Solving some problems of algebra, analysis, and mathematical physics using computer algebra systems (original) (raw)

Interval Computations: Introduction, Uses, and Resources

1996

Interval analysis is a broad Þeld in which rigorous mathematics is associated with with scientiÞc computing. A number,of researchers worldwide have produced a voluminous literature on the subject. This article introduces interval arithmetic and its interaction with established mathematical theory. The article provides pointers to traditional literature collections, as well as electronic resources. Some successful scientiÞc and engineering applications

Interval Mathematics as a Potential Weapon against Uncertainty

Mathematics of Uncertainty Modeling in the Analysis of Engineering and Science Problems, 2014

This chapter is devoted to introducing the theories of interval algebra to people who are interested in applying the interval methods to uncertainty analysis in science and engineering. In view of this purpose, we shall introduce the key concepts of the algebraic theories of intervals that form the foundations of the interval techniques as they are now practised, provide a historical and epistemological background of interval mathematics and uncertainty in science and technology, and finally describe some typical applications that clarify the need for interval computations to cope with uncertainty in a wide variety of scientific disciplines. Keywords. Interval mathematics, Uncertainty, Quantitative Knowledge, Reliability, Complex interval arithmetic, Machine interval arithmetic, Interval automatic differentiation, Computer graphics, Ray tracing, Interval root isolation. Recommended Citation: Hend Dawood. Interval Mathematics as a Potential Weapon against Uncertainty. In S. Chakraverty, editor, Mathematics of Uncertainty Modeling in the Analysis of Engineering and Science Problems. chapter 1, pages 1-38. IGI Global, Hershey, PA, 2014. ISBN 978-1-4666-4991-0. The final publication is available at IGI Global via http://dx.doi.org/10.4018/978-1-4666-4991-0.ch001

Postbinary calculations as a machine-assisted realization of real interval calculations

The World Academy of Research in Science and Engineering

The following article deals with confidence, reliability and precision of computer calculations, and the ways of achievement thereof. Postbinary calculations are regarded as perspective ones since they are capable of comprising the realization of the interval analysis methods. Here are the examples of postbinary code (tetracode) decoding to the borders of real interval. The modified floating-point formats of numbers used for storage and processing of tetracodes are suggested, they enable providing the reliability of computer calculations.

Theories of Interval Arithmetic: Mathematical Foundations and Applications

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.

Verified numerical computation for nonlinear equations

Japan Journal of Industrial and Applied Mathematics, 2009

After the introduction basic properties of interval arithmetic are discussed and different approaches are repeated by which one can compute verified numerical approximations for a solution of a nonlinear equation.

Interval Arithmetic, Affine Arithmetic, Taylor Series Methods: Why, What Next?

Numerical Algorithms, 2000

In interval computations, the range of each intermediate result r is described by an interval r. To decrease excess interval width, we can keep some information on how r depends on the input x = (x 1 ; : : : ; x n ). There are several successful methods of approximating this dependence; in these methods, the dependence is approximated by linear functions (a ne arithmetic) or by general polynomials (Taylor series methods). Why linear functions and polynomials? What other classes can we try? These questions are answered in this paper.

Invisible Mathematics: Numerical Analysis and its Role in Mathematical Application

The obtaining of numerical results in mathematical applications, the so-called "number crunching" is quite often taken for granted, both in "ordinary life" and even in relatively more sophisticated settings in science and engineering. We blithely expect our calculators and computers to produce numerical answers, flawlessly and unambiguously. Whatever mathematics is lurking behind those calculations is hidden, obscured, invisible. This invisible mathematics is known as numerical analysis.