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Nondiscrete induction and iterative processes. (English) Zbl 0549.41001
Research Notes in Mathematics, 103. Boston-London-Melbourne: Pitman Advanced Publishing Program. IX, 207 p. £9.95 (1984).
This book links up some ideas from functional analysis to the study of iterative processes in numerical mathematics. One of the starting points is to introduce rates of convergence which are functions instead of real numbers (in itself this is not new). For instance the rate of convergence of a certain Newton process can be described by the function \(\omega (r)=r^ 2/(r^ 2+a^ 2)^{1/2}\) with constant \(a>0\); r measures the approximation. If \(0<a<r\) then \(\omega\) (r)\(\approx r\), if \(0<r<a\) then \(\omega (r)\approx r^ 2/a\). The second case describes the quadratic convergence of the Newton process when r is small, roughly speaking, when we are near enough to the solution. But the rate of convergence \(\omega\) (r) also describes the approximation process in the initial stage of iteration.
This idea is then tied in with the induction theorem and the closed graph theorem in functional analysis. For a given problem one does not start by constructing an approximation process but by generating a suitable rate of convergence, a function in a certain set; this leads naturally to the construction of iterations.
The book starts by giving a clear discussion of three examples. The sections 3-6 are the most interesting from the point of view of applications: here one discusses rates of convergence, the secant method (Regula Falsi) and Newton’s method. In section 7 an eigenvalue problem in linear algebra is discussed. The remaining sections introduce factorization theorems, transitivity in operator algebras, stability of openness and an interesting modification of Newton’s method by J. Moser. Some technical aspects are elaborated upon in three appendices.
This is an intriguing book. It seems to be of interest to establish its importance in some research problems.
Reviewer: F.Verhulst

41-02 Research exposition (monographs, survey articles) pertaining to approximations and expansions
41A25 Rate of convergence, degree of approximation
65D99 Numerical approximation and computational geometry (primarily algorithms)