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# Dawood H. Theories of Interval Arithmetic: Mathematical Foundations and Applications

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Saarbrücken: LAP LAMBERT Academic Publishing, 2011. - 128p.
Scientists are, all the time, in a struggle with uncertainty which is always a threat to a trustworthy scientific knowledge. A very simple and natural idea, to defeat uncertainty, is that of enclosing uncertain measured values in real closed intervals. On the basis of this idea, interval arithmetic is constructed. The idea of calculating with intervals is not completely new in mathematics: the concept has been known since Archimedes, who used guaranteed lower and upper bounds to compute his constant Pi. Interval arithmetic is now a broad field in which rigorous mathematics is associated with scientific computing. This connection makes it possible to solve uncertainty problems that cannot be efficiently solved by floating-point arithmetic. Today, application areas of interval methods include electrical engineering, control theory, remote sensing, experimental and computational physics, chaotic systems, celestial mechanics, signal processing, computer graphics, robotics, and computer-assisted proofs. The purpose of this book is to be a concise but informative introduction to the theories of interval arithmetic as well as to some of their computational and scientific applications.
Table of contents :
Preface.
Notations and Conventions.
Prologue: A Weapon Against Uncertainty.
What Interval Arithmetic is and Why it is Considered.
A History Against Uncertainty.
The Classical Theory of Interval Arithmetic.
Algebraic Operations for Interval Numbers.
Point Operations for Interval Numbers.
Algebraic Properties of Interval Arithmetic.
Complex Interval Arithmetic.
Algebraic Operations for Complex Interval Numbers.
Point Operations for Complex Interval Numbers.
Algebraic Properties of Complex Interval Arithmetic.
Alternate Theories of Interval Arithmetic.
The Interval Dependency Problem.
Constraint Interval Arithmetic.
Modal Interval Arithmetic.
Computational Applications of Interval Arithmetic.
Estimates of the Image of Real Functions.
Bounding the Error Term in Taylor's Series.
Estimates of Definite Integrals.
Hardware Implementation of Interval Arithmetic.
Machine Interval Arithmetic.
Rounded-Outward Interval Arithmetic.
Rounded-Upward Interval Arithmetic.
An Interval Adder.
An Interval Squarrer.
Epilogue: What is Next?
A View to the Future of Interval Computations.
More Scientific Applications of Interval Arithmetic.
Current and Future Research in Interval Arithmetic.
Suggestions for Further Reading.
A Verilog Description for the 4-by-4 Bit Multiplier.
A Verilog Description for the Interval Squarrer.
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