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Scientific Computation on Mathematical Problems and Conjectures

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by Richard S. Varga

1990 / vi + 122 pages / Softcover / ISBN: 978-0-898712-57-5 / List Price $56.00 / SIAM/CBMS Member Price $39.20 / Order Code CB60

"Let me state at the outset that this monograph is a gem. It contains six completely independent chapters devoted to six different areas. The unifying theme is that all of the problems are drawn from classical analysis with a strong tilt toward approximation theory. The unique part of the monograph is how high-precision numerical calculations can play a creative role in this area. In fact, it is possibly fair to say that the monograph is a sequence of challenging and interesting problems, where both hard analysis and sophisticated computing techniques are brought to bear to develop rigorous proof of interesting conjectures, or in some cases, used to establish counterexamples." – George J. Fix, SIAM Review, June 1993 

"This book contains fascinating accounts of how some longstanding mathematical problems could be solved after an accumulation of efforts by many mathematicians and the use of highly accurate floating- point computations. At the same time it shows that there are still a lot of unsolved mathematical problems, the solution of which may require deep mathematics, but probably also fast computers and highly-accurate numerical software." – Herman J.J. te Riele, Mathematical Reviews, Issue 92b

Studies the use of scientific computation as a tool in attacking a number of mathematical problems and conjectures. In this case, scientific computation refers primarily to computations that are carried out with a large number of significant digits, for calculations associated with a variety of numerical techniques such as the (second) Remez algorithm in polynomial and rational approximation theory, Richardson extrapolation of sequences of numbers, the accurate finding of zeros of polynomials of large degree, and the numerical approximation of integrals by quadrature techniques.

The goal of this book is not to delve into the specialized field dealing with the creation of robust and reliable software needed to implement these high-precision calculations, but rather to emphasize the enormous power that existing software brings to the mathematician's arsenal of weapons for attacking mathematical problems and conjectures.

Scientific Computation on Mathematical Problems and Conjectures includes studies of the Bernstein Conjecture of 1913 in polynomial approximation theory, the "1/9" Conjecture of 1977 in rational approximation theory, the famous Riemann Hypothesis of 1859, and the Polya Conjecture of 1927. The emphasis of this monograph rests strongly on the interplay between hard analysis and high-precision calculations.

Chapter 1: The Bernstein Conjecture in Approximation Theory; Chapter 2: The "1/9" Conjecture and Its Recent Solution; Chapter 3: Theoretical and Computational Aspects of the Riemann Hypothesis; Chapter 4: Asymptotics for the Zeros of the Partial Sums of exp(z); Chapter 5: Real vs. Complex Best Rational Approximation; Chapter 6: Generalizations of Jensen's Inequality for Polynomials Having Concentration at Low Degrees.

ISBN: 9780898712575 

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