In this chapter, we offer advice for dealing with some common problems that you may encounter. We also list and describe the most common mistakes that MATLAB users make. Finally, we offer some simple but useful techniques for debugging your M-files.

Common Problems

Problems manifest themselves in various ways: totally unexpected or plainly wrong output appears; MATLAB produces an error message (or at least a warning); MATLAB refuses to process an input line; something that worked earlier stops working; or, worst of all, the computer freezes. Fortunately, these problems are often caused by several easily identifiable and correctable mistakes. What follows is a description of some common problems, together with a presentation of likely causes, suggested solutions, and illustrative examples. We also refer to places in the book where related issues are discussed.

Here is a list of the problems:

• Wrong or unexpected output

• Syntax error

• Spelling error

• Error or warning messages when plotting

• A previously saved M-file evaluates differently

• Computer won't respond.

Wrong or Unexpected Output

There are many possible causes for this problem, but they are likely to be among the following.

CAUSE: Forgetting to clear or reset variables.

SOLUTION: Clear or initialize variables before using them, especially in a long session.

In this chapter we describe more of MATLAB's graphics commands and the most common ways of manipulating and customizing graphics. For an overview of commands, type help graphics (for general graphics commands), help graph2d (for two-dimensional graphics commands), help graph3d (for three-dimensional graphics commands), and help specgraph (for specialized graphing commands).

We have already discussed the commands plot and ezplot in Chapter 2. We will begin this chapter by discussing more uses of these commands, as well as some of the other most commonly used plotting commands. Then we will discuss methods for customizing and manipulating graphics. Finally, we will introduce some commands and techniques for creating and modifying images and sounds.

Two-Dimensional Plots

Often one wants to draw a curve in the x-y plane, but with y not given explicitly as a function of x. There are two main techniques for plotting such curves: parametric plotting and contour or implicit plotting.

In this chapter, we present examples showing you how to apply MATLAB to problems in several disciplines. Each example is presented in the form of a MATLAB M-file, published to LATEX. We modified the default style sheet for publishing to LATEX to adjust the spacing between input, output, and text, and to allow formatting of mathematical formulas not just as displayed equations but also within a paragraph. We also made a few minor adjustments to the published LATEX code to improve line breaks. Finally, because publish does not produce italic text, we used bold text instead in places. These examples are illustrations of the kinds of polished, integrated documents that you can create with MATLAB. The examples are:

• Illuminating a Room

• Mortgage Payments

• Monte Carlo Simulation

• Population Dynamics

• Linear Economic Models

• Linear Programming

• The 360° Pendulum

• ☆ Numerical Solution of the Heat Equation

• ☆ A Model of Traffic Flow

We have not explained all the MATLAB commands that we use; you can learn about the new commands from the online help. Simulink is used in A Model of Traffic Flow and as an optional accessory in Population Dynamics and Numerical Solution of the Heat Equation. The example on Linear Programming also requires an M-file found (in slightly different forms) in the Simulink and Optimization toolboxes. The examples require different levels of mathematical background and expertise in other subjects.

In this chapter, we will introduce you to the tools you need in order to begin using MATLAB effectively. These include the following: some relevant information on computer platforms and software; installation protocols; how to launch MATLAB, enter commands and use online help; a roster of MATLAB's various windows; and finally, how to exit the program. We know you are anxious to get started using MATLAB, so we will keep this chapter brief. After you complete it, you can go immediately to Chapter 2 to find concrete and simple instructions for using MATLAB to do mathematics. We describe the MATLAB interface more elaborately in Chapter 3.

Platforms and Versions

It is likely that you will use MATLAB on a computer running Microsoft Windows or on some form of a UNIX operating system (such as Linux). Some previous versions of MATLAB (Releases 11 and 12) did not support Macintosh, but the most current versions (Releases 13 and 14) do. If you are using a Macintosh, you should find that our instructions for Windows will suffice for most of your needs. Like MATLAB 6 (Releases 12 and 13), and unlike earlier versions, MATLAB 7 (Release 14) looks virtually identical on these different platforms. For definitiveness, we shall assume that the reader is using a Windows computer. In those very few instances where our instructions must be tailored differently for Linux, UNIX, or Macintosh users, we shall point it out clearly.

To be able to read or work on matrix computing, a reader must have completed a course on linear algebra. The inclusion of this Appendix A is to review some selected topics from basic linear algebra for reference purposes.

Starting from this chapter, we shall first describe various preconditioning techniques that are based on manipulation of a given matrix. These are classified into four categories: direct matrix extraction (or operator splitting type), inverse approximation (or inverse operator splitting type), multilevel Schur complements and multi-level operator splitting (multilevel methods).

The acoustic scattering modelling provides a typical example of utilizing a boundary element method to derive a dense matrix application as shown in Chapter 1. Such a physical problem is only a simple model of the full wave equations or the Maxell equations from electromagnetism. The challenges are:

1. (i) the underlying system is dense and non-Hermitian;

2. (ii) the kernel of a boundary integral operator is highly oscillatory for high wavenumbers, implying that a large linear system must be solved. The oscillation means that the fast multipole method and the fast wavelet methods are not immediately applicable.

This chapter reviews the recent work on using preconditioned iterative solvers for such linear systems arising from acoustic scattering modelling and points out the various challenges for future research work. We consider the following.

Matrix computing arises in the solution of almost all linear and nonlinear systems of equations. As the computer power upsurges and high resolution simulations are attempted, a method can reach its applicability limits quickly and hence there is a constant demand for new and fast matrix solvers. Preconditioning is the key to a successful iterative solver. It is the intention of this book to present a comprehensive exposition of the many useful preconditioning techniques.

Preconditioning equations mainly serve for an iterative method and are often solved by a direct solver (occasionally by another iterative solver). Therefore it is inevitable to address direct solution techniques for both sparse and dense matrices. While fast solvers are frequently associated with iterative solvers, for special problems, a direct solver can be competitive. Moreover, there are situations where preconditioning is also needed for a direct solution method. This clearly demonstrates the close relationship between a direct and an iterative method.