A comprehensive, self-contained primer on validated numerics
This textbook provides a comprehensive introduction to the theory and practice of validated numerics, an emerging new field that combines the strengths of scientific computing and pure mathematics. In numerous fields ranging from pharmaceutics and engineering to weather prediction and robotics, fast and precise computations are essential. Based on the theory of set-valued analysis, a new suite of numerical methods is developed, producing efficient and reliable solvers for numerous problems in nonlinear analysis. Validated numerics yields rigorous computations that can find all possible solutions to a problem while taking into account all possible sources of error—fast, and with guaranteed accuracy.
Validated Numerics offers a self-contained primer on the subject, guiding readers from the basics to more advanced concepts and techniques. This book is an essential resource for those entering this fast-developing field, and it is also the ideal textbook for graduate students and advanced undergraduates needing an accessible introduction to the subject. Validated Numerics features many examples, exercises, and computer labs using MATLAB/C++, as well as detailed appendixes and an extensive bibliography for further reading.
Provides a comprehensive, self-contained introduction to validated numerics
Requires no advanced mathematics or programming skills
Features many examples, exercises, and computer labs
Includes code snippets that illustrate implementation
Suitable as a textbook for graduate students and advanced undergraduates
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Yes, you can access Validated Numerics by Warwick Tucker in PDF and/or ePUB format, as well as other popular books in Mathematics & Computer Science General. We have over one million books available in our catalogue for you to explore.
In This Chapter, we give an elementary overview of how (and what type of) numbers are represented, stored, and manipulated in a computer. This will provide insight as to why some computations produce grossly incorrect results. This topic is covered much more extensively in, for example, [Mu09], [Hi96], [Ov01], and [Wi63].
1.1 POSITIONAL SYSTEMS
Our everyday decimal number system is a positional system in base 10. Since computer arithmetic is often built on positional systems in other bases (e.g., 2 or 16),1 we will begin this section by recalling how real numbers are represented in a positional system with an arbitrary integer base β ≥ 2. Setting aside practical restrictions, such as the finite storage capabilities of a computer, any real number can be expressed as an infinite string
where bn, bn–1, ... are integers in the range [0, β – 1], and σ
{0, 1} provides the sign of the number. The real number corresponding to (1.1) is
If the number ends in an infinite number of consecutive zeros we omit them in the expression (1.1). Thus we write (12.25)10 instead of (12.25000…)10. Also, we omit any zeros preceding the integer part (–1)σ (bnbn–1…b0)β. Thus we write (12.25)10 instead of (0012.25)10, and (0.0025)10 instead of (000.0025)10. Allowing for either leading or trailing extra zeros is called padding and is not common practice since it leads to redundancies in the representation.
Even without padding, the positional system is slightly flawed. No matter what base we choose, there are still real numbers that do not have a unique representation. For example, the decimal number (12.2549999 …)10 is equal to (12.255)10, and the binary number (100.01101111 …)2 is equal to (100.0111)2. This redundancy, however, can be avoided if we add the requirement that 0 ≤ bi ≤ β – 2 for infinitely many i.
Exercise 1.1.Prove that any real number x ≠ 0 has a unique representation (1.1) in a positional system (allowing no padding) with integer base β ≥ 2 under the two conditions (a) 0 ≤ bi ≤ β – 1 for all i, and (b) 0 ≤ bi ≤ β – 2 for infinitely many i.
Exercise 1.2.What is the correct way to represent zero in a positional system allowing no padding?
1.2 FLOATING POINT NUMBERS
When expressing a real number on the form (1.1), the placement of the decimal2 point is crucial. The floating point number system provides a more convenient way to represent real numbers. A floating point number is a real number on the form
where(–1)σ is the sign of x, m is called the mantissa,3 and e is called the exponent of x. Writing numbers in floating point notation frees us from the burden of keeping track of the decimal point: it al...
