OVER AND
OVER AGAIN

Relations of mathematical objects to each other are expressed by transformations. The repeated application of a transformation over and over again - i.e., its iteration - can lead to solutions of equations, as in Newton's methods for finding roots, or Picard's method for solving differential equations. This book studies a treasure trove of iterations, in number theory, analysis and geometry, and applies them to various problems, many of them taken from international and national Mathematical Olympiad competitions.

The eminent mathematician and expositor Philip Davis has said about this book:

Among topics treated are classical and not so classical inequalities. Sharkovskii's theorem, interpolation, Bernstein polynomials, Bézier curves and surfaces and splines.

Most of the book requires only high school mathematics; the last part requires elementary calculus. This book would be an excellent supplement to courses in calculus, differential equations, numerical analysis, approximation theory and computer-aided geometric design.