Over and over again / by Gengzhe Chang and Thomas W. Sederberg.

Format:

Book

Author/Creator:

Chang, Gengzhe.

Contributor:

Sederberg, Thomas W.

Series:

Anneli Lax New Mathematical Library ; 39

Language:

English

Subjects (All):

Iterative methods (Mathematics).

Transformations (Mathematics).

Physical Description:

1 online resource (xiv, 309 pages) : digital, PDF file(s).

Edition:

1st ed.

Place of Publication:

Washington, DC : Mathematical Association of America, 1997.

Language Note:

English

Summary:

Suitable as supplemental reading in courses in differential and integral calculus, numerical analysis, approximation theory and computer-aided geometric design. 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 leads to solution of equations, as in Newton's method 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 applied them to various problems, many of them taken from international and national Mathematical Olympiad competitions. Among topics treated are classical and not so classical inequalities, Sharkovskii's theorem, interpolation, Bernstein polynomials, BŽzier 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.

Contents:

Transformations and their iteration

Arithmetic and geometric means

Isoperimetric inequality for triangles

Isoperimetric quotient

Colored marbles

Candy for school children

Sugar rather than candy

Checkers on a circle

Decreasing sets of positive integers

Matrix manipulations

Nested traingles

Morley's theorem and Napoleon's theorem

Complex numbers in geometry

Birth of an IMO problem

Barycentric coordinates

Douglas-Neumann theorem

Lagrange interpolation

The Isoperimetric problem

Formulas for iterates

Convergent orbits

Finding roots by iteration

Chebyshev polynomials

Sharkovskii's theorem

Variation diminishing matrices

Approximation by Bernstein polynomials

Properties of Bernstein polynomials

Bézier curves

Cubic interpolatory splines

Moving averages

Approximation of surfaces

Properties of triangular patches

Convexity of patches.

Notes:

Title from publisher's bibliographic system (viewed on 02 Oct 2015).

Includes bibliographical references and index.

Description based on print version record.

ISBN:

0-88385-953-X

OCLC:

929120343