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Analysis for Computer Scientists

Foundations, Methods, and Algorithms

Authors: Univ.-Prof. Dr. Dr. Michael Oberguggenberger, Univ.-Prof. Dr. Alexander Ostermann

Publisher: Springer International Publishing

Book Series : Undergraduate Topics in Computer Science

Part of: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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About this book

This easy-to-follow textbook/reference presents a concise introduction to mathematical analysis from an algorithmic point of view, with a particular focus on applications of analysis and aspects of mathematical modelling. The text describes the mathematical theory alongside the basic concepts and methods of numerical analysis, enriched by computer experiments using MATLAB, Python, Maple, and Java applets. This fully updated and expanded new edition also features an even greater number of programming exercises.

Topics and features: describes the fundamental concepts in analysis, covering real and complex numbers, trigonometry, sequences and series, functions, derivatives, integrals, and curves; discusses important applications and advanced topics, such as fractals and L-systems, numerical integration, linear regression, and differential equations; presents tools from vector and matrix algebra in the appendices, together with further information on continuity; includes added material on hyperbolic functions, curves and surfaces in space, second-order differential equations, and the pendulum equation (NEW); contains experiments, exercises, definitions, and propositions throughout the text; supplies programming examples in Python, in addition to MATLAB (NEW); provides supplementary resources at an associated website, including Java applets, code source files, and links to interactive online learning material.

Addressing the core needs of computer science students and researchers, this clearly written textbook is an essential resource for undergraduate-level courses on numerical analysis, and an ideal self-study tool for professionals seeking to enhance their analysis skills.

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Table of Contents

Frontmatter

Chapter 1. Numbers

Abstract

The commonly known rational numbers (fractions) are not sufficient for a rigorous foundation of mathematical analysis. The historical development shows that for issues concerning analysis, the rational numbers have to be extended to the real numbers. For clarity we introduce the real numbers as decimal numbers with an infinite number of decimal places. We illustrate exemplarily how the rules of calculation and the order relation extend from the rational to the real numbers in a natural way. A further section is dedicated to floating point numbers, which are implemented in most programming languages as approximations to the real numbers. In particular, we will discuss optimal rounding and in connection with this the relative machine accuracy.

Michael Oberguggenberger, Alexander Ostermann

Chapter 2. Real-Valued Functions

Abstract

The notion of a function is the mathematical way of formalising the idea that one or more independent quantities are assigned to one or more dependent quantities. Functions in general and their investigation are at the core of analysis. They help to model dependencies of variable quantities, from simple planar graphs, curves and surfaces in space to solutions of differential equations or the algorithmic construction of fractals. One the one hand, this chapter serves to introduce the basic concepts. On the other hand, the most important examples of real-valued, elementary functions are discussed in an informal way. These include the power functions, the exponential functions and their inverses. Trigonometric functions will be discussed in Chap. 3, complex-valued functions in Chap. 4.

Michael Oberguggenberger, Alexander Ostermann

Chapter 3. Trigonometry

Abstract

Trigonometric functions play a major role in geometric considerations as well as in the modelling of oscillations. We introduce these functions at the right-angled triangle and extend them periodically to \(\mathbb R\) using the unit circle. Furthermore, we will discuss the inverse functions of the trigonometric functions in this chapter. As an application we will consider the transformation between Cartesian and polar coordinates.

Michael Oberguggenberger, Alexander Ostermann

Chapter 4. Complex Numbers

Abstract

Complex numbers are not just useful when solving polynomial equations but play an important role in many fields of mathematical analysis. With the help of complex functions transformations of the plane can be expressed, solution formulas for differential equations can be obtained, and matrices can be classified. Not least, fractals can be defined by complex iteration processes. In this section we introduce complex numbers and then discuss some elementary complex functions, like the complex exponential function. Applications can be found in Chaps. 9 (fractals) and 20 (systems of differential equations) and in Appendix B (normal form of matrices).

Michael Oberguggenberger, Alexander Ostermann

Chapter 5. Sequences and Series

Abstract

The concept of a limiting process at infinity is one of the central ideas of mathematical analysis. It forms the basis for all its essential concepts, like continuity, differentiability, series expansions of functions, integration, etc. The transition from the discrete to the continuous constitutes the modelling strength of mathematical analysis. Discrete models of physical, technical or economic processes can often be better and easily understood, provided that the number of their atoms—their discrete building blocks—is sufficiently big, if they are approximated by a continuous model with the help of a limiting process. The transition from difference equations for biological growth processes in discrete time to differential equations in continuous time or the description of share prices by stochastic processes in continuous time are examples for that. The majority of models in physics are field models, that is, they are expressed in a continuous space and time structure. Even though the models are discretised again in numerical approximations, the continuous model is still helpful as a background, for example for the derivation of error estimates.

Michael Oberguggenberger, Alexander Ostermann

Chapter 6. Limits and Continuity of Functions

Abstract

In this section we extend the notion of the limit of a sequence to the concept of the limit of a function. Hereby we obtain a tool which enables us to investigate the behaviour of graphs of functions in the neighbourhood of chosen points. Moreover, limits of functions form the basis of one of the central themes in mathematical analysis, namely differentiation (Chap. 7). In order to derive certain differentiation formulas some elementary limits are needed, for instance, limits of trigonometric functions. The property of continuity of a function has far-reaching consequences like, for instance, the intermediate value theorem, according to which a continuous function which changes its sign in an interval has a zero. Not only does this theorem allow one to show the solvability of equations, it also provides numerical procedures to approximate the solutions. Further material on continuity can be found in Appendix C.

Michael Oberguggenberger, Alexander Ostermann

Chapter 7. The Derivative of a Function

Abstract

Starting from the problem to define the tangent to the graph of a function, we introduce the derivative of a function. Two points on the graph can always be joined by a secant, which is a good model for the tangent whenever these points are close to each other. In a limiting process, the secant (discrete model) is replaced by the tangent (continuous model). Differential calculus, which is based on this limiting process, has become one of the most important building blocks of mathematical modelling.

Michael Oberguggenberger, Alexander Ostermann

Chapter 8. Applications of the Derivative

Abstract

This chapter is devoted to some applications of the derivative which form part of the basic skills in modelling. We start with a discussion of features of graphs. More precisely, we use the derivative to describe geometric properties like maxima, minima and monotonicity. Even though plotting functions with MATLAB or maple is simple, understanding the connection with the derivative is important, for example, when a function with given properties is to be chosen from a particular class of functions.

Michael Oberguggenberger, Alexander Ostermann

Chapter 9. Fractals and L-systems

Abstract

In geometry objects are often defined by explicit rules and transformations which can easily be translated into mathematical formulas. For example, a circle is the set of all points which are at a fixed distance r from a centre (a, b).

$$ K = \{(x, y) \in \mathbb R^2 \;;\; (x - a)^2 + (y -b)^2 = r^2\} $$

or

$$ K = \{(x, y) \in \mathbb R^2 \;;\; x = a + r \cos \varphi ,\ y = b + r \sin \varphi , \ 0 \le \varphi < 2\pi \}. $$

Michael Oberguggenberger, Alexander Ostermann

Chapter 10. Antiderivatives

Abstract

The derivative of a function \(y = F(x)\) describes its local rate of change, i.e. the change \(\varDelta y\) of the y-value with respect to the change \(\varDelta x\) of the x-value in the limit \(\varDelta x \rightarrow 0\); more precisely.

$$ f(x) = F'(x) = \lim _{\varDelta x \rightarrow 0}\frac{\varDelta y}{\varDelta x} = \lim _{\varDelta x \rightarrow 0}\frac{F(x + \varDelta x) - F(x)}{\varDelta x}. $$

Michael Oberguggenberger, Alexander Ostermann

Chapter 11. Definite Integrals

Abstract

In the introduction to Chap. 10 the notion of the definite integral of a function f on an interval [a, b] was already mentioned. It arises from summing up expressions of the form \(f(x)\varDelta x\) and taking limits. Such sums appear in many applications including the calculation of areas, surface areas and volumes as well as the calculation of lengths of curves. This chapter employs the notion of Riemann integrals as the basic concept of definite integration. Riemann’s approach provides an intuitive concept in many applications, as will be elaborated in examples at the end of the chapter.

Michael Oberguggenberger, Alexander Ostermann

Chapter 12. Taylor Series

Abstract

Approximations of complicated functions by simpler functions play a vital part in applied mathematics. Starting with the concept of linear approximation we discuss the approximation of a function by Taylor polynomials and by Taylor series in this chapter. As important applications we will use Taylor series to compute limits of functions and to analyse various approximation formulas.

Michael Oberguggenberger, Alexander Ostermann

Chapter 13. Numerical Integration

Abstract

The fundamental theorem of calculus suggests the following approach to the calculation of definite integrals: One determines an antiderivative F of the integrand f and computes from that the value of the integral

$$ \int ^b_a \! f(x) \, \mathrm{d}x = F(b) - F(a). $$

In practice, however, it is difficult and often even impossible to find an antiderivative F as a combination of elementary functions. Apart from that, antiderivatives can also be fairly complex, as the example \(\int x^{100} \sin x \,\mathrm{d}x\) shows. Finally, in concrete applications the integrand is often given numerically and not by an explicit formula. In all these cases one reverts to numerical methods. In this chapter the basic concepts of numerical integration (quadrature formulas and their order) are introduced and explained. By means of instructive examples we analyse the achievable accuracy for the Gaussian quadrature formulas and the required computational effort.

Michael Oberguggenberger, Alexander Ostermann

Chapter 14. Curves

Abstract

The graph of a function \(y = f(x)\) represents a curve in the plane. This concept, however, is too tight to represent more intricate curves, like loops, self-intersections or even curves of fractal dimension. The aim of this chapter is to introduce the concept of parametrised curves and to study, in particular, the case of differentiable curves. For the visualisation of the trajectory of a curve, the notions of velocity vector, moving frame and curvature are important. The chapter contains a collection of geometrically interesting examples of curves and several of their construction principles. Further, the computation of the arc length of differentiable curves is discussed, and an example of a continuous, bounded curve of infinite length is given. The chapter ends with a short outlook on spatial curves. For the vector algebra used in this chapter, we refer to Appendix A.

Michael Oberguggenberger, Alexander Ostermann

Chapter 15. Scalar-Valued Functions of Two Variables

Abstract

This chapter is devoted to differential calculus of functions of two variables. In particular we will study geometrical objects such as tangents and tangent planes, maxima and minima, as well as linear and quadratic approximations. The restriction to two variables has been made for simplicity of presentation. All ideas in this and the next chapter can easily be extended (although with slightly more notational effort) to the case of n variables. We begin by studying the graph of a function with the help of vertical cuts and level sets. As a further tool we introduce partial derivatives, which describe the rate of change of the function in the direction of the coordinate axes. Finally the notion of the Fréchet derivative allows us to define the tangent plane to the graph. As for functions of one variable the Taylor formula plays a central role. We use it, e.g., to determine extrema of functions of two variables. In the entire chapter D denotes a subset of \(\mathbb R^2\), and

$$ f:D\subset \mathbb R^2 \rightarrow \mathbb R: (x,y) \mapsto z=f(x, y) $$

denotes a scalar-valued function of two variables. Details of vector and matrix algebra used in this chapter can be found in Appendices A and B.

Michael Oberguggenberger, Alexander Ostermann

Chapter 16. Vector-Valued Functions of Two Variables

Abstract

In this section we briefly touch upon the theory of vector-valued functions in several variables. To simplify matters we limit ourselves again to the case of two variables. First we define vector fields in the plane and extend the notions of continuity and differentiability to vector-valued functions. Then we discuss Newton’s method in two variables. As an application we compute a common zero of two nonlinear functions. Finally, as an extension of Sect. 15.1, we show how smooth surfaces can be described mathematically with the help of parameterisations.

Michael Oberguggenberger, Alexander Ostermann

Chapter 17. Integration of Functions of Two Variables

Abstract

In Sect. 11.3 we have shown how to calculate the volume of solids of revolution. If there is no rotational symmetry, however, one needs an extension of integral calculus to functions of two variables. This arises, for example, if one wants to find the volume of a solid that lies between a domain D in the (x, y)-plane and the graph of a non-negative function \(z = f(x, y)\). In this section we will extend the notion of Riemann integrals from Chap. 11 to double integrals of functions of two variables. Important tools for the computation of double integrals are their representation as iterated integrals and the transformation formula (change of coordinates). The integration of functions of several variables occurs in numerous applications, a few of which we will discuss.

Michael Oberguggenberger, Alexander Ostermann

Chapter 18. Linear Regression

Abstract

Linear regression is one of the most important methods of data analysis. It serves the determination of model parameters, model fitting, assessing the importance of influencing factors, and prediction, in all areas of human, natural and economic sciences. Computer scientists who work closely with people from these areas will definitely come across regression models.

Michael Oberguggenberger, Alexander Ostermann

Chapter 19. Differential Equations

Abstract

In this chapter we discuss the theory of initial value problems for ordinary differential equations. We limit ourselves to scalar equations here; systems will be discussed in the next chapter. After presenting the general definition of a differential equation and the geometric significance of its direction field, we start with a detailed discussion of first-order linear equations. As important applications we discuss the modelling of growth and decay processes. Subsequently, we investigate questions of existence and (local) uniqueness of the solution of general differential equations and discuss the method of power series. We also study the qualitative behaviour of solutions close to an equilibrium point. Finally, we discuss the solution of second-order linear problems with constant coefficients.

Michael Oberguggenberger, Alexander Ostermann

Chapter 20. Systems of Differential Equations

Abstract

Systems of differential equations, often called differentiable dynamical systems, play a vital role in modelling time-dependent processes in mechanics, meteorology, biology, medicine, economics and other sciences. We limit ourselves to two-dimensional systems, whose solutions (trajectories) can be graphically represented as curves in the plane. The first section introduces linear systems, which can be solved analytically as will be shown. In many applications, however, nonlinear systems are required. In general, their solution cannot be given explicitly. Here it is of primary interest to understand the qualitative behaviour of solutions. In the second section of this chapter, we touch upon the rich qualitative theory of dynamical systems. The third section is devoted to analysing the mathematical pendulum in various ways. Numerical methods will be discussed in Chap. 21.

Michael Oberguggenberger, Alexander Ostermann

Chapter 21. Numerical Solution of Differential Equations

Abstract

As we have seen in the last two chapters, only particular classes of differential equations can be solved analytically. Especially for nonlinear problems one has to rely on numerical methods. In this chapter we discuss several variants of Euler’s method as a prototype. Motivated by the Taylor expansion of the analytical solution we deduce Euler approximations and study their stability properties. In this way we introduce the reader to several important aspects of the numerical solution of differential equations. We point out, however, that for most real-life applications one has to use more sophisticated numerical methods.

Michael Oberguggenberger, Alexander Ostermann

Backmatter

Title: Analysis for Computer Scientists
Authors: Univ.-Prof. Dr. Dr. Michael Oberguggenberger
Univ.-Prof. Dr. Alexander Ostermann
Copyright Year: 2018
Publisher: Springer International Publishing
Electronic ISBN: 978-3-319-91155-7
Print ISBN: 978-3-319-91154-0
DOI: https://doi.org/10.1007/978-3-319-91155-7

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