Introduction

Our second numerical laboratory assignment is the computation of an area under a given curve to a desired accuracy. Unlike the calculation of the areas of various polygons, the computation of the area of a circle, an ellipse, or the area under the curve y = 1/log x from x = 2 to x = 7, say, is not at all trivial and requires methods of integral calculus or numerical approximations. Such areas arise not only in a geometrical context but also in various applications in engineering, biology, and statistics. In keeping with our policy of making the material as accessible as possible to precalculus students, without sacrificing rigor, we shall somewhat limit the generality so that results can be proved by elementary means and generalizations pointed out.

We shall henceforth be interested in the computation of the area under the graph of a positive function y = f(x), from x = a to x = b, but limit ourselves for the time being to monotonic (increasing or decreasing) or convex functions.

Rectangular approximations

Although the ensuing analysis is carried out in terms of a general, positive, monotonic function f(x), a ≤ x ≤ b, it is advisable that the laboratory participants bear in mind a concrete example such as f(x) = 1/x, 1 ≤ x ≤ 2.

Introduction

In the spirit of the mathematical laboratory, special attention is accorded to the elimination of traditional mathematical tables, which students have always used as “black boxes,” without the faintest understanding of their origin and construction. Thus, these tables were antieducational tools that the students aquired without any mathematical enlightenment and used as “cookbook recipes.” Now that calculators and microcomputers are used in mathematical education, the danger arises of replacing one set of black boxes with another. Of course, we are not advocating the introduction of these new electronic black boxes merely for using, say, the logarithmic built-in function of the computer (or pressing the “log” key on a pocket calculator). What we do advocate is to teach students what is behind such built-in functions as part of the material covered in the mathematical laboratory. This subject fits naturally into the environment of the laboratory and reveals the “story behind the key.” The attainment of this objective is the subject of this chapter and the next.

We might ask whether students should be allowed to use the built-in functions before (and during) learning how they were built in. We feel that no harm can result from such a practice, so long as the students are told expressly that their “ignorant” use of the built-in functions is temporary. Before long, the secrets held by the computer keys will be revealed.

Built-in functions

The term computer library functions refers to the collection of built-in functions – sin x, In x, ex, arctan x, to name but a few – that were installed in the computer's permanent memory. These built-in functions, of course, are efficient approximations of the abstract mathematical entities they represent. By efficiency we mean that every evaluation is performed with utmost speed and yields all the correct significant figures that are available on the computing device used. The construction of such built-in functions usually entails lengthy, computationally expensive preparations, which, however, are carried out only once. The first preparatory step is to reduce, as much as possible, the interval [a, b] in which the given f(x) is to be approximated (examples of this strategy can be found in Sections 2.2 and 7.6). To guarantee the desired correct significant figures, we also must control the relative error in the approximation of f(x) in [a, b]. This issue will be discussed in Section 8.3.

We shall concentrate on polynomial approximations, making use of the results obtained in Chapter 7, and shall consider the possibility of constructing rational approximations in Sections 8.5 and 8.6. We start with polynomial approximation of trigonometric functions in the same spirit with which we treated In x in Chapter 7.

The mathematical laboratory

Accumulated experience has shown that early emphasis on algorithmic thinking, augmented by actual computing, is indispensable in mathematical education. Recognizing the cardinal importance of the individual, active involvement of every student in the computational work (as opposed to mere demonstration by the teacher), we advocate the use of mathematical laboratories equipped with microcomputers. Optimally, a special room should be set aside for the mathematical laboratory. Failing that, physics or biology laboratories can be used since they tend to create the proper atmosphere. A pair of students is assigned to each microcomputer, as to a microscope in a biology laboratory, and spends a few hours a week working with the microcomputer in the laboratory.

The mere presence of an increasing number of microcomputers in various educational institutions, even those at which a programming language such as True-Basic or Pascal is taught, in no way constitutes a new mode of teaching and learning. The full potential of microcomputers and proper courseware should be harnessed to improve the state of the art in education. Moreover, a new role will be played by the mathematics teacher when traditional “chalk-and-talk” methods are augmented by active participation as a laboratory instructor. The numerous advantages of such computer-aided teaching of mathematics are detailed in Section 1.2.

The laboratory work will center around specific assignments, or modules, to be carried out by the participants at their own pace.

Numerical Mathematics – A Laboratory Approach is a unique book that introduces the computational microcomputer laboratory as a vehicle for teaching algorithmic aspects of mathematics. This is achieved through a sequence of laboratory assignments, presupposing no previous knowledge of calculus or linear algebra, where the “chalk and talk” lecturer turns into a laboratory instructor. The computational assignments cover basic numerical topics that should be part of the mathematical education in the era of microcomputers.

In writing this book at the precalculus and pre–linear algebra level, we were mainly addressing an audience of four groups: first-year university students of mathematics, sciences, and engineering who have had no exposure to systematic calculus; students at teachers' training colleges who will be tomorrow's teachers of mathematics and computer science; superior high-school mathematics students; and scientific programmers at all levels. Various parts of this book were successfully tested on classes representative of each of these groups and subsequently modified. The material was received enthusiastically by high-school students who were members of Tel Aviv University's Math Club, some of whom are now faculty members of the School of Mathematical Sciences. The material was also welcomed by members of New York University's summer program for talented high-school students (held every summer at the Courant Institute of Mathematical Sciences and directed by Henry Mullish), and by several classes of in-service or future mathematics teachers at Tel Aviv University and at New York University.

Introduction

In this chapter we present an algorithmic approach to the solution of systems of linear equations, another typical subject for the mathematical laboratory. No knowledge of matrices, vectors, and their underlying theory is presupposed, and thus the laboratory participants can handle this material even before the study of linear algebra.

After the development of an algorithm for the solution of “naive” systems of linear equations, special attention will be paid to problematic cases in which unrealistic answers with huge errors might be obtained. In particular, we shall discuss reasons for loss of accuracy, sensitivity to minor changes in the data, pivoting, scaling, and computational efficiency. By elaborating on each of these points by means of appropriate examples, we hope to present this traditionally abstract mathematical subject in a concrete, practical way that will be more meaningful to many students.

Coefficient tables

Systems of linear equations arise naturally in many practical areas such as mixing liquids, work and power calculations, electrical circuit computations, and marketing problems. It is particularly useful to demonstrate the subject under consideration by means of 3 × 3 systems (three equations and three unknowns). Such systems are not too large and cumbersome, but nevertheless constitute a case in which a pattern is revealed. Occasionally, when it is necessary for clarity, 4 × 4 and 2 × 2 systems will also be used.

SIGNALS

When describing the behaviour of a real-time process, we may wish to include instantaneous observable events that are not synchronisations. These signals may make it easier to describe and analyse certain aspects of behaviour, providing useful reference points in a history of the system. For example, an audible bell might form part of the user interface to a telephone network, even though the bell may ring without the cooperation of the user. This is incompatible with our existing view of communication.

In some cases, suitable environmental assumptions will allow us to describe such behaviour within the existing timed failures model. However, if we intend that these signals should be used to trigger other events or behaviours, then we must extend our semantic model to include an element of broadcast communication: some output events may occur without the cooperation of the environment.

In our model, signal events will occur as soon as they become available, and will propagate through parallel combination. A process may ignore any signal â performed by another process, unless it is waiting to perform the corresponding synchronisation a. If this is the case, then both â and a will occur. Of these, only the signal will be observed outside the parallel combination; it makes no sense to propagate a synchronisation.

REAL-TIME SYSTEMS

As computing devices become faster and more powerful, we find ourselves increasingly dependent upon systems which are difficult to understand and prone to failure. The failure of a commercial banking system or a company database may be expensive and inconvenient. The failure of an aircraft control system or a railway signalling network may result in injury or death. As the consequences of system failure become ever more severe, we must find ways to make these applications of computing technology safer and more reliable.

Over the past twenty-five years, mathematical techniques have been developed for the specification and implementation of computing systems. Formal methods have been used in the design and analysis of transformational systems—in which results are computed from a given set of inputs—and have been shown to reduce design costs and improve reliability. However, many of the systems in which safety is a primary concern are real-time systems, and cannot easily be viewed in a transformational setting.

Real-time systems maintain a continuous interaction with their environment and are often subject to complex timing constraints. They may also be required to perform several tasks concurrently. To reason about such systems we require a mathematical formalism that supports a treatment of timed concurrency. In this thesis we explore and extend one such formalism, the theory of Communicating Sequential Processes, first introduced by Hoare (1985).

OTHER APPROACHES

A wide variety of formal methods have been proposed for the specification and development of real-time systems, based upon process algebras, temporal logics, and timed programming languages. Although much research has been carried out, a consensus has yet to emerge concerning the applicability of the various formalisms to different types of system. A successful development method is likely to involve some combination of the features mentioned above. A notation that is well-suited to requirements capture is unlikely to be an efficient programming language, and vice versa.

Quantitative Temporal Logics

Hooman and Widom (1989) present a compositional proof system relating an occam-like language to a quantitative temporal logic, similar to the one developed by Koymans and de Roever (1983). Although the system description language is somewhat limited, it is clear that quantitative temporal logics are useful assertion languages; Jackson (1990) shows how such a logic may be employed as a specification language for timed CSP. It would be interesting to see the proof system applied in the development of a large, complex system.

Shasha et al. (1983) use a quantitative temporal logic to prove the correctness of a carrier-sense broadcast protocol, similar to the one described in chapter seven. By assuming a simplified version of the service provided by the physical layer, and an internal specification of the data link layer, the authors are able to establish that certain desirable properties hold of the network.