.