AMS Chelsea Publishing 1964; 310 pp; hardcover Reprint/Revision History: reprinted 1982 ISBN10: 0828403120 ISBN13: 9780828403122 List Price: US$44 Member Price: US$39.60 Order Code: CHEL/312
 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 Most of the material is new and unpublished.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 ... " Readership Graduate students and research mathematicians. Reviews "There is a vast amount of material in the book and a great deal is either new or presented in new form ... Although the mathematical treatment is rigorous throughout, attention is definitely focused on the computational aspects of the topic ... Many examples are provided to show how wellknown I.F. are special cases of the general results ... The author has certainly succeeded in presenting a systematic account of a large class of known I.F. and this makes the work an interesting basic text as well as a valuable reference book."  Mathematical Reviews 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 fixedpoint 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
OnePoint 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 onepoint iteration functions
 5.5 The coefficients of the error series of \(E_s\)
OnePoint Iteration Functions With Memory  6.1 Interpolatory iteration functions
 6.2 Derivativeestimated onepoint iteration functions with memory
 6.3 Discussion of onepoint 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 Onepoint 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 vectorvalued iteration functions by inverse interpolation
 11.3 Error estimates for some vectorvalued iteration functions
 11.4 Vectorvalued iteration functions which require no derivative evaluations
A Compilation of Iteration Functions  12.1 Introduction
 12.2 Onepoint iteration functions
 12.3 Onepoint 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
