Computable calculus. With 1 CD-ROM (Windows 95, 98, 2000, Windows NT) (Q2734826)

Computable calculus deals with computability in classical mathematics and gives an explicit meaning to the phrase ``constructively possible''. The discussions related to these topics are also called ``constructive mathematics'', ``computable mathematics'' or ``effective analysis'', etc. The founding work of computable calculus is the remarkable paper of \textit{A. M. Turing} [``On computable numbers, with an application to the Entscheidungsproblem'' Proc. Lond. Math. Soc., Ser. II. 42, 230--265 (1936; JFM 62.1059.03; Zbl 0016.09701)] where the notion of computable real number as well as the Turing machine are introduced. According to Turing, a real number is computable if arbitrarily precise approximations of this number can be computed by a (Turing) machine. The computable mathematics can be developed in the framework of classical mathematics. Namely, the universe is the same. But every object is classified into computable or non-computable. Thus, computable real numbers, computable real functions, computable real sets, computable operators, etc. are introduced and discussed (see, e.g., [\textit{M. B. Pour-El} and \textit{J. I. Richards}. ``Computability in analysis and physics'' (1989; Zbl 0678.03027)]; [\textit{K. Weihrauch}, ``Computable analysis. An introduction'' (2000; Zbl 0956.68056)]). Of course, computable mathematics can also be approached independently from classical mathematics. In this case, only the ``constructive objects'' are considered. The Russian School of Markov (cf. [\textit{B. A. Kushner}, ``Markov's constructive analysis; a participant's view'', Theor. Comput. Sci. 219, No. 1--2, 267--285 (1999; Zbl 0916.68053)] and constructive mathematics of \textit{E. Bishop} and \textit{D. Bridges} [``Constructive analysis'' (1985; Zbl 0656.03042)] are examples of this approach. Both of them apply the non-classical logics and hence are not very easy to access. The book under review belongs essentially to the second approach. The purpose is ``to make computable analysis more easily accessible'' and ``to present a version of constructively defined calculus, or computable analysis''. After an informal introduction in Chapter 1, the real numbers and the arithmetical operations on real numbers are introduced in Chapter 2. Chapter 3 discusses briefly some examples of solvable and nonsolvable problems. In Chapter 4, the sequences of real numbers and some basic properties of functions are investigated. The formal definition of the computability is given in Chapter 5 by introducing the ``ideal computer'', a kind of register machine. Details about the ideal computer and their applications are given in this chapter. The limits are considered in Chapter 6. Several interesting results which do not hold in classical mathematics are shown here. E.g, there is a bounded increasing sequence that does not converge to a limit (Specker, 1949); Any real function defined on an interval \(I\) is continuous at every interior point of \(I\) (Ceitin, 1959), and so on. Uniformly continuous functions are discussed in Chapter 7, where it is shown that there is a function which is uniformly continuous on \([0;1]\) but does not assume its least upper bound (Zaslavsky, 1955). Chapters 8 and 9 contribute to derivative and Riemann integral, respectively. Chapters 10 and 11 consider functions of two variables and the differential equation of the form \(y\,'=f(x,y)\), respectively. The last Chapter 12 presents some subsidiary information about the ideal computer and the simulation software included in the accompanying CD that can simulate an ideal computer on PC. This book is a very good introduction to the computable analysis. Without any presumption on the computability theory and elementary calculus, the reader can explore the computable calculus directly. However, for those readers who are familiar with classical calculus it might be helpful to remind that the nonexistence in computable calculus means usually the noncomputability.