Unformatted text preview: MATLAB
The Language of Technical Computing Getting Started with MATLAB
Version 7 How to Contact The MathWorks:
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Other product or brand names are trademarks or registered trademarks of their respective holders. Printing History: December 1996 May 1997 September 1998 September 2000 June 2001 July 2002 August 2002 June 2004 First printing Second printing Third printing Fourth printing Online only Online only Fifth printing Sixth printing For MATLAB 5 For MATLAB 5.1 For MATLAB 5.3 Revised for MATLAB 6 (Release 12) Revised for MATLAB 6.1 (Release 12.1) Revised for MATLAB 6.5 (Release 13) Revised for MATLAB 6.5 Revised for MATLAB 7.0 (Release 14) Contents
Introduction 1
What Is MATLAB? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 The MATLAB System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 MATLAB Documentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 MATLAB Online Help . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 Starting and Quitting MATLAB . . . . . . . . . . . . . . . . . . . . . . . . Starting MATLAB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Quitting MATLAB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . MATLAB Desktop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
16 16 16 17 Matrices and Arrays 2
Matrices and Magic Squares . . . . . . . . . . . . . . . . . . . . . . . . . . . Entering Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . sum, transpose, and diag . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Subscripts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Colon Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The magic Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Expressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Examples of Expressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
22 23 24 26 27 28 210 210 210 211 211 213 i Working with Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Generating Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The load Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . MFiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Concatenation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Deleting Rows and Columns . . . . . . . . . . . . . . . . . . . . . . . . . . . More About Matrices and Arrays . . . . . . . . . . . . . . . . . . . . . . Linear Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Arrays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Multivariate Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Scalar Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Logical Subscripting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The find Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Controlling Command Window Input and Output . . . . . . . The format Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Suppressing Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Entering Long Statements . . . . . . . . . . . . . . . . . . . . . . . . . . . . Command Line Editing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214 214 215 215 216 217 218 218 221 224 225 226 227 228 228 230 230 230 Graphics 3
Overview of MATLAB Plotting . . . . . . . . . . . . . . . . . . . . . . . . . 32 The Plotting Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 Graph Components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 Figure Tools . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 Arranging Graphs Within a Figure . . . . . . . . . . . . . . . . . . . . . 313 Selecting Plot Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314 Editing Plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316 Plot Editing Mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316 Using Functions to Edit Graphs . . . . . . . . . . . . . . . . . . . . . . . . 319 ii Contents Examples  Using MATLAB Plotting Tools . . . . . . . . . . . . . 320 Modifying the Graph Data Source . . . . . . . . . . . . . . . . . . . . . . 327 Preparing Graphs for Presentation . . . . . . . . . . . . . . . . . . . . Modify the Graph to Enhance the Presentation . . . . . . . . . . . Printing the Graph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exporting the Graph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Basic Plotting Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Creating a Plot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Multiple Data Sets in One Graph . . . . . . . . . . . . . . . . . . . . . . . Specifying Line Styles and Colors . . . . . . . . . . . . . . . . . . . . . . . Plotting Lines and Markers . . . . . . . . . . . . . . . . . . . . . . . . . . . . Imaginary and Complex Data . . . . . . . . . . . . . . . . . . . . . . . . . . Adding Plots to an Existing Graph . . . . . . . . . . . . . . . . . . . . . . Figure Windows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Multiple Plots in One Figure . . . . . . . . . . . . . . . . . . . . . . . . . . . Controlling the Axes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Axis Labels and Titles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Saving Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
329 330 333 335 338 338 340 341 341 343 344 346 346 348 349 351 Mesh and Surface Plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352 Visualizing Functions of Two Variables . . . . . . . . . . . . . . . . . . 352 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 358 Reading and Writing Images . . . . . . . . . . . . . . . . . . . . . . . . . . . 359 Printing Graphics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 360 Handle Graphics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Using the Handle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Graphics Objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Setting Object Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Specifying the Axes or Figure . . . . . . . . . . . . . . . . . . . . . . . . . . Finding the Handles of Existing Objects . . . . . . . . . . . . . . . . .
362 362 363 365 368 369 Animations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371 Erase Mode Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371 Creating Movies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373 iii Programming 4
Flow Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . if . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . switch and case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . for . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . while . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . continue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . break . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . try  catch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . return . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
42 42 44 44 45 45 46 47 47 Other Data Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 Multidimensional Arrays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 Cell Arrays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 410 Characters and Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412 Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415 Scripts and Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Scripts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Types of Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Global Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Passing String Arguments to Functions . . . . . . . . . . . . . . . . . . The eval Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Function Handles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Function Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Vectorization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Preallocation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
418 419 420 422 424 425 426 427 427 430 430 iv Contents Creating Graphical User Interfaces 5
What Is GUIDE? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 Laying Out a GUI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 Starting GUIDE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 The Layout Editor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 Programming a GUI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 Desktop Tools and Development Environment 6
Desktop Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 Arranging the Desktop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 Start Button . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 Command Window and Command History . . . . . . . . . . . . . . . 65 Command Window . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 Command History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 Help Browser . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 Current Directory Browser and Search Path . . . . . . . . . . . 610 Current Directory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 610 Search Path . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 611 Workspace Browser and Array Editor . . . . . . . . . . . . . . . . . . 612 Workspace Browser . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 612 Array Editor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 613 v Editor/Debugger . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 614 MLint Code Check and Profiler Reports . . . . . . . . . . . . . . . 616 MLint Code Check Report . . . . . . . . . . . . . . . . . . . . . . . . . . . . 616 Profiler Report . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 618 Other Development Environment Features . . . . . . . . . . . . . 619 Index vi Contents 1
Introduction 1 Introduction What Is MATLAB?
MATLAB is a highperformance language for technical computing. It integrates computation, visualization, and programming in an easytouse environment where problems and solutions are expressed in familiar mathematical notation. Typical uses include Math and computation Algorithm development Data acquisition Modeling, simulation, and prototyping Data analysis, exploration, and visualization Scientific and engineering graphics Application development, including graphical user interface building MATLAB is an interactive system whose basic data element is an array that does not require dimensioning. This allows you to solve many technical computing problems, especially those with matrix and vector formulations, in a fraction of the time it would take to write a program in a scalar noninteractive language such as C or Fortran. The name MATLAB stands for matrix laboratory. MATLAB was originally written to provide easy access to matrix software developed by the LINPACK and EISPACK projects. Today, MATLAB engines incorporate the LAPACK and BLAS libraries, embedding the state of the art in software for matrix computation. MATLAB has evolved over a period of years with input from many users. In university environments, it is the standard instructional tool for introductory and advanced courses in mathematics, engineering, and science. In industry, MATLAB is the tool of choice for highproductivity research, development, and analysis. MATLAB features a family of addon applicationspecific solutions called toolboxes. Very important to most users of MATLAB, toolboxes allow you to learn and apply specialized technology. Toolboxes are comprehensive collections of MATLAB functions (Mfiles) that extend the MATLAB environment to solve particular classes of problems. Areas in which toolboxes are available include signal processing, control systems, neural networks, fuzzy logic, wavelets, simulation, and many others. 12 What Is MATLAB? The MATLAB System
The MATLAB system consists of five main parts:
Development Environment. This is the set of tools and facilities that help you use MATLAB functions and files. Many of these tools are graphical user interfaces. It includes the MATLAB desktop and Command Window, a command history, an editor and debugger, and browsers for viewing help, the workspace, files, and the search path. The MATLAB Mathematical Function Library. This is a vast collection of computational algorithms ranging from elementary functions, like sum, sine, cosine, and complex arithmetic, to more sophisticated functions like matrix inverse, matrix eigenvalues, Bessel functions, and fast Fourier transforms.
The MATLAB Language. This is a highlevel matrix/array language with control flow statements, functions, data structures, input/output, and objectoriented programming features. It allows both "programming in the small" to rapidly create quick and dirty throwaway programs, and "programming in the large" to create large and complex application programs. Graphics. MATLAB has extensive facilities for displaying vectors and matrices as graphs, as well as annotating and printing these graphs. It includes highlevel functions for twodimensional and threedimensional data visualization, image processing, animation, and presentation graphics. It also includes lowlevel functions that allow you to fully customize the appearance of graphics as well as to build complete graphical user interfaces on your MATLAB applications. The MATLAB Application Program Interface (API). This is a library that allows you to write C and Fortran programs that interact with MATLAB. It includes facilities for calling routines from MATLAB (dynamic linking), calling MATLAB as a computational engine, and for reading and writing MATfiles. 13 1 Introduction MATLAB Documentation
MATLAB provides extensive documentation, in both printed and online format, to help you learn about and use all of its features. If you are a new user, start with this Getting Started book. It covers all the primary MATLAB features at a high level, including many examples. The MATLAB online help provides taskoriented and reference information about MATLAB features. MATLAB documentation is also available in printed form and in PDF format. MATLAB Online Help
To view the online documentation, select MATLAB Help from the Help menu in MATLAB. The MATLAB documentation is organized into these main topics: Desktop Tools and Development Environment  Startup and shutdown, the desktop, and other tools that help you use MATLAB Mathematics  Mathematical operations and data analysis Programming  The MATLAB language and how to develop MATLAB applications Graphics  Tools and techniques for plotting, graph annotation, printing, and programming with Handle Graphics 3D Visualization  Visualizing surface and volume data, transparency, and viewing and lighting techniques Creating Graphical User Interfaces  GUIbuilding tools and how to write callback functions External Interfaces/API  MEXfiles, the MATLAB engine, and interfacing to Java, COM, and the serial port MATLAB also includes reference documentation for all MATLAB functions: Functions  By Category  Lists all MATLAB functions grouped into categories Handle Graphics Property Browser  Provides easy access to descriptions of graphics object properties External Interfaces/API Reference  Covers those functions used by the MATLAB external interfaces, providing information on syntax in the calling language, description, arguments, return values, and examples 14 MATLAB Documentation The MATLAB online documentation also includes Examples  An index of examples included in the documentation Release Notes  New features and known problems in the current release Printable Documentation  PDF versions of the documentation suitable for printing For more information about using the Help browser, see Chapter 6, "Desktop Tools and Development Environment." 15 1 Introduction Starting and Quitting MATLAB
Starting MATLAB
On Windows platforms, start MATLAB by doubleclicking the MATLAB shortcut icon on your Windows desktop. On UNIX platforms, start MATLAB by typing matlab at the operating system prompt. You can customize MATLAB startup. For example, you can change the directory in which MATLAB starts or automatically execute MATLAB statements in a script file named startup.m. For More Information See "Starting MATLAB" in the Desktop Tools and Development Environment documentation. Quitting MATLAB
To end your MATLAB session, select File > Exit MATLAB in the desktop, or type quit in the Command Window. You can run a script file named finish.m each time MATLAB quits that, for example, executes functions to save the workspace, or displays a quit confirmation dialog box. For More Information See "Quitting MATLAB" in the Desktop Tools and Development Environment documentation. 16 Starting and Quitting MATLAB MATLAB Desktop
When you start MATLAB, the MATLAB desktop appears, containing tools (graphical user interfaces) for managing files, variables, and applications associated with MATLAB. The following illustration shows the default desktop. You can customize the arrangement of tools and documents to suit your needs. For more information about the desktop tools, see Chapter 6, "Desktop Tools and Development Environment." Enter MATLAB functions at the Command Window prompt. The Command History maintains a record of the MATLAB functions you ran. 17 1 Introduction 18 2
Matrices and Arrays Matrices and Magic Squares (p. 22) Expressions (p. 210) Working with Matrices (p. 214) Enter matrices, perform matrix operations, and access matrix elements. Work with variables, numbers, operators, functions, and expressions. Generate matrices, load matrices, create matrices from Mfiles and concatenation, and delete matrix rows and columns. Use matrices for linear algebra, work with arrays, multivariate data, scalar expansion, and logical subscripting, and use the find function. Change output format, suppress output, enter long lines, and edit at the command line. More About Matrices and Arrays (p. 218) Controlling Command Window Input and Output (p. 228) 2 Matrices and Arrays Matrices and Magic Squares
In MATLAB, a matrix is a rectangular array of numbers. Special meaning is sometimes attached to 1by1 matrices, which are scalars, and to matrices with only one row or column, which are vectors. MATLAB has other ways of storing both numeric and nonnumeric data, but in the beginning, it is usually best to think of everything as a matrix. The operations in MATLAB are designed to be as natural as possible. Where other programming languages work with numbers one at a time, MATLAB allows you to work with entire matrices quickly and easily. A good example matrix, used throughout this book, appears in the Renaissance engraving Melencolia I by the German artist and amateur mathematician Albrecht Drer. 22 Matrices and Magic Squares This image is filled with mathematical symbolism, and if you look carefully, you will see a matrix in the upper right corner. This matrix is known as a magic square and was believed by many in Drer's time to have genuinely magical properties. It does turn out to have some fascinating characteristics worth exploring. Entering Matrices
The best way for you to get started with MATLAB is to learn how to handle matrices. Start MATLAB and follow along with each example. You can enter matrices into MATLAB in several different ways: Enter an explicit list of elements. Load matrices from external data files. Generate matrices using builtin functions. Create matrices with your own functions in Mfiles. Start by entering Drer's matrix as a list of its elements. You only have to follow a few basic conventions: Separate the elements of a row with blanks or commas. Use a semicolon, ; , to indicate the end of each row. Surround the entire list of elements with square brackets, [ ]. 23 2 Matrices and Arrays To enter Drer's matrix, simply type in the Command Window
A = [16 3 2 13; 5 10 11 8; 9 6 7 12; 4 15 14 1] MATLAB displays the matrix you just entered.
A = 16 5 9 4 3 10 6 15 2 11 7 14 13 8 12 1 This matrix matches the numbers in the engraving. Once you have entered the matrix, it is automatically remembered in the MATLAB workspace. You can refer to it simply as A. Now that you have A in the workspace, take a look at what makes it so interesting. Why is it magic? sum, transpose, and diag
You are probably already aware that the special properties of a magic square have to do with the various ways of summing its elements. If you take the sum along any row or column, or along either of the two main diagonals, you will always get the same number. Let us verify that using MATLAB. The first statement to try is
sum(A) MATLAB replies with
ans = 34 34 34 34 When you do not specify an output variable, MATLAB uses the variable ans, short for answer, to store the results of a calculation. You have computed a row vector containing the sums of the columns of A. Sure enough, each of the columns has the same sum, the magic sum, 34. How about the row sums? MATLAB has a preference for working with the columns of a matrix, so the easiest way to get the row sums is to transpose the matrix, compute the column sums of the transpose, and then transpose the result. The transpose operation is denoted by an apostrophe or single quote, '. It flips a matrix about its main diagonal and it turns a row vector into a column vector. 24 Matrices and Magic Squares So
A' produces
ans = 16 3 2 13 5 10 11 8 9 6 7 12 4 15 14 1 And
sum(A')' produces a column vector containing the row sums
ans = 34 34 34 34 The sum of the elements on the main diagonal is obtained with the sum and the diag functions.
diag(A) produces
ans = 16 10 7 1 and
sum(diag(A)) produces
ans = 34 25 2 Matrices and Arrays The other diagonal, the socalled antidiagonal, is not so important mathematically, so MATLAB does not have a readymade function for it. But a function originally intended for use in graphics, fliplr, flips a matrix from left to right.
sum(diag(fliplr(A))) ans = 34 You have verified that the matrix in Drer's engraving is indeed a magic square and, in the process, have sampled a few MATLAB matrix operations. The following sections continue to use this matrix to illustrate additional MATLAB capabilities. Subscripts
The element in row i and column j of A is denoted by A(i,j). For example, A(4,2) is the number in the fourth row and second column. For our magic square, A(4,2) is 15. So to compute the sum of the elements in the fourth column of A, type
A(1,4) + A(2,4) + A(3,4) + A(4,4) This produces
ans = 34 but is not the most elegant way of summing a single column. It is also possible to refer to the elements of a matrix with a single subscript, A(k). This is the usual way of referencing row and column vectors. But it can also apply to a fully twodimensional matrix, in which case the array is regarded as one long column vector formed from the columns of the original matrix. So, for our magic square, A(8) is another way of referring to the value 15 stored in A(4,2). If you try to use the value of an element outside of the matrix, it is an error.
t = A(4,5) Index exceeds matrix dimensions. 26 Matrices and Magic Squares On the other hand, if you store a value in an element outside of the matrix, the size increases to accommodate the newcomer.
X = A; X(4,5) = 17 X = 16 5 9 4 3 10 6 15 2 11 7 14 13 8 12 1 0 0 0 17 The Colon Operator
The colon, :, is one of the most important MATLAB operators. It occurs in several different forms. The expression
1:10 is a row vector containing the integers from 1 to 10,
1 2 3 4 5 6 7 8 9 10 To obtain nonunit spacing, specify an increment. For example,
100:7:50 is
100 93 86 79 72 65 58 51 and
0:pi/4:pi is
0 0.7854 1.5708 2.3562 3.1416 Subscript expressions involving colons refer to portions of a matrix.
A(1:k,j) is the first k elements of the jth column of A. So
sum(A(1:4,4)) 27 2 Matrices and Arrays computes the sum of the fourth column. But there is a better way. The colon by itself refers to all the elements in a row or column of a matrix and the keyword end refers to the last row or column. So
sum(A(:,end)) computes the sum of the elements in the last column of A.
ans = 34 Why is the magic sum for a 4by4 square equal to 34? If the integers from 1 to 16 are sorted into four groups with equal sums, that sum must be
sum(1:16)/4 which, of course, is
ans = 34 The magic Function
MATLAB actually has a builtin function that creates magic squares of almost any size. Not surprisingly, this function is named magic.
B = magic(4) B = 16 5 9 4 2 11 7 14 3 10 6 15 13 8 12 1 This matrix is almost the same as the one in the Drer engraving and has all the same "magic" properties; the only difference is that the two middle columns are exchanged. To make this B into Drer's A, swap the two middle columns.
A = B(:,[1 3 2 4]) 28 Matrices and Magic Squares This says, for each of the rows of matrix B, reorder the elements in the order 1, 3, 2, 4. It produces
A = 16 5 9 4 3 10 6 15 2 11 7 14 13 8 12 1 Why would Drer go to the trouble of rearranging the columns when he could have used MATLAB ordering? No doubt he wanted to include the date of the engraving, 1514, at the bottom of his magic square. 29 2 Matrices and Arrays Expressions
Like most other programming languages, MATLAB provides mathematical expressions, but unlike most programming languages, these expressions involve entire matrices. The building blocks of expressions are "Variables" on page 210 "Numbers" on page 210 "Operators" on page 211 "Functions" on page 211 See also "Examples of Expressions" on page 213. Variables
MATLAB does not require any type declarations or dimension statements. When MATLAB encounters a new variable name, it automatically creates the variable and allocates the appropriate amount of storage. If the variable already exists, MATLAB changes its contents and, if necessary, allocates new storage. For example,
num_students = 25 creates a 1by1 matrix named num_students and stores the value 25 in its single element. Variable names consist of a letter, followed by any number of letters, digits, or underscores. MATLAB uses only the first 31 characters of a variable name. MATLAB is case sensitive; it distinguishes between uppercase and lowercase letters. A and a are not the same variable. To view the matrix assigned to any variable, simply enter the variable name. Numbers
MATLAB uses conventional decimal notation, with an optional decimal point and leading plus or minus sign, for numbers. Scientific notation uses the letter e to specify a poweroften scale factor. Imaginary numbers use either i or j as a suffix. Some examples of legal numbers are
3 9.6397238 1i 99 1.60210e20 3.14159j 0.0001 6.02252e23 3e5i 210 Expressions All numbers are stored internally using the long format specified by the IEEE floatingpoint standard. Floatingpoint numbers have a finite precision of roughly 16 significant decimal digits and a finite range of roughly 10308 to 10+308. Operators
Expressions use familiar arithmetic operators and precedence rules.
+ Addition Subtraction Multiplication Division Left division (described in "Matrices and Linear Algebra" in the MATLAB documentation) Power Complex conjugate transpose Specify evaluation order * / \ ^ ' ( ) Functions
MATLAB provides a large number of standard elementary mathematical functions, including abs, sqrt, exp, and sin. Taking the square root or logarithm of a negative number is not an error; the appropriate complex result is produced automatically. MATLAB also provides many more advanced mathematical functions, including Bessel and gamma functions. Most of these functions accept complex arguments. For a list of the elementary mathematical functions, type
help elfun For a list of more advanced mathematical and matrix functions, type
help specfun help elmat 211 2 Matrices and Arrays Some of the functions, like sqrt and sin, are built in. Builtin functions are part of the MATLAB core so they are very efficient, but the computational details are not readily accessible. Other functions, like gamma and sinh, are implemented in Mfiles. There are some differences between builtin functions and other functions. For example, for builtin functions, you cannot see the code. For other functions, you can see the code and even modify it if you want. Several special functions provide values of useful constants.
pi i j eps realmin realmax Inf NaN 3.14159265... Imaginary unit, Same as i Floatingpoint relative precision, 1 = 2 52 1022 1023 Smallest floatingpoint number, 2 Largest floatingpoint number, ( 2 )2 Infinity Notanumber Infinity is generated by dividing a nonzero value by zero, or by evaluating well defined mathematical expressions that overflow, i.e., exceed realmax. Notanumber is generated by trying to evaluate expressions like 0/0 or InfInf that do not have well defined mathematical values. The function names are not reserved. It is possible to overwrite any of them with a new variable, such as
eps = 1.e6 and then use that value in subsequent calculations. The original function can be restored with
clear eps 212 Expressions Examples of Expressions
You have already seen several examples of MATLAB expressions. Here are a few more examples, and the resulting values.
rho = (1+sqrt(5))/2 rho = 1.6180 a = abs(3+4i) a = 5 z = sqrt(besselk(4/3,rhoi)) z = 0.3730+ 0.3214i huge = exp(log(realmax)) huge = 1.7977e+308 toobig = pi*huge toobig = Inf 213 2 Matrices and Arrays Working with Matrices
This section introduces you to other ways of creating matrices: "Generating Matrices" on page 214 "The load Function" on page 215 "MFiles" on page 215 "Concatenation" on page 216 "Deleting Rows and Columns" on page 217 Generating Matrices
MATLAB provides four functions that generate basic matrices.
zeros ones rand randn All zeros All ones Uniformly distributed random elements Normally distributed random elements Here are some examples.
Z = zeros(2,4) Z = 0 0 0 0 F = 5*ones(3,3) F = 5 5 5 5 5 5 0 0 0 0 5 5 5 N = fix(10*rand(1,10)) N = 9 2 6 4 R = randn(4,4) 8 7 4 0 8 4 214 Working with Matrices R = 0.6353 0.6014 0.5512 1.0998 0.0860 2.0046 0.4931 0.4620 0.3210 1.2366 0.6313 2.3252 1.2316 1.0556 0.1132 0.3792 The load Function
The load function reads binary files containing matrices generated by earlier MATLAB sessions, or reads text files containing numeric data. The text file should be organized as a rectangular table of numbers, separated by blanks, with one row per line, and an equal number of elements in each row. For example, outside of MATLAB, create a text file containing these four lines.
16.0 5.0 9.0 4.0 3.0 10.0 6.0 15.0 2.0 11.0 7.0 14.0 13.0 8.0 12.0 1.0 Store the file under the name magik.dat. Then the statement
load magik.dat reads the file and creates a variable, magik, containing our example matrix. An easy way to read data into MATLAB in many text or binary formats is to use Import Wizard. MFiles
You can create your own matrices using Mfiles, which are text files containing MATLAB code. Use the MATLAB Editor or another text editor to create a file containing the same statements you would type at the MATLAB command line. Save the file under a name that ends in .m. For example, create a file containing these five lines.
A = [ ... 16.0 3.0 5.0 10.0 9.0 6.0 4.0 15.0 2.0 11.0 7.0 14.0 13.0 8.0 12.0 1.0 ]; 215 2 Matrices and Arrays Store the file under the name magik.m. Then the statement
magik reads the file and creates a variable, A, containing our example matrix. Concatenation
Concatenation is the process of joining small matrices to make bigger ones. In fact, you made your first matrix by concatenating its individual elements. The pair of square brackets, , is the concatenation operator. For an example, start with the 4by4 magic square, A, and form
B = [A A+32; A+48 A+16] The result is an 8by8 matrix, obtained by joining the four submatrices.
B = 16 5 9 4 64 53 57 52 3 10 6 15 51 58 54 63 2 11 7 14 50 59 55 62 13 8 12 1 61 56 60 49 48 37 41 36 32 21 25 20 35 42 38 47 19 26 22 31 34 43 39 46 18 27 23 30 45 40 44 33 29 24 28 17 This matrix is halfway to being another magic square. Its elements are a rearrangement of the integers 1:64. Its column sums are the correct value for an 8by8 magic square.
sum(B) ans = 260 260 260 260 260 260 260 260 But its row sums, sum(B')', are not all the same. Further manipulation is necessary to make this a valid 8by8 magic square. 216 Working with Matrices Deleting Rows and Columns
You can delete rows and columns from a matrix using just a pair of square brackets. Start with
X = A; Then, to delete the second column of X, use
X(:,2) = This changes X to
X = 16 5 9 4 2 11 7 14 13 8 12 1 If you delete a single element from a matrix, the result is not a matrix anymore. So, expressions like
X(1,2) = result in an error. However, using a single subscript deletes a single element, or sequence of elements, and reshapes the remaining elements into a row vector. So
X(2:2:10) = results in
X = 16 9 2 7 13 12 1 217 2 Matrices and Arrays More About Matrices and Arrays
This section shows you more about working with matrices and arrays, focusing on "Linear Algebra" on page 218 "Arrays" on page 221 "Multivariate Data" on page 224 "Scalar Expansion" on page 225 "Logical Subscripting" on page 226 "The find Function" on page 227 Linear Algebra
Informally, the terms matrix and array are often used interchangeably. More precisely, a matrix is a twodimensional numeric array that represents a linear transformation. The mathematical operations defined on matrices are the subject of linear algebra. Drer's magic square
A = [16 5 9 4 3 10 6 15 2 11 7 14 13 8 12 1 ] provides several examples that give a taste of MATLAB matrix operations. You have already seen the matrix transpose, A'. Adding a matrix to its transpose produces a symmetric matrix.
A + A' ans = 32 8 11 17 8 20 17 23 11 17 14 26 17 23 26 2 The multiplication symbol, *, denotes the matrix multiplication involving inner products between rows and columns. Multiplying the transpose of a matrix by the original matrix also produces a symmetric matrix. 218 More About Matrices and Arrays A'*A ans = 378 212 206 360 212 370 368 206 206 368 370 212 360 206 212 378 The determinant of this particular matrix happens to be zero, indicating that the matrix is singular.
d = det(A) d = 0 The reduced row echelon form of A is not the identity.
R = rref(A) R = 1 0 0 0 0 1 0 0 0 0 1 0 1 3 3 0 Since the matrix is singular, it does not have an inverse. If you try to compute the inverse with
X = inv(A) you will get a warning message
Warning: Matrix is close to singular or badly scaled. Results may be inaccurate. RCOND = 9.796086e018. Roundoff error has prevented the matrix inversion algorithm from detecting exact singularity. But the value of rcond, which stands for reciprocal condition estimate, is on the order of eps, the floatingpoint relative precision, so the computed inverse is unlikely to be of much use. 219 2 Matrices and Arrays The eigenvalues of the magic square are interesting.
e = eig(A) e = 34.0000 8.0000 0.0000 8.0000 One of the eigenvalues is zero, which is another consequence of singularity. The largest eigenvalue is 34, the magic sum. That is because the vector of all ones is an eigenvector.
v = ones(4,1) v = 1 1 1 1 A*v ans = 34 34 34 34 When a magic square is scaled by its magic sum,
P = A/34 the result is a doubly stochastic matrix whose row and column sums are all 1.
P = 0.4706 0.1471 0.2647 0.1176 0.0882 0.2941 0.1765 0.4412 0.0588 0.3235 0.2059 0.4118 0.3824 0.2353 0.3529 0.0294 220 More About Matrices and Arrays Such matrices represent the transition probabilities in a Markov process. Repeated powers of the matrix represent repeated steps of the process. For our example, the fifth power
P^5 is
0.2507 0.2497 0.2500 0.2496 0.2495 0.2501 0.2498 0.2506 0.2494 0.2502 0.2499 0.2505 0.2504 0.2500 0.2503 0.2493 This shows that as k approaches infinity, all the elements in the k th power, k p , approach 1 / 4 . Finally, the coefficients in the characteristic polynomial
poly(A) are
1 34 64 2176 0 This indicates that the characteristic polynomial det(A I ) is 34 64 + 2176 The constant term is zero, because the matrix is singular, and the coefficient of the cubic term is 34, because the matrix is magic!
4 3 2 Arrays
When they are taken away from the world of linear algebra, matrices become twodimensional numeric arrays. Arithmetic operations on arrays are done element by element. This means that addition and subtraction are the same for arrays and matrices, but that multiplicative operations are different. MATLAB uses a dot, or decimal point, as part of the notation for multiplicative array operations. 221 2 Matrices and Arrays The list of operators includes
+ .* ./ .\ .^ .' Addition Subtraction Elementbyelement multiplication Elementbyelement division Elementbyelement left division Elementbyelement power Unconjugated array transpose If the Drer magic square is multiplied by itself with array multiplication
A.*A the result is an array containing the squares of the integers from 1 to 16, in an unusual order.
ans = 256 25 81 16 9 100 36 225 4 121 49 196 169 64 144 1 Building Tables
Array operations are useful for building tables. Suppose n is the column vector
n = (0:9)'; Then
pows = [n n.^2 2.^n] 222 More About Matrices and Arrays builds a table of squares and powers of 2.
pows = 0 1 2 3 4 5 6 7 8 9 0 1 4 9 16 25 36 49 64 81 1 2 4 8 16 32 64 128 256 512 The elementary math functions operate on arrays element by element. So
format short g x = (1:0.1:2)'; logs = [x log10(x)] builds a table of logarithms.
logs = 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 0 0.04139 0.07918 0.11394 0.14613 0.17609 0.20412 0.23045 0.25527 0.27875 0.30103 223 2 Matrices and Arrays Multivariate Data
MATLAB uses columnoriented analysis for multivariate statistical data. Each column in a data set represents a variable and each row an observation. The (i,j)th element is the ith observation of the jth variable. As an example, consider a data set with three variables: Heart rate Weight Hours of exercise per week For five observations, the resulting array might look like
D = [ 72 81 69 82 75 134 201 156 148 170 3.2 3.5 7.1 2.4 1.2 ] The first row contains the heart rate, weight, and exercise hours for patient 1, the second row contains the data for patient 2, and so on. Now you can apply many MATLAB data analysis functions to this data set. For example, to obtain the mean and standard deviation of each column
mu = mean(D), sigma = std(D) mu = 75.8 sigma = 5.6303 161.8 3.48 25.499 2.2107 For a list of the data analysis functions available in MATLAB, type
help datafun If you have access to the Statistics Toolbox, type
help stats 224 More About Matrices and Arrays Scalar Expansion
Matrices and scalars can be combined in several different ways. For example, a scalar is subtracted from a matrix by subtracting it from each element. The average value of the elements in our magic square is 8.5, so
B = A  8.5 forms a matrix whose column sums are zero.
B = 7.5 3.5 0.5 4.5 sum(B) ans = 0 5.5 1.5 2.5 6.5 6.5 2.5 1.5 5.5 4.5 0.5 3.5 7.5 0 0 0 With scalar expansion, MATLAB assigns a specified scalar to all indices in a range. For example,
B(1:2,2:3) = 0 zeroes out a portion of B.
B = 7.5 3.5 0.5 4.5 0 0 2.5 6.5 0 0 1.5 5.5 4.5 0.5 3.5 7.5 225 2 Matrices and Arrays Logical Subscripting
The logical vectors created from logical and relational operations can be used to reference subarrays. Suppose X is an ordinary matrix and L is a matrix of the same size that is the result of some logical operation. Then X(L) specifies the elements of X where the elements of L are nonzero. This kind of subscripting can be done in one step by specifying the logical operation as the subscripting expression. Suppose you have the following set of data.
x = [2.1 1.7 1.6 1.5 NaN 1.9 1.8 1.5 5.1 1.8 1.4 2.2 1.6 1.8]; The NaN is a marker for a missing observation, such as a failure to respond to an item on a questionnaire. To remove the missing data with logical indexing, use isfinite(x), which is true for all finite numerical values and false for NaN and Inf.
x = x(isfinite(x)) x = 2.1 1.7 1.6 1.5 1.9 1.8 1.5 5.1 1.8 1.4 2.2 1.6 1.8 Now there is one observation, 5.1, which seems to be very different from the others. It is an outlier. The following statement removes outliers, in this case those elements more than three standard deviations from the mean.
x = x(abs(xmean(x)) <= 3*std(x)) x = 2.1 1.7 1.6 1.5 1.9 1.8 1.5 1.8 1.4 2.2 1.6 1.8 For another example, highlight the location of the prime numbers in Drer's magic square by using logical indexing and scalar expansion to set the nonprimes to 0. (See "The magic Function" on page 28.)
A(~isprime(A)) = 0 A = 0 5 0 0 3 0 0 0 2 11 7 0 13 0 0 0 226 More About Matrices and Arrays The find Function
The find function determines the indices of array elements that meet a given logical condition. In its simplest form, find returns a column vector of indices. Transpose that vector to obtain a row vector of indices. For example, start again with Drer's magic square. (See "The magic Function" on page 28.)
k = find(isprime(A))' picks out the locations, using onedimensional indexing, of the primes in the magic square.
k = 2 5 9 10 11 13 Display those primes, as a row vector in the order determined by k, with
A(k) ans = 5 3 2 11 7 13 When you use k as a lefthandside index in an assignment statement, the matrix structure is preserved.
A(k) = NaN A = 16 NaN 9 4 NaN 10 6 15 NaN NaN NaN 14 NaN 8 12 1 227 2 Matrices and Arrays Controlling Command Window Input and Output
So far, you have been using the MATLAB command line, typing functions and expressions, and seeing the results printed in the Command Window. This section describes "The format Function" on page 228 "Suppressing Output" on page 230 "Entering Long Statements" on page 230 "Command Line Editing" on page 230 The format Function
The format function controls the numeric format of the values displayed by MATLAB. The function affects only how numbers are displayed, not how MATLAB computes or saves them. Here are the different formats, together with the resulting output produced from a vector x with components of different magnitudes. Note To ensure proper spacing, use a fixedwidth font, such as Courier. x = [4/3 1.2345e6] format short 1.3333 0.0000 format short e 1.3333e+000 format short g 1.3333 1.2345e006 1.2345e006 228 Controlling Command Window Input and Output format long 1.33333333333333 format long e 1.333333333333333e+000 format long g 1.33333333333333 format bank 1.33 format rat 4/3 format hex 3ff5555555555555 3eb4b6231abfd271 1/810045 0.00 1.2345e006 1.234500000000000e006 0.00000123450000 If the largest element of a matrix is larger than 103 or smaller than 103, MATLAB applies a common scale factor for the short and long formats. In addition to the format functions shown above
format compact suppresses many of the blank lines that appear in the output. This lets you view more information on a screen or window. If you want more control over the output format, use the sprintf and fprintf functions. 229 2 Matrices and Arrays Suppressing Output
If you simply type a statement and press Return or Enter, MATLAB automatically displays the results on screen. However, if you end the line with a semicolon, MATLAB performs the computation but does not display any output. This is particularly useful when you generate large matrices. For example,
A = magic(100); Entering Long Statements
If a statement does not fit on one line, use an ellipsis (three periods), ..., followed by Return or Enter to indicate that the statement continues on the next line. For example,
s = 1 1/2 + 1/3 1/4 + 1/5  1/6 + 1/7 ...  1/8 + 1/9  1/10 + 1/11  1/12; Blank spaces around the =, +, and  signs are optional, but they improve readability. Command Line Editing
Various arrow and control keys on your keyboard allow you to recall, edit, and reuse statements you have typed earlier. For example, suppose you mistakenly enter
rho = (1 + sqt(5))/2 You have misspelled sqrt. MATLAB responds with
Undefined function or variable 'sqt'. Instead of retyping the entire line, simply press the key. The statement you typed is redisplayed. Use the key to move the cursor over and insert the missing r. Repeated use of the key recalls earlier lines. Typing a few characters and then the key finds a previous line that begins with those characters. You can also copy previously executed statements from the Command History. For more information, see "Command History" on page 66. Following is the list of arrow and control keys you can use in the Command Window. If the preference you select for Command line key bindings is Emacs (MATLAB standard), you can also use the Ctrl+key combinations 230 Controlling Command Window Input and Output shown. See also general keyboard shortcuts for desktop tools in the MATLAB Development Environment documentation.
Key Control Key for Emacs (MATLAB standard) Preference Operation Ctrl+P Ctrl+N Ctrl+B Ctrl+F Ctrl+ Ctrl+ Home End Ctrl+Home Ctrl+End Esc Delete Backspace Ctrl+U Ctrl+D Ctrl+H Ctrl+K Shift+Home Shift+End Ctrl+A Ctrl+E Recall previous line. Works only at command line. Recall next line. Works only at command line if you previously used the up arrow or Ctrl+P. Move back one character. Move forward one character. Move right one word. Move left one word. Move to beginning of command line. Move to end of command line. Move to top of Command Window. Move to end of Command Window. Clear command line. Delete character at cursor in command line. Delete character before cursor in command line. Cut contents (kill) to end of command line. Highlight to beginning of command line. Highlight to end of last line. Can start at any line in the Command Window. 231 2 Matrices and Arrays 232 3
Graphics Overview of MATLAB Plotting (p. 32) Editing Plots (p. 316) Mesh and Surface Plots (p. 352) Images (p. 358) Printing Graphics (p. 360) Handle Graphics (p. 362) Animations (p. 371) Create plots, include multiple data sets, specify property values, and save figures. Edit plots interactively and using functions, and use the property editor. Visualize functions of two variables. Work with images. Print and export figures. Work with graphics objects and set object properties. Create moving graphics. 3 Graphics Overview of MATLAB Plotting
MATLAB provides a wide variety of techniques to display data graphically. Interactive tools enable you to manipulate graphs to achieve results that reveal the most information about your data. You can also annotate and print graphs for presentations, or export graphs to standard graphics formats for presentation in web browsers or other media. For More Information "Graphics" and "3D Visualization" in the MATLAB documentation provide indepth coverage of MATLAB graphics and visualization tools. Access these topics from the Help browser. The Plotting Process
The process of visualizing data typically involves a series of operations. This section provides a "big picture" view of the plotting process and contains links to sections that have examples and specific details about performing each operation. Creating a Graph
The type of graph you choose to create depends on the nature of your data and what you want to reveal about the data. MATLAB predefines many graph types, such as line, bar, histogram, and pie graphs. There are also 3D graphs, such as surfaces, slice planes, and streamlines. There are two basic ways to create graphs in MATLAB: Use plotting tools to create graphs interactively. See "Examples  Using MATLAB Plotting Tools" on page 320. Use the command interface to enter commands in the Command Window or create plotting programs. See "Basic Plotting Functions" on page 338. You might find it useful to combine both approaches. For example, you might issue a plotting command to create a graph and then modify the graph using one of the interactive tools. 32 Overview of MATLAB Plotting Exploring Data
Once you create a graph, you can extract specific information about the data, such as the numeric value of a peak in a plot, the average value of a series of data, or you can perform data fitting. For More Information See "Data Exploration Tools" in the MATLAB documentation. Editing the Graph Components
Graphs are composed of objects, which have properties you can change. These properties affect the way the various graph components look and behave. For example, the axes used to define the coordinate system of the graph has properties that define the limits of each axis, the scale, color, etc. The line used to create a line graph has properties such as color, type of marker used at each data point (if any), line style, etc. Note that the data used to create a line graph are properties of the line. You can, therefore, change the data without actually creating a new graph. See "Editing Plots" on page 316. Annotating Graphs
Annotations are the text, arrows, callouts, and other labels added to graphs to help viewers see what is important about the data. You typically add annotations to graphs when you want to show them to other people or when you want to save them for later reference. For More Information See "Annotating Graphs" in the MATLAB documentation or select Annotating Graphs from the figure Help menu. Printing and Exporting Graphs
You can print your graph on any printer connected to your computer. The print previewer enables you to view how your graph will look when printed. It enables you to add headers, footers, a date, and so on. The page setup dialog 33 3 Graphics lets you control the size, layout, and other characteristics of the graph (select Page Setup from the figure File menu). Exporting a graph means creating a copy of it in a standard graphics file format, such as TIF, JPEG, or EPS. You can then import the file into a word processor, include it in an HTML document, or edit it in a drawing package select Export Setup from the figure File menu). For More Information See the print command reference page and "Printing and Exporting" in the MATLAB documentation or select Printing and Exporting from the figure Help menu. Saving Graphs to Reload into MATLAB
There are two ways to save graphs that enable you to save the work you have invested in their preparation: Save the graph as a FIGfile (select Save from the figure File menu). Generate MATLAB code that can recreate the graph (select Generate MFile from the figure File menu).
FIGFiles. FIGfiles are a binary format that saves a figure in its current state. This means that all graphics objects and property settings are stored in the file when you create it. You can reload the file into a different MATLAB session, even if you are running MATLAB on a different type of computer. When you load a FIGfile, MATLAB creates a new figure in the same state as the one you saved. Note that the states of any figure tools (i.e., any items on the toolbars) are not saved in a FIGfile; only the contents of the graph are saved.
Generated Code. You can use the MATLAB Mcode generator to create code that recreates the graph. Unlike a FIGfile, the generated code does not contain any data. You must pass the data to the generated function when you run the code. Note that studying the generating code for a graph is a good way to learn how to program with MATLAB. 34 Overview of MATLAB Plotting For More Information See the print command reference page and "Saving Your Work" in the MATLAB documentation. Graph Components
MATLAB displays graphs in a special window known as a figure. To create a graph, you need to define a coordinate system. Therefore every graph is placed within axes, which are contained by the figure. The actual visual representation of the data is achieved with graphics objects like lines and surfaces. These objects are drawn within the coordinate system defined by the axes, which MATLAB automatically creates specifically to accommodate the range of the data. The actual data is stored as properties of the graphics objects. See "Handle Graphics" on page 362 for more information about graphics object properties. The following picture shows the basic components of a typical graph. 35 3 Graphics Figure window displays graphs. y = 1.5cos(x) + 4e0.01xcos(x) + e0.07xsin(3x) 20 15 Axes define a coordinate system for the graph.
Y Axis 10 5 0 Line plot represents data. 5 10 15 20 10 5 0 5 10 15 X Axis 20 25 30 35 40 36 Overview of MATLAB Plotting Figure Tools
The figure is equipped with sets of tools that operate on graphs. The figure Tools menu provides access to many graph tools. For More Information See "MATLAB Plotting Tools" in the MATLAB documentation or select Plotting Tools from the figure Help menu. 37 3 Graphics Access to Tools
You can access the figure toolbars and the plotting tools from the View menu, as shown in the following picture. Enable the figure toolbars and the plotting tools from the View menu. Figure Toolbars
Figure toolbars provide easy access to many graph modification features. There are three toolbars. When you place the cursor over a particular tool, a text box pops up with the tool name. The following picture shows the three toolbars displayed with the cursor over the Data Cursor tool. For More Information See "Figure Toolbars" in the MATLAB documentation. 38 Overview of MATLAB Plotting Plotting Tools
Plotting tools are attached to figures and create an environment for creating graphs. These tools enable you to do the following: Select from a wide variety of graph types Set the properties of graphics objects Annotate graphs with text, arrows, etc. Create and arrange subplots in the figure Drag and drop data into graphs Display the plotting tools from the View menu or by clicking in the figure toolbar, as shown in the following picture.
Enable plotting tools from the View menu or toolbar. There are three components to the plotting tools: Figure Palette  Specify and arrange subplots, access workspace variables for plotting or editing, and add annotations. Plot Browser  Select objects in the graphics hierarchy, control visibility, and add data to axes. Property Editor  Change key properties of the selected object. Click Inspector for access to all object properties. 39 3 Graphics The following picture shows a figure with the plotting tools enabled. 310 Overview of MATLAB Plotting Plotting Tools and MATLAB Commands
You can enable the plotting tools on any graph, even if you created it using MATLAB commands. For example, suppose you create the following graph.
t = 0:pi/20:2*pi; y = exp(sin(t)); plotyy(t,y,t,y,'plot','stem') xlabel('X Axis') ylabel('Plot Y Axis') title('Two Y Axes') Two Y Axes 3 3 2.5 2.5 2 2 Plot Y Axis 1.5 1.5 1 1 0.5 0.5 0 0 1 2 3 X Axis 4 5 6 7 0 This graph contains two yaxes, one for each plot type (lineseries and stem graphs). The plotting tools make it easy to select any of the objects that the graph contains and set their properties. 311 3 Graphics For example, adding a label for the yaxis that corresponds to the stem plot is easily accomplished by selecting that axes in the Plot Browser and setting the YLabel property in the Property Editor. 312 Overview of MATLAB Plotting Arranging Graphs Within a Figure
You can place a number of axes within a figure by selecting the layout you want from the Figure Palette. For example, the following picture shows how to specify four 2D axes in the figure.
Click to add one axes to bottom of current layout. Click and drag right to specify axes layout. Select the axes you want to target for plotting. You can also use the subplot command to create multiple axes. 313 3 Graphics Selecting Plot Types
You can use the Plot Catalog to select from a variety of techniques for plotting data. To access the Plot Catalog,
1 Select the variables you want to plot in the Figure Palette. 2 Rightclick to display the context menu. 3 Select More Plots to display the catalog. 314 Overview of MATLAB Plotting MATLAB displays the Plot Catalog with the selected variables ready to plot, once you select a plot type.
Specify variables to plot. See a description of each plot type. Select a category of graphs and then choose a specific type. 315 3 Graphics Editing Plots
MATLAB automatically formats graphs by setting the scale of the axes, adding tick marks along axes, and using colors and line styles to distinguish the data plotted in the graph. However, if you are creating graphics for presentation, you can change the default formatting or add descriptive labels, titles, legends and other annotations to help explain your data. Plot Editing Mode
Plot editing mode enables you to perform pointandclick editing of the graphics objects in your graph. Enabling Plot Edit Mode
To enable plot edit mode, click the arrowhead in the figure toolbar:
Enable plot edit mode. You can also select Edit Plot from the figure Tools menu. Setting Object Properties
Once you have enabled plot edit mode, you can select objects by clicking on them in the graph. Selection handles appear and indicate that the object is selected. Select multiple objects using Shift+click. Rightclick with the pointer over the selected object to display the object's context menu: 316 Editing Plots The context menu provides quick access to the most commonly used operations and properties. Using the Property Editor
In plot edit mode, doubleclicking on an object in a graph starts the Property Editor with that object's major properties displayed. The Property Editor provides access to the most used object properties. It is updated to display the properties of whatever object you select. Click to display Property Inspector. 317 3 Graphics Accessing All Properties  Property Inspector
The Property Inspector is a tool that enables you to access all object properties. If you do not find the property you want to set in the Property Editor, click the Inspector button to display the Property Inspector. You can also use the inspect command to start the Property Inspector. The following picture shows the Property Inspector displaying the properties of a graph's axes. It lists each property and provides a text field or other appropriate device (such as a color picker) from which you can set the value of the property. As you select different objects, the Property Inspector is updated to display the properties of the current object. 318 Editing Plots Using Functions to Edit Graphs
If you prefer to work from the MATLAB command line, or if you are creating an Mfile, you can use MATLAB commands to edit the graphs you create. You can use the set and get commands to change the properties of the objects in a graph. For more information about using graphics commands, see "Handle Graphics" on page 362. 319 3 Graphics Examples  Using MATLAB Plotting Tools
Suppose you want to graph the function y = x3 over the x domain 1 to 1. The first step is to generate the data to plot. It is simple to evaluate a function because MATLAB can distribute arithmetic operations over all elements of a multivalued variable. For example, the following statement creates a variable x that contains values ranging from 1 to 1 in increments of 0.1 (you could also use the linspace function to generate data for x). The second statement raises each value in x to the third power and stores these values in y.
x = 1:.1:1; % Define the range of x y = x.^3; % Raise each element in x to the third power Now that you have generated some data, you can plot it using the MATLAB plotting tools. To start the plotting tools, type
plottools 320 Examples  Using MATLAB Plotting Tools MATLAB displays a figure with plotting tools attached.
Variables in workspace Figure plotting area 321 3 Graphics Plotting Two Variables
A simple line graph is a suitable way to display x as the independent variable and y as the dependent variable. To do this, select both variables (click to select, then Shiftclick to select again), then rightclick to display the context menu. Select plot(x,y) from the menu. MATLAB creates the line graph in the figure area. The black squares indicate that the line is selected and you can edit its properties with the Property Editor. 322 Examples  Using MATLAB Plotting Tools Changing the Appearance
Next change the line properties so that the graph displays only the data point. Use the Property Editor to set following properties: Line to no line Marker to o Marker size to 4.0 Marker fill color to red Set Line to no line. Set Marker to o . Set Marker size to 4.0. Set Marker fill color to red. 323 3 Graphics Adding More Data to the Graph
You can add more data to the graph by defining more variables or by specifying an expression that MATLAB uses to generate data for the plot. This second approach makes it easy to explore variations of the data already plotted. To add data to the graph, select the axes in the Plot Browser and click the Add Data button. When you are using the plotting tools, MATLAB always adds data to the existing graph, instead of replacing the graph, as it would if you issued repeated plotting commands. That is, the plotting tools are in a hold on state. 1. Select axes. 3. Enter expression 2. Click Add Data. The picture above shows how to configure the Add Data to Axes dialog to create a line plot of y = x4, which is added to the existing plot of y = x3. 324 Examples  Using MATLAB Plotting Tools Changing the Type of Graph
The plotting tools enable you to easily view your data with a variety of plot types. The following picture shows the same data as above converted to stem plots. To change the plot type,
1 Select the plotted data in the Plot Browser. 2 Select Stem in the Plot Type menu. 325 3 Graphics Select both sets of data. Select Stem as the Plot Type. 326 Examples  Using MATLAB Plotting Tools Modifying the Graph Data Source
You can link graph data to variables in your workspace. When you change the values contained in the variables, you can then update the graph to use the new data without having to create a new graph. (See also the refresh function.) Suppose you have the following data:
x = linspace(pi,pi,25); % 25 points between  and y = sin(x); Using the plotting tools, create a graph of y = sin(x).
plottools MATLAB sets the data source to the selected variables. 327 3 Graphics New Values for the Data Source
The data that defines the graph is linked to the x and y variables in the base workspace. If you assign new values to these variables and click the Refresh Data button, MATLAB updates the graph to use the new data.
x = linspace(2*pi,2*pi,25); % 25 points between 2pi and 2pi y = sin(x); % Recalculate y based on the new x values Click Refresh Data to update the plot. 328 Preparing Graphs for Presentation Preparing Graphs for Presentation
Suppose you plot the following data and want to create a graph that presents certain information about the data.
x = 10:.005:40; y = [1.5*cos(x)+4*exp(.01*x).*cos(x)+exp(.07*x).*sin(3*x)]; plot(x,y) This picture shows the graph created by the code above. 20 15 10 5 0 5 10 15 20 10 5 0 5 10 15 20 25 30 35 40 Now suppose you want to save copies of the graph by Printing the graph on a local printer so you have a copy for your notebook Exporting the graph to an Encapsulated PostScript (EPS) file to incorporate into a word processor document 329 3 Graphics Modify the Graph to Enhance the Presentation
To obtain a better view, zoom in on the graph using horizontal zoom. After enabling zoom mode from the figure toolbar, rightclick to display the context menu. Select Horizontal Zoom (2D Plots Only) from the Zoom Options. Leftclick to zoom in on a region of the graph and use the panning tool to position the points of interest where you want them on the graph. Label some key points using data tips. 330 Preparing Graphs for Presentation Finally, add text annotations, axis labels, and a title. You can add the title and axis labels using the following commands:
title ('y = 1.5cos(x) + 4e^{0.01x}cos(x) + e^{0.07x}sin(3x)') xlabel('X Axis') ylabel('Y Axis') Note that the text string passed to the title command uses TEX syntax to produce the exponents. See the text String property for more information on using TEX syntax to produce mathematical symbols. The graph is now ready to print and export. 331 3 Graphics 332 Preparing Graphs for Presentation Printing the Graph
Before printing the graph, use the print previewer to see how the graph will be laid out on the page. Display the graph in the print previewer by selecting Print Preview from the figure File menu. Click the Header button to add some descriptive text to the top of the page. 333 3 Graphics Next click the Page Setup button. The Page Setup dialog enables you to set a number of properties that control how the page is printed. Note that MATLAB recalculates the values of the axes tick marks because the printed graph is larger than the one displayed on the computer screen. To force MATLAB to use the same tick marks and limits, select Keep screen limits and ticks from the Axes and Figure tab in the Page Setup dialog. Click OK to accept the setting and dismiss the dialog. Click Print on the print previewer to send the graph to your default printer. The Page Setup dialog provides many other options for controlling how printed graphs look. Click Help for more information. 334 Preparing Graphs for Presentation Exporting the Graph
Exporting a graph is the process of creating a standard graphics file format of the graph (such as EPS or TIFF), which you can then import into other applications like word processors, drawing packages, etc. This example exports the graph as an EPS file with the following requirements: The size of the picture when imported into the word processor document should be four inches wide and three inches high. All the text in the figure should have a size of 8 points. Specifying the Size of the Graph
To set the size, use the Export Setup dialog (select Export Setup from the figure File menu).
Set the size of the graph in the exported file. 335 3 Graphics Specifying the Font Size
To set the font size of all the text in the graph, select Fonts in the Export Setup dialog's Properties selector. Then select Use fixed font size and enter 8 in the text box. 336 Preparing Graphs for Presentation Selecting the File Format
Once you finish setting options for the exported graph, click the Export button. MATLAB displays a Save As dialog that enables you to specify a name for the file as well as select the type of file format you want to use. For this example, select EPS, as shown in the following picture. Select EPS from the dropdown menu. You can import the saved file into any application that supports EPS files. The
Save as type dropdown menu lists other options for file types. You can also use the print command to print figures on your local printer or to export graphs to standard file types. For More Information See the print command reference page and "Printing and Exporting" in the MATLAB documentation or select Printing and Exporting from the figure Help menu. 337 3 Graphics Basic Plotting Functions
This section describes important graphics functions and provides examples of some typical applications. The plotting tools, described in previous sections, make use of MATLAB plotting functions and use these functions to generate code for graphs. "Creating a Plot" on page 338 "Multiple Data Sets in One Graph" on page 340 "Specifying Line Styles and Colors" on page 341 "Plotting Lines and Markers" on page 341 "Imaginary and Complex Data" on page 343 "Adding Plots to an Existing Graph" on page 344 "Figure Windows" on page 346 "Multiple Plots in One Figure" on page 346 "Controlling the Axes" on page 348 "Axis Labels and Titles" on page 349 "Saving Figures" on page 351 Creating a Plot
The plot function has different forms, depending on the input arguments. If y is a vector, plot(y) produces a piecewise linear graph of the elements of y versus the index of the elements of y. If you specify two vectors as arguments, plot(x,y) produces a graph of y versus x. For example, these statements use the colon operator to create a vector of x values ranging from 0 to 2, compute the sine of these values, and plot the result.
x = 0:pi/100:2*pi; y = sin(x); plot(x,y) 338 Basic Plotting Functions Now label the axes and add a title. The characters \pi create the symbol . See text strings for more symbols.
xlabel('x = 0:2\pi') ylabel('Sine of x') title('Plot of the Sine Function','FontSize',12) Plot of the Sine Function
1 0.8 0.6 0.4 0.2 Sine of x 0 0.2 0.4 0.6 0.8 1 0 1 2 3 x = 0:2 4 5 6 7 339 3 Graphics Multiple Data Sets in One Graph
Multiple xy pair arguments create multiple graphs with a single call to plot. MATLAB automatically cycles through a predefined (but user settable) list of colors to allow discrimination among sets of data. See the axes ColorOrder and LineStyleOrder properties. For example, these statements plot three related functions of x, with each curve in a separate distinguishing color.
x = 0:pi/100:2*pi; y = sin(x); y2 = sin(x.25); y3 = sin(x.5); plot(x,y,x,y2,x,y3) The legend command provides an easy way to identify the individual plots.
legend('sin(x)','sin(x.25)','sin(x.5)') 1 sin(x) sin(x.25) sin(x.5) 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 1 0 1 2 3 4 5 6 7 340 Basic Plotting Functions For More Information See "Defining the Color of Lines for Plotting" in "Axes Properties" in the MATLAB documentation. Specifying Line Styles and Colors
It is possible to specify color, line styles, and markers (such as plus signs or circles) when you plot your data using the plot command.
plot(x,y,'color_style_marker') color_style_marker is a string containing from one to four characters (enclosed in single quotation marks) constructed from a color, a line style, and a marker type: Color strings are 'c', 'm', 'y', 'r', 'g', 'b', 'w', and 'k'. These correspond to cyan, magenta, yellow, red, green, blue, white, and black. Line style strings are '' for solid, '' for dashed, ':' for dotted, '.' for dashdot. Omit the line style for no line. The marker types are '+', 'o', '*', and 'x', and the filled marker types are 's' for square, 'd' for diamond, '^' for up triangle, 'v' for down triangle, '>' for right triangle, '<' for left triangle, 'p' for pentagram, 'h' for hexagram, and none for no marker. You can also edit color, line style, and markers interactively. See "Editing Plots" on page 316 for more information. Plotting Lines and Markers
If you specify a marker type but not a line style, MATLAB draws only the marker. For example,
plot(x,y,'ks') plots black squares at each data point, but does not connect the markers with a line. 341 3 Graphics The statement
plot(x,y,'r:+') plots a red dotted line and places plus sign markers at each data point. Placing Markers at Every Tenth Data Point
You might want to use fewer data points to plot the markers than you use to plot the lines. This example plots the data twice using a different number of points for the dotted line and marker plots.
x1 = 0:pi/100:2*pi; x2 = 0:pi/10:2*pi; plot(x1,sin(x1),'r:',x2,sin(x2),'r+') 1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 1 0 1 2 3 4 5 6 7 342 Basic Plotting Functions Imaginary and Complex Data
When the arguments to plot are complex, the imaginary part is ignored except when you pass plot a single complex argument. For this special case, the command is a shortcut for a graph of the real part versus the imaginary part. Therefore,
plot(Z) where Z is a complex vector or matrix, is equivalent to
plot(real(Z),imag(Z)) For example,
t = 0:pi/10:2*pi; plot(exp(i*t),'o') axis equal draws a 20sided polygon with little circles at the vertices. The command axis equal makes the individual tickmark increments on the x and yaxes the same length, which makes this plot more circular in appearance.
1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 1 1 0.5 0 0.5 1 343 3 Graphics Adding Plots to an Existing Graph
The hold command enables you to add plots to an existing graph. When you type
hold on MATLAB does not replace the existing graph when you issue another plotting command; it adds the new data to the current graph, rescaling the axes if necessary. For example, these statements first create a contour plot of the peaks function, then superimpose a pseudocolor plot of the same function.
[x,y,z] = peaks; pcolor(x,y,z) shading interp hold on contour(x,y,z,20,'k') hold off The hold on command causes the pcolor plot to be combined with the contour plot in one figure. 344 Basic Plotting Functions For More Information See "Creating Specialized Plots" in the MATLAB documentation for information on a variety of graph types. 345 3 Graphics Figure Windows
Graphing functions automatically open a new figure window if there are no figure windows already on the screen. If a figure window exists, MATLAB uses that window for graphics output. If there are multiple figure windows open, MATLAB targets the one that is designated the "current figure" (the last figure used or clicked in). To make an existing figure window the current figure, you can click the mouse while the pointer is in that window or you can type
figure(n) where n is the number in the figure title bar. The results of subsequent graphics commands are displayed in this window. To open a new figure window and make it the current figure, type
figure Clearing the Figure for a New Plot
When a figure already exists, most plotting commands clear the axes and use this figure to create the new plot. However, these commands do not reset figure properties, such as the background color or the colormap. If you have set any figure properties in the previous plot, you might want to use the clf command with the reset option,
clf reset before creating your new plot to restore the figure's properties to their defaults. For More Information See "Figure Properties" and " Graphics Window  the Figure" in the MATLAB documentation for information on figures. Multiple Plots in One Figure
The subplot command enables you to display multiple plots in the same window or print them on the same piece of paper. Typing
subplot(m,n,p) 346 Basic Plotting Functions partitions the figure window into an mbyn matrix of small subplots and selects the pth subplot for the current plot. The plots are numbered along first the top row of the figure window, then the second row, and so on. For example, these statements plot data in four different subregions of the figure window.
t = 0:pi/10:2*pi; [X,Y,Z] = cylinder(4*cos(t)); subplot(2,2,1); mesh(X) subplot(2,2,2); mesh(Y) subplot(2,2,3); mesh(Z) subplot(2,2,4); mesh(X,Y,Z) 5 5 0 0 5 40 20 0 10 0 30 20 5 40 20 0 10 0 30 20 1 1 0.5 0.5 0 40 20 0 10 0 30 20 0 5 0 5 5 5 0 347 3 Graphics Controlling the Axes
The axis command provides a number of options for setting the scaling, orientation, and aspect ratio of graphs. You can also set these options interactively. See "Editing Plots" on page 316 for more information. Setting Axis Limits
By default, MATLAB finds the maxima and minima of the data and chooses the axis limits to span this range. The axis command enables you to specify your own limits:
axis([xmin xmax ymin ymax]) or for threedimensional graphs,
axis([xmin xmax ymin ymax zmin zmax]) Use the command
axis auto to reenable MATLAB automatic limit selection. Setting Axis Aspect Ratio
axis also enables you to specify a number of predefined modes. For example, axis square makes the xaxis and yaxis the same length.
axis equal makes the individual tick mark increments on the xaxes and yaxes the same length. This means
plot(exp(i*[0:pi/10:2*pi])) followed by either axis square or axis equal turns the oval into a proper circle.
axis auto normal returns the axis scaling to its default automatic mode. 348 Basic Plotting Functions Setting Axis Visibility
You can use the axis command to make the axis visible or invisible.
axis on makes the axes visible. This is the default.
axis off makes the axes invisible. Setting Grid Lines
The grid command toggles grid lines on and off. The statement
grid on turns the grid lines on, and
grid off turns them back off again. For More Information See the axis and axes reference pages and "Axes Properties" in the MATLAB documentation. Axis Labels and Titles
The xlabel, ylabel, and zlabel commands add x, y, and zaxis labels. The title command adds a title at the top of the figure and the text function inserts text anywhere in the figure. 349 3 Graphics You can produce mathematical symbols using LaTeX notation in the text string, as the following example illustrates.
t = pi:pi/100:pi; y = sin(t); plot(t,y) axis([pi pi 1 1]) xlabel('\pi \leq {\itt} \leq \pi') ylabel('sin(t)') title('Graph of the sine function') text(1,1/3,'{\itNote the odd symmetry.}')
Graph of the sine function 1 0.8 0.6 0.4 0.2 sin(t) 0 0.2 Note the odd symmetry. 0.4 0.6 0.8 1 3 2 1 0  t 1 2 3 You can also set these options interactively. See "Editing Plots" on page 316 for more information. Note that the location of the text string is defined in axes units (i.e., the same units as the data). See the annotation function for a way to place text in normalized figure units. 350 Basic Plotting Functions Saving Figures
Save a figure by selecting Save from the File menu to display a file save dialog. MATLAB saves the data it needs to recreate the figure and its contents (i.e., the entire graph) in a file with a .fig extension. To save a figure using a standard graphics format, such as TIFF, for use with other applications, select Export Setup from the File menu. You can also save from the command line  use the saveas command, including any options to save the figure in a different format. See "Exporting the Graph" on page 335 for an example. Saving Workspace Data
You can save the variables in your workspace using the Save Workspace As item in the figure File menu. You can reload saved data using the Import Data item in the figure File menu. MATLAB supports a variety of data file formats, including MATLAB data files, which have a .mat extension. Generating MCode to Recreate a Figure
You can generate MATLAB code that recreates a figure and the graph it contains by selecting the Generate MFile item from the figure File menu. This option is particularly useful if you have developed a graph using plotting tools and want to create a similar graph using the same or different data. Saving Figures That Are Compatible with Previous Version of MATLAB
Create backwardcompatible FIGfiles by following these two steps. Ensure that any plotting functions used to create the contents of the figure are called with the 'v6' argument, where applicable. Use the 'v6' option with the hgsave command. For More Information See Plot Objects and Backward Compatibility in the MATLAB documentation more information. 351 3 Graphics Mesh and Surface Plots
MATLAB defines a surface by the zcoordinates of points above a grid in the xy plane, using straight lines to connect adjacent points. The mesh and surf plotting functions display surfaces in three dimensions. mesh produces wireframe surfaces that color only the lines connecting the defining points. surf displays both the connecting lines and the faces of the surface in color. The figure colormap and figure properties determine how MATLAB colors the surface. Visualizing Functions of Two Variables
To display a function of two variables, z = f (x,y), Generate X and Y matrices consisting of repeated rows and columns, respectively, over the domain of the function. Use X and Y to evaluate and graph the function. The meshgrid function transforms the domain specified by a single vector or two vectors x and y into matrices X and Y for use in evaluating functions of two variables. The rows of X are copies of the vector x and the columns of Y are copies of the vector y. Example  Graphing the sinc Function
This example evaluates and graphs the twodimensional sinc function, sin(r)/r, between the x and y directions. R is the distance from the origin, which is at the center of the matrix. Adding eps (a MATLAB command that returns a small floatingpoint number) avoids the indeterminate 0/0 at the origin.
[X,Y] = meshgrid(8:.5:8); R = sqrt(X.^2 + Y.^2) + eps; Z = sin(R)./R; mesh(X,Y,Z,'EdgeColor','black') 352 Mesh and Surface Plots 1 0.5 0 0.5 10 5 0 5 10 10 5 5 0 10 By default, MATLAB colors the mesh using the current colormap. However, this example uses a singlecolored mesh by specifying the EdgeColor surface property. See the surface reference page for a list of all surface properties. You can create a mesh with seethrough faces by disabling hidden line removal.
hidden off See the hidden reference page for more information on this option. Example  Colored Surface Plots
A surface plot is similar to a mesh plot except that MATLAB colors the rectangular faces of the surface. The color of each faces is determined by the values of Z and the colormap (a colormap is an ordered list of colors). These statements graph the sinc function as a surface plot, specify a colormap, and add a color bar to show the mapping of data to color.
surf(X,Y,Z) colormap hsv colorbar 353 3 Graphics 1 1 0.8 0.5 0.6 0.4 0 0.2 0.5 10 5 0 5 10 10 5 5 0 0.2 10 0 See the colormap reference page for information on colormaps. For More Information See "Creating 3D Graphs" in the MATLAB documentation for more information on surface plots. Transparent Surfaces
You can make the faces of a surface transparent to a varying degree. Transparency (referred to as the alpha value) can be specified for the whole object or can be based on an alphamap, which behaves in a way analogous to colormaps. For example,
surf(X,Y,Z) colormap hsv alpha(.4) 354 Mesh and Surface Plots produces a surface with a face alpha value of 0.4. Alpha values range from 0 (completely transparent) to 1 (not transparent). For More Information See "Transparency" in the MATLAB documentation for more information on using this feature. Surface Plots with Lighting
Lighting is the technique of illuminating an object with a directional light source. In certain cases, this technique can make subtle differences in surface shape easier to see. Lighting can also be used to add realism to threedimensional graphs. This example uses the same surface as the previous examples, but colors it red and removes the mesh lines. A light object is then added to the left of the "camera" (the camera is the location in space from where you are viewing the surface).
surf(X,Y,Z,'FaceColor','red','EdgeColor','none') camlight left; lighting phong 355 3 Graphics Manipulating the Surface
The figure toolbar and the camera toolbar provide ways to explore 3D graphics interactively. Display the camera toolbar by selecting Camera Toolbar from the figure View menu. The following picture shows both toolbars with the Rotate 3D tool selected. These tools enables you to move the camera around the surface object, zoom, add lighting, and perform other viewing operations without issuing commands. 356 Mesh and Surface Plots The following picture shows the surface viewed by orbiting the camera toward the bottom using Rotate 3D. A scene light has been added to illuminate the underside of the surface, which is not lit by the light added in the previous section. For More Information See "Lighting as a Visualization Tool" and "View Control with the Camera Toolbar" in the MATLAB documentation for information on these techniques. 357 3 Graphics Images
Twodimensional arrays can be displayed as images, where the array elements determine brightness or color of the images. For example, the statements
load durer whos Name X caption map Size 648x509 2x28 128x3 Bytes Class double array char array double array 2638656 112 3072 load the file durer.mat, adding three variables to the workspace. The matrix X is a 648by509 matrix and map is a 128by3 matrix that is the colormap for this image. MATfiles, such as durer.mat, are binary files that can be created on one platform and later read by MATLAB on a different platform. The elements of X are integers between 1 and 128, which serve as indices into the colormap, map. Then
image(X) colormap(map) axis image reproduces Drer's etching shown at the beginning of this book. A highresolution scan of the magic square in the upper right corner is available in another file. Type
load detail and then use the up arrow key on your keyboard to reexecute the image, colormap, and axis commands. The statement
colormap(hot) adds some twentieth century colorization to the sixteenth century etching. The function hot generates a colormap containing shades of reds, oranges, and yellows. Typically a given image matrix has a specific colormap associated with it. See the colormap reference page for a list of other predefined colormaps. 358 Images Reading and Writing Images
You can read standard image files (TIFF, JPEG, BMP, etc.) into MATLAB using the imread function. The type of data returned by imread depends on the type of image you are reading. You can write MATLAB data to a variety of standard image formats using the imwrite function. See the reference pages for these functions for more information and examples. For More Information See "Displaying BitMapped Images" in the MATLAB documentation for information on the image processing capabilities of MATLAB. 359 3 Graphics Printing Graphics
You can print a MATLAB figure directly on a printer connected to your computer or you can export the figure to one of the standard graphics file formats supported by MATLAB. There are two ways to print and export figures: Use the Print or Export Setup options under the File menu Use the print command to print or export the figure See "Preparing Graphs for Presentation" on page 329 for an example. Printing from the Menu
There are four menu options under the File menu that pertain to printing: The Page Setup option displays a dialog box that enables you to adjust characteristics of the figure on the printed page. The Print Setup option displays a dialog box that sets printing defaults, but does not actually print the figure. The Print Preview option enables you to view the figure the way it will look on the printed page. The Print option displays a dialog box that lets you select standard printing options and print the figure. Generally, use Print Preview to determine whether the printed output is what you want. If not, use the Page Setup dialog box to change the output settings. Select the Page Setup dialog box Help button to display information on how to set up the page. Exporting Figure to Graphics Files
The Export Setup option in the File menu enables you to set a variety of figure characteristics, such as size and font type, as well as apply predefined templates to achieve standardlooking graphics files. After setup, you can export the figure to a number of standard graphics file formats. 360 Printing Graphics Using the Print Command
The print command provides more flexibility in the type of output sent to the printer and allows you to control printing from Mfiles. The result can be sent directly to your default printer or stored in a specified file. A wide variety of output formats, including TIFF, JPEG, and PostScript, is available. For example, this statement saves the contents of the current figure window as color Encapsulated Level 2 PostScript in the file called magicsquare.eps. It also includes a TIFF preview, which enables most word processors to display the picture.
print depsc2 tiff magicsquare.eps To save the same figure as a TIFF file with a resolution of 200 dpi, use the command
print dtiff r200 magicsquare.tiff If you type print on the command line,
print MATLAB prints the current figure on your default printer. For More Information See the print reference page and "Printing and Exporting" in the MATLAB documentation for more information on printing. 361 3 Graphics Handle Graphics
Handle Graphics refers to a system of graphics objects that MATLAB uses to implement graphing and visualization functions. Each object created has a fixed set of properties. You can use these properties to control the behavior and appearance of your graph. When you call a plotting function, MATLAB creates the graph using various graphics objects, such as a figure window, axes, lines, text, and so on. MATLAB enables you to query the value of each property and set the values of most properties. For example, the following statement creates a figure with a white background color and without displaying the figure toolbar.
figure('Color','white','Toolbar','none') Using the Handle
Whenever MATLAB creates a graphics object, it assigns an identifier (called a handle) to the object. You can use this handle to access the object's properties with the set and get functions. For example, the following statements create a graph and return a handle to a lineseries object in h.
x = 1:10; y = x.^3; h = plot(x,y); You can use the handle h to set the properties of the lineseries object. For example, you can set its Color property:
set(h,'Color','red') You can also specify properties when you call the plotting function.
h = plot(x,y,'Color','red'); When you query the lineseries properties,
get(h,'LineWidth') MATLAB returns the answer:
ans = 0.5000 362 Handle Graphics Use the handle to see what properties a particular object contains.
get(h) Graphics Objects
Graphics objects are the basic elements used to display graphs and user interface components. These objects are organized into a hierarchy, as shown by the following diagram.
Root Figure UI Objects Axes Hidden Annotation Axes Core Objects Plot Objects Group Objects Annotation Objects Key Graphics Objects
When you call a function to create a graph, MATLAB creates a hierarchy of graphics objects. For example, calling the plot function creates the following graphics objects: Lineseries plot objects  Represent the data passed to the plot function Axes  Provide a frame of reference and scaling for the plotted lineseries Text  Label the axes tick marks and are used for titles and annotations Figures  Are the windows that contain axes toolbars, menus, etc. Different types of graphs use different objects to represent data: however, all data objects are contained in axes and all objects (except root) are contained in figures. The root is an abstract object that primarily stores information about your computer or MATLAB state. You cannot create an instance of the root object. 363 3 Graphics For More Information See "Handle Graphics Objects" in the MATLAB documentation for information on graphics objects. User interface objects are used to create graphical user interfaces (GUIs). These objects include components like push buttons, editable text boxes, and list boxes. For More Information See "Creating Graphical User Interfaces" in the MATLAB documentation for more information on user interface objects. Creating Objects
Plotting functions (like plot and surf) call the appropriate lowlevel function to draw their respective graph. For information about an object's properties, see the Handle Graphics Property Browser in the MATLAB online documentation. Commands for Working with Objects
This table lists commands commonly used when working with objects.
Function allchild ancestor copyobj delete findall findobj Purpose Find all children of specified objects Find ancestor of graphics object Copy graphics object Delete an object Find all graphics objects (including hidden handles) Find the handles of objects having specified property values Return the handle of the current axes gca 364 Handle Graphics Function gcf gco get ishandle set Purpose Return the handle of the current figure Return the handle of the current object Query the values of an object's properties True if value is valid object handle Set the values of an object's properties Setting Object Properties
All object properties have default values. However, you might find it useful to change the settings of some properties to customize your graph. There are two ways to set object properties: Specify values for properties when you create the object. Set the property value on an object that already exists. Setting Properties from Plotting Commands
You can specify object property value pairs as arguments to many plotting functions, such as plot, mesh, and surf. For example, plotting commands that create lineseries or surfaceplot objects enable you to specify property name/property value pairs as arguments. The command
surf(x,y,z,'FaceColor','interp',... 'FaceLighting','gouraud') plots the data in the variables x, y, and z using a surfaceplot object with interpolated face color and employing the Gouraud face light technique. You can set any of the object's properties this way. Setting Properties of Existing Objects
To modify the property values of existing objects, you can use the set command or the Property Editor. This section describes how to use the set command. See "Using the Property Editor" on page 317 for more information. 365 3 Graphics Most plotting functions return the handles of the objects that they create so you can modify the objects using the set command. For example, these statements plot a fivebyfive matrix (creating five lineseries, one per column) and then set the Marker property to a square and the MarkerFaceColor property to green.
h = plot(magic(5)); set(h,'Marker','s','MarkerFaceColor','g') In this case, h is a vector containing five handles, one for each of the five lineseries in the graph. The set statement sets the Marker and MarkerFaceColor properties of all lineseries to the same values. Setting Multiple Property Values
If you want to set the properties of each lineseries to a different value, you can use cell arrays to store all the data and pass it to the set command. For example, create a plot and save the lineseries handles.
h = plot(magic(5)); Suppose you want to add different markers to each lineseries and color the marker's face color the same color as the lineseries. You need to define two cell arrays  one containing the property names and the other containing the desired values of the properties. The prop_name cell array contains two elements.
prop_name(1) = {'Marker'}; prop_name(2) = {'MarkerFaceColor'}; The prop_values cell array contains 10 values: five values for the Marker property and five values for the MarkerFaceColor property. Notice that prop_values is a twodimensional cell array. The first dimension indicates which handle in h the values apply to and the second dimension indicates which property the value is assigned to.
prop_values(1,1) prop_values(1,2) prop_values(2,1) prop_values(2,2) prop_values(3,1) prop_values(3,2) prop_values(4,1) prop_values(4,2) = = = = = = = = {'s'}; {get(h(1),'Color')}; {'d'}; {get(h(2),'Color')}; {'o'}; {get(h(3),'Color')}; {'p'}; {get(h(4),'Color')}; 366 Handle Graphics prop_values(5,1) = {'h'}; prop_values(5,2) = {get(h(5),'Color')}; The MarkerFaceColor is always assigned the value of the corresponding line's color (obtained by getting the lineseries Color property with the get command). After defining the cell arrays, call set to specify the new property values.
set(h,prop_name,prop_values) 25 20 15 10 5 0 1 1.5 2 2.5 3 3.5 4 4.5 5 367 3 Graphics Specifying the Axes or Figure
MATLAB always creates an axes or figure if one does not exist when you issue a plotting command. However, when you are creating a graphics Mfile, it is good practice to explicitly create and specify the parent axes and figure, particularly if others will use your program. Specifying the parent prevents the following problems: Your Mfile overwrites the graph in the current figure. Note that a figure becomes the current figure whenever a user clicks on it. The current figure might be in an unexpected state and not behave as your program expects. The following examples shows a simple Mfile that plots a function and the mean of the function over the specified range.
function myfunc(x) % x = 10:.005:40; Here's a value you can use for x y = [1.5*cos(x) + 6*exp(.1*x) + exp(.07*x).*sin(3*x)]; ym = mean(y); hfig = figure('Name','Function and Mean',... 'Pointer','fullcrosshair'); hax = axes('Parent',hfig); plot(hax,x,y) hold on plot(hax,[min(x) max(x)],[ym ym],'Color','red') hold off ylab = get(hax,'YTick'); set(hax,'YTick',sort([ylab ym])) title ('y = 1.5cos(x) + 6e^{0.1x} + e^{0.07x}sin(3x)') xlabel('X Axis'); ylabel('Y Axis') 368 Handle Graphics y = 1.5cos(x) + 6e0.1x + e0.07xsin(3x) 20 15 10 5 3.1596 Y Axis 0 5 10 15 20 10 5 0 5 10 15 X Axis 20 25 30 35 40 Finding the Handles of Existing Objects
The findobj function enables you to obtain the handles of graphics objects by searching for objects with particular property values. With findobj you can specify the values of any combination of properties, which makes it easy to pick one object out of many. findobj also recognizes regular expressions (regexp). For example, you might want to find the blue line with square marker having blue face color.You can also specify which figures or axes to search, if there are more than one. The following sections provide examples illustrating how to use findobj. Finding All Objects of a Certain Type
Because all objects have a Type property that identifies the type of object, you can find the handles of all occurrences of a particular type of object. For example,
h = findobj('Type','patch'); finds the handles of all patch objects. 369 3 Graphics Finding Objects with a Particular Property
You can specify multiple properties to narrow the search. For example,
h = findobj('Type','line','Color','r','LineStyle',':'); finds the handles of all red dotted lines. Limiting the Scope of the Search
You can specify the starting point in the object hierarchy by passing the handle of the starting figure or axes as the first argument. For example,
h = findobj(gca,'Type','text','String','\pi/2'); finds the string /2 only within the current axes. Using findobj as an Argument
Because findobj returns the handles it finds, you can use it in place of the handle argument. For example,
set(findobj('Type','line','Color','red'),'LineStyle',':') finds all red lines and sets their line style to dotted. 370 Animations Animations
MATLAB provides three ways of generating moving, animated graphics: "Erase Mode Method" on page 371  Continually erase and then redraw the objects on the screen, making incremental changes with each redraw. "Creating Movies" on page 373  Save a number of different pictures and then play them back as a movie. Using AVI files. See avifile for more information and examples. Erase Mode Method
Using the EraseMode property is appropriate for long sequences of simple plots where the change from frame to frame is minimal. Here is an example showing simulated Brownian motion. Specify a number of points, such as
n = 20 and a temperature or velocity, such as
s = .02 The best values for these two parameters depend upon the speed of your particular computer. Generate n random points with (x,y) coordinates between 1/2 and +1/2.
x = rand(n,1)0.5; y = rand(n,1)0.5; Plot the points in a square with sides at 1 and +1. Save the handle for the vector of points and set its EraseMode to xor. This tells the MATLAB graphics system not to redraw the entire plot when the coordinates of one point are changed, but to restore the background color in the vicinity of the point using an exclusive or operation.
h = plot(x,y,'.'); axis([1 1 1 1]) axis square grid off set(h,'EraseMode','xor','MarkerSize',18) 371 3 Graphics Now begin the animation. Here is an infinite while loop, which you can eventually exit by typing Ctrl+c. Each time through the loop, add a small amount of normally distributed random noise to the coordinates of the points. Then, instead of creating an entirely new plot, simply change the XData and YData properties of the original plot.
while 1 drawnow x = x + s*randn(n,1); y = y + s*randn(n,1); set(h,'XData',x,'YData',y) end See how long it takes for one of the points to get outside the square and how long before all the points are outside the square. 1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 1 1 0.5 0 0.5 1 372 Animations Creating Movies
If you increase the number of points in the Brownian motion example to n = 300 and s = .02, the motion is no longer very fluid; it takes too much time to draw each time step. It becomes more effective to save a predetermined number of frames as bitmaps and to play them back as a movie. First, decide on the number of frames,
nframes = 50; Next, set up the first plot as before, except using the default EraseMode (normal).
x = rand(n,1)0.5; y = rand(n,1)0.5; h = plot(x,y,'.'); set(h,'MarkerSize',18); axis([1 1 1 1]) axis square grid off Generate the movie and use getframe to capture each frame.
for k = 1:nframes x = x + s*randn(n,1); y = y + s*randn(n,1); set(h,'XData',x,'YData',y) M(k) = getframe; end Finally, play the movie 30 times.
movie(M,30) 373 3 Graphics 374 4
Programming Flow Control (p. 42) Other Data Structures (p. 48) Scripts and Functions (p. 418) Use flow control constructs including if, switch and case, for, while, continue, and break. Work with multidimensional arrays, cell arrays, character and text data, and structures. Write scripts and functions, use global variables, pass string arguments to functions, use eval to evaluate text expressions, vectorize code, preallocate arrays, reference functions using handles, and use functions that operate on functions. 4 Programming Flow Control
MATLAB has several flow control constructs: "if" on page 42 "switch and case" on page 44 "for" on page 44 "while" on page 45 "continue" on page 45 "break" on page 46 "try  catch" on page 47 "return" on page 47 if
The if statement evaluates a logical expression and executes a group of statements when the expression is true. The optional elseif and else keywords provide for the execution of alternate groups of statements. An end keyword, which matches the if, terminates the last group of statements. The groups of statements are delineated by the four keywords  no braces or brackets are involved. The MATLAB algorithm for generating a magic square of order n involves three different cases: when n is odd, when n is even but not divisible by 4, or when n is divisible by 4. This is described by
if rem(n,2) ~= 0 M = odd_magic(n) elseif rem(n,4) ~= 0 M = single_even_magic(n) else M = double_even_magic(n) end In this example, the three cases are mutually exclusive, but if they weren't, the first true condition would be executed. 42 Flow Control It is important to understand how relational operators and if statements work with matrices. When you want to check for equality between two variables, you might use
if A == B, ... This is legal MATLAB code, and does what you expect when A and B are scalars. But when A and B are matrices, A == B does not test if they are equal, it tests where they are equal; the result is another matrix of 0's and 1's showing elementbyelement equality. In fact, if A and B are not the same size, then A == B is an error. The proper way to check for equality between two variables is to use the isequal function,
if isequal(A,B), ... Here is another example to emphasize this point. If A and B are scalars, the following program will never reach the unexpected situation. But for most pairs of matrices, including our magic squares with interchanged columns, none of the matrix conditions A > B, A < B, or A == B is true for all elements and so the else clause is executed.
if A > B 'greater' elseif A < B 'less' elseif A == B 'equal' else error('Unexpected situation') end Several functions are helpful for reducing the results of matrix comparisons to scalar conditions for use with if, including
isequal isempty all any 43 4 Programming switch and case
The switch statement executes groups of statements based on the value of a variable or expression. The keywords case and otherwise delineate the groups. Only the first matching case is executed. There must always be an end to match the switch. The logic of the magic squares algorithm can also be described by
switch (rem(n,4)==0) + (rem(n,2)==0) case 0 M = odd_magic(n) case 1 M = single_even_magic(n) case 2 M = double_even_magic(n) otherwise error('This is impossible') end Note Unlike the C language switch statement, MATLAB switch does not fall through. If the first case statement is true, the other case statements do not execute. So, break statements are not required. for
The for loop repeats a group of statements a fixed, predetermined number of times. A matching end delineates the statements.
for n = 3:32 r(n) = rank(magic(n)); end r The semicolon terminating the inner statement suppresses repeated printing, and the r after the loop displays the final result. 44 Flow Control It is a good idea to indent the loops for readability, especially when they are nested.
for i = 1:m for j = 1:n H(i,j) = 1/(i+j); end end while
The while loop repeats a group of statements an indefinite number of times under control of a logical condition. A matching end delineates the statements. Here is a complete program, illustrating while, if, else, and end, that uses interval bisection to find a zero of a polynomial.
a = 0; fa = Inf; b = 3; fb = Inf; while ba > eps*b x = (a+b)/2; fx = x^32*x5; if sign(fx) == sign(fa) a = x; fa = fx; else b = x; fb = fx; end end x The result is a root of the polynomial x3  2x  5, namely
x = 2.09455148154233 The cautions involving matrix comparisons that are discussed in the section on the if statement also apply to the while statement. continue
The continue statement passes control to the next iteration of the for loop or while loop in which it appears, skipping any remaining statements in the body 45 4 Programming of the loop. In nested loops, continue passes control to the next iteration of the for loop or while loop enclosing it. The example below shows a continue loop that counts the lines of code in the file magic.m, skipping all blank lines and comments. A continue statement is used to advance to the next line in magic.m without incrementing the count whenever a blank line or comment line is encountered.
fid = fopen('magic.m','r'); count = 0; while ~feof(fid) line = fgetl(fid); if isempty(line)  strncmp(line,'%',1) continue end count = count + 1; end disp(sprintf('%d lines',count)); break
The break statement lets you exit early from a for loop or while loop. In nested loops, break exits from the innermost loop only. Here is an improvement on the example from the previous section. Why is this use of break a good idea?
a = 0; fa = Inf; b = 3; fb = Inf; while ba > eps*b x = (a+b)/2; fx = x^32*x5; if fx == 0 break elseif sign(fx) == sign(fa) a = x; fa = fx; else b = x; fb = fx; end end x 46 Flow Control try  catch
The general form of a trycatch statement sequence is
try statement ... statement catch statement ... statement end In this sequence the statements between try and catch are executed until an error occurs. The statements between catch and end are then executed. Use lasterr to see the cause of the error. If an error occurs between catch and end, MATLAB terminates execution unless another trycatch sequence has been established. return
return terminates the current sequence of commands and returns control to the invoking function or to the keyboard. return is also used to terminate keyboard mode. A called function normally transfers control to the function that invoked it when it reaches the end of the function. You can insert a return statement within the called function to force an early termination and to transfer control to the invoking function. 47 4 Programming Other Data Structures
This section introduces you to some other data structures in MATLAB, including "Multidimensional Arrays" on page 48 "Cell Arrays" on page 410 "Characters and Text" on page 412 "Structures" on page 415 Multidimensional Arrays
Multidimensional arrays in MATLAB are arrays with more than two subscripts. One way of creating a multidimensional array is by calling zeros, ones, rand, or randn with more than two arguments. For example,
R = randn(3,4,5); creates a 3by4by5 array with a total of 3x4x5 = 60 normally distributed random elements. A threedimensional array might represent threedimensional physical data, say the temperature in a room, sampled on a rectangular grid. Or it might represent a sequence of matrices, A(k), or samples of a timedependent matrix, A(t). In these latter cases, the (i, j)th element of the kth matrix, or the tkth matrix, is denoted by A(i,j,k). MATLAB and Drer's versions of the magic square of order 4 differ by an interchange of two columns. Many different magic squares can be generated by interchanging columns. The statement
p = perms(1:4); generates the 4! = 24 permutations of 1:4. The kth permutation is the row vector p(k,:). Then
A = magic(4); M = zeros(4,4,24); for k = 1:24 M(:,:,k) = A(:,p(k,:)); end 48 Other Data Structures stores the sequence of 24 magic squares in a threedimensional array, M. The size of M is
size(M) ans = 4 4 24
13 8 16 5 9 4 2 11 7 14 3 10 6 15 . ..
16 16 16 5 9 4 10 2 6 11 15 7 14 28 12 8 3 121 10 1 6 15 3 11 13 117 13 14 7 8 14 12 1 2 10 3 106 15 6 15 13 8 12 1 12 1 Note The order of the matrices shown in this illustration might differ from your results. The perms function always returns all permutations of the input vector, but the order of the permutations might be different for different MATLAB versions. The statement
sum(M,d) computes sums by varying the dth subscript. So
sum(M,1) 49 4 Programming is a 1by4by24 array containing 24 copies of the row vector
34 34 34 34 and
sum(M,2) is a 4by1by24 array containing 24 copies of the column vector
34 34 34 34 Finally,
S = sum(M,3) adds the 24 matrices in the sequence. The result has size 4by4by1, so it looks like a 4by4 array.
S = 204 204 204 204 204 204 204 204 204 204 204 204 204 204 204 204 Cell Arrays
Cell arrays in MATLAB are multidimensional arrays whose elements are copies of other arrays. A cell array of empty matrices can be created with the cell function. But, more often, cell arrays are created by enclosing a miscellaneous collection of things in curly braces, {}. The curly braces are also used with subscripts to access the contents of various cells. For example,
C = {A sum(A) prod(prod(A))} produces a 1by3 cell array. The three cells contain the magic square, the row vector of column sums, and the product of all its elements. When C is displayed, you see
C = [4x4 double] [1x4 double] [20922789888000] 410 Other Data Structures This is because the first two cells are too large to print in this limited space, but the third cell contains only a single number, 16!, so there is room to print it. Here are two important points to remember. First, to retrieve the contents of one of the cells, use subscripts in curly braces. For example, C{1} retrieves the magic square and C{3} is 16. Second, cell arrays contain copies of other arrays, not pointers to those arrays. If you subsequently change A, nothing happens to C. You can use threedimensional arrays to store a sequence of matrices of the same size. Cell arrays can be used to store a sequence of matrices of different sizes. For example,
M = cell(8,1); for n = 1:8 M{n} = magic(n); end M produces a sequence of magic squares of different order.
M = [ [ [ [ [ [ [ [ 2x2 3x3 4x4 5x5 6x6 7x7 8x8 1] double] double] double] double] double] double] double] 411 4 Programming 64 9 17 40 32 41 49 8 2 55 47 26 34 23 15 58 3 54 46 27 35 22 14 59 61 12 20 37 29 44 52 5 60 13 21 36 28 45 53 4 6 51 43 30 38 19 11 62 7 50 42 31 39 18 10 63 57 16 24 33 25 48 56 1 16 5 9 4 8 3 4 1 4 1 3 2 1 5 9 2 11 7 14 6 7 2 3 10 6 15 You can retrieve our old friend with
M{4} Characters and Text
Enter text into MATLAB using single quotes. For example,
s = 'Hello' The result is not the same kind of numeric matrix or array you have been dealing with up to now. It is a 1by5 character array. 412 .. .
13 8 12 1 Other Data Structures Internally, the characters are stored as numbers, but not in floatingpoint format. The statement
a = double(s) converts the character array to a numeric matrix containing floatingpoint representations of the ASCII codes for each character. The result is
a = 72 101 108 108 111 The statement
s = char(a) reverses the conversion. Converting numbers to characters makes it possible to investigate the various fonts available on your computer. The printable characters in the basic ASCII character set are represented by the integers 32:127. (The integers less than 32 represent nonprintable control characters.) These integers are arranged in an appropriate 6by16 array with
F = reshape(32:127,16,6)'; The printable characters in the extended ASCII character set are represented by F+128. When these integers are interpreted as characters, the result depends on the font currently being used. Type the statements
char(F) char(F+128) and then vary the font being used for the Command Window. Select Preferences from the File menu to change the font. If you include tabs in lines of code, use a fixedwidth font, such as Monospaced, to align the tab positions on different lines. Concatenation with square brackets joins text variables together into larger strings. The statement
h = [s, ' world'] joins the strings horizontally and produces
h = Hello world 413 4 Programming The statement
v = [s; 'world'] joins the strings vertically and produces
v = Hello world Note that a blank has to be inserted before the 'w' in h and that both words in v have to have the same length. The resulting arrays are both character arrays; h is 1by11 and v is 2by5. To manipulate a body of text containing lines of different lengths, you have two choices  a padded character array or a cell array of strings. When creating a character array, you must make each row of the array the same length. (Pad the ends of the shorter rows with spaces.) The char function does this padding for you. For example,
S = char('A','rolling','stone','gathers','momentum.') produces a 5by9 character array.
S = A rolling stone gathers momentum. Alternatively, you can store the text in a cell array. For example,
C = {'A';'rolling';'stone';'gathers';'momentum.'} creates a 5by1 cell array that requires no padding because each row of the array can have a different length.
C = 'A' 'rolling' 'stone' 'gathers' 'momentum.' 414 Other Data Structures You can convert a padded character array to a cell array of strings with
C = cellstr(S) and reverse the process with
S = char(C) Structures
Structures are multidimensional MATLAB arrays with elements accessed by textual field designators. For example,
S.name = 'Ed Plum'; S.score = 83; S.grade = 'B+' creates a scalar structure with three fields.
S = name: 'Ed Plum' score: 83 grade: 'B+' Like everything else in MATLAB, structures are arrays, so you can insert additional elements. In this case, each element of the array is a structure with several fields. The fields can be added one at a time,
S(2).name = 'Toni Miller'; S(2).score = 91; S(2).grade = 'A'; or an entire element can be added with a single statement.
S(3) = struct('name','Jerry Garcia',... 'score',70,'grade','C') Now the structure is large enough that only a summary is printed.
S = 1x3 struct array with fields: name score grade 415 4 Programming There are several ways to reassemble the various fields into other MATLAB arrays. They are all based on the notation of a commaseparated list. If you type
S.score it is the same as typing
S(1).score, S(2).score, S(3).score This is a commaseparated list. Without any other punctuation, it is not very useful. It assigns the three scores, one at a time, to the default variable ans and dutifully prints out the result of each assignment. But when you enclose the expression in square brackets,
[S.score] it is the same as
[S(1).score, S(2).score, S(3).score] which produces a numeric row vector containing all the scores.
ans = 83 91 70 Similarly, typing
S.name just assigns the names, one at a time, to ans. But enclosing the expression in curly braces,
{S.name} creates a 1by3 cell array containing the three names.
ans = 'Ed Plum' 'Toni Miller' 'Jerry Garcia' And
char(S.name) 416 Other Data Structures calls the char function with three arguments to create a character array from the name fields,
ans = Ed Plum Toni Miller Jerry Garcia Dynamic Field Names
The most common way to access the data in a structure is by specifying the name of the field that you want to reference. Another means of accessing structure data is to use dynamic field names. These names express the field as a variable expression that MATLAB evaluates at runtime. The dotparentheses syntax shown here makes expression a dynamic field name:
structName.(expression) Index into this field using the standard MATLAB indexing syntax. For example, to evaluate expression into a field name and obtain the values of that field at columns 1 through 25 of row 7, use
structName.(expression)(7,1:25) Dynamic Field Names Example. The avgscore function shown below computes an average test score, retrieving information from the testscores structure using dynamic field names: function avg = avgscore(testscores, student, first, last) for k = first:last scores(k) = testscores.(student).week(k); end avg = sum(scores)/(last  first + 1); You can run this function using different values for the dynamic field student:
avgscore(testscores, 'Ann Lane', 1, 20) ans = 83.5000 avgscore(testscores, 'William King', 1, 20) ans = 92.1000 417 4 Programming Scripts and Functions
Topics covered in this section are "Scripts" on page 419 "Functions" on page 420 "Global Variables" on page 424 "Passing String Arguments to Functions" on page 425 "The eval Function" on page 426 "Function Handles" on page 427 "Function Functions" on page 427 "Vectorization" on page 430 "Preallocation" on page 430 MATLAB is a powerful programming language as well as an interactive computational environment. Files that contain code in the MATLAB language are called Mfiles. You create Mfiles using a text editor, then use them as you would any other MATLAB function or command. There are two kinds of Mfiles: Scripts, which do not accept input arguments or return output arguments. They operate on data in the workspace. Functions, which can accept input arguments and return output arguments. Internal variables are local to the function. If you're a new MATLAB programmer, just create the Mfiles that you want to try out in the current directory. As you develop more of your own Mfiles, you will want to organize them into other directories and personal toolboxes that you can add to your MATLAB search path. If you duplicate function names, MATLAB executes the one that occurs first in the search path. To view the contents of an Mfile, for example, myfunction.m, use
type myfunction 418 Scripts and Functions Scripts
When you invoke a script, MATLAB simply executes the commands found in the file. Scripts can operate on existing data in the workspace, or they can create new data on which to operate. Although scripts do not return output arguments, any variables that they create remain in the workspace, to be used in subsequent computations. In addition, scripts can produce graphical output using functions like plot. For example, create a file called magicrank.m that contains these MATLAB commands:
% Investigate the rank of magic squares r = zeros(1,32); for n = 3:32 r(n) = rank(magic(n)); end r bar(r) Typing the statement
magicrank causes MATLAB to execute the commands, compute the rank of the first 30 magic squares, and plot a bar graph of the result. After execution of the file is complete, the variables n and r remain in the workspace. 419 4 Programming 35 30 25 20 15 10 5 0 0 5 10 15 20 25 30 35 Functions
Functions are Mfiles that can accept input arguments and return output arguments. The names of the Mfile and of the function should be the same. Functions operate on variables within their own workspace, separate from the workspace you access at the MATLAB command prompt. A good example is provided by rank. The Mfile rank.m is available in the directory
toolbox/matlab/matfun You can see the file with
type rank 420 Scripts and Functions Here is the file.
function r = rank(A,tol) % RANK Matrix rank. % RANK(A) provides an estimate of the number of linearly % independent rows or columns of a matrix A. % RANK(A,tol) is the number of singular values of A % that are larger than tol. % RANK(A) uses the default tol = max(size(A)) * norm(A) * eps. s = svd(A); if nargin==1 tol = max(size(A)') * max(s) * eps; end r = sum(s > tol); The first line of a function Mfile starts with the keyword function. It gives the function name and order of arguments. In this case, there are up to two input arguments and one output argument. The next several lines, up to the first blank or executable line, are comment lines that provide the help text. These lines are printed when you type
help rank The first line of the help text is the H1 line, which MATLAB displays when you use the lookfor command or request help on a directory. The rest of the file is the executable MATLAB code defining the function. The variable s introduced in the body of the function, as well as the variables on the first line, r, A and tol, are all local to the function; they are separate from any variables in the MATLAB workspace. This example illustrates one aspect of MATLAB functions that is not ordinarily found in other programming languages  a variable number of arguments. The rank function can be used in several different ways.
rank(A) r = rank(A) r = rank(A,1.e6) 421 4 Programming Many Mfiles work this way. If no output argument is supplied, the result is stored in ans. If the second input argument is not supplied, the function computes a default value. Within the body of the function, two quantities named nargin and nargout are available which tell you the number of input and output arguments involved in each particular use of the function. The rank function uses nargin, but does not need to use nargout. Types of Functions
MATLAB offers several different types of functions to use in your programming. Anonymous Functions
An anonymous function is a simple form of MATLAB function that does not require an Mfile. It consists of a single MATLAB expression and any number of input and output arguments. You can define an anonymous function right at the MATLAB command line, or within an Mfile function or script. This gives you a quick means of creating simple functions without having to create Mfiles each time. The syntax for creating an anonymous function from an expression is
f = @(arglist)expression The statement below creates an anonymous function that finds the square of a number. When you call this function, MATLAB assigns the value you pass in to variable x, and then uses x in the equation x.^2:
sqr = @(x) x.^2; To execute the sqr function defined above, type
a = sqr(5) a = 25 Primary and Subfunctions
All functions that are not anonymous must be defined within an Mfile. Each Mfile has a required primary function that appears first in the file, and any number of subfunctions that follow the primary. Primary functions have a wider scope than subfunctions. That is, primary functions can be invoked from outside of their Mfile (from the MATLAB command line or from functions in 422 Scripts and Functions other Mfiles) while subfunctions cannot. Subfunctions are visible only to the primary function and other subfunctions within their own Mfile. The rank function shown in the section on "Functions" on page 420 is an example of a primary function. Private Functions
A private function is a type of primary Mfile function. Its unique characteristic is that it is visible only to a limited group of other functions. This type of function can be useful if you want to limit access to a function, or when you choose not to expose the implementation of a function. Private functions reside in subdirectories with the special name private. They are visible only to functions in the parent directory. For example, assume the directory newmath is on the MATLAB search path. A subdirectory of newmath called private can contain functions that only the functions in newmath can call. Because private functions are invisible outside the parent directory, they can use the same names as functions in other directories. This is useful if you want to create your own version of a particular function while retaining the original in another directory. Because MATLAB looks for private functions before standard Mfile functions, it will find a private function named test.m before a nonprivate Mfile named test.m. Nested Functions
You can define functions within the body of any MATLAB Mfile function. These are said to be nested within the outer function. A nested function contains any or all of the components of any other Mfile function. In this example, function y is nested in function x:
function x = A(p1, p2) ... B(p2) function y = B(p3) ... end ... end 423 4 Programming Like other functions, a nested function has its own workspace where variables used by the function are stored. But it also has access to the workspaces of all functions in which it is nested. So, for example, a variable that has a value assigned to it by the primary function can be read or overwritten by a function nested at any level within the primary. Similarly, a variable that is assigned in a nested function can be read or overwritten by any of the functions containing that function. Function Overloading
Overloaded functions act the same way as overloaded functions in most computer languages. Overloaded functions are useful when you need to create a function that responds to different types of inputs accordingly. For instance, you might want one of your functions to accept both doubleprecision and integer input, but to handle each type somewhat differently. You can make this difference invisible to the user by creating two separate functions having the same name, and designating one to handle double types and one to handle integers. When you call the function, MATLAB chooses which Mfile to dispatch to based on the type of the input arguments. Global Variables
If you want more than one function to share a single copy of a variable, simply declare the variable as global in all the functions. Do the same thing at the command line if you want the base workspace to access the variable. The global declaration must occur before the variable is actually used in a function. Although it is not required, using capital letters for the names of global variables helps distinguish them from other variables. For example, create an Mfile called falling.m.
function h = falling(t) global GRAVITY h = 1/2*GRAVITY*t.^2; Then interactively enter the statements
global GRAVITY GRAVITY = 32; y = falling((0:.1:5)'); 424 Scripts and Functions The two global statements make the value assigned to GRAVITY at the command prompt available inside the function. You can then modify GRAVITY interactively and obtain new solutions without editing any files. Passing String Arguments to Functions
You can write MATLAB functions that accept string arguments without the parentheses and quotes. That is, MATLAB interprets
foo a b c as
foo('a','b','c') However, when you use the unquoted form, MATLAB cannot return output arguments. For example,
legend apples oranges creates a legend on a plot using the strings apples and oranges as labels. If you want the legend command to return its output arguments, then you must use the quoted form.
[legh,objh] = legend('apples','oranges'); In addition, you cannot use the unquoted form if any of the arguments is not a string. Constructing String Arguments in Code
The quoted form enables you to construct string arguments within the code. The following example processes multiple data files, August1.dat, August2.dat, and so on. It uses the function int2str, which converts an integer to a character, to build the filename.
for d = 1:31 s = ['August' int2str(d) '.dat']; load(s) % Code to process the contents of the dth file end 425 4 Programming A Cautionary Note
While the unquoted syntax is convenient, it can be used incorrectly without causing MATLAB to generate an error. For example, given a matrix A,
A = 0 6 5 6 2 20 1 16 10 The eig command returns the eigenvalues of A.
eig(A) ans = 3.0710 2.4645+17.6008i 2.464517.6008i The following statement is not allowed because A is not a string; however, MATLAB does not generate an error.
eig A ans = 65 MATLAB actually takes the eigenvalue of the ASCII numeric equivalent of the letter A (which is the number 65). The eval Function
The eval function works with text variables to implement a powerful text macro facility. The expression or statement
eval(s) uses the MATLAB interpreter to evaluate the expression or execute the statement contained in the text string s. The example of the previous section could also be done with the following code, although this would be somewhat less efficient because it involves the full interpreter, not just a function call. 426 Scripts and Functions for d = 1:31 s = ['load August' int2str(d) '.dat']; eval(s) % Process the contents of the dth file end Function Handles
You can create a handle to any MATLAB function and then use that handle as a means of referencing the function. A function handle is typically passed in an argument list to other functions, which can then execute, or evaluate, the function using the handle. Construct a function handle in MATLAB using the at sign, @, before the function name. The following example creates a function handle for the sin function and assigns it to the variable fhandle.
fhandle = @sin; You can call a function by means of its handle in the same way that you would call the function using its name. The syntax is
fhandle(arg1, arg2, ...); The function plot_fhandle, shown below, receives a function handle and data, generates yaxis data using the function handle, and plots it:
function x = plot_fhandle(fhandle, data) plot(data, fhandle(data)) When you call plot_fhandle with a handle to the sin function and the argument shown below, the resulting evaluation produces a sine wave plot.
plot_fhandle(@sin, pi:0.01:pi) Function Functions
A class of functions called "function functions" works with nonlinear functions of a scalar variable. That is, one function works on another function. The function functions include Zero finding Optimization 427 4 Programming Quadrature Ordinary differential equations MATLAB represents the nonlinear function by a function Mfile. For example, here is a simplified version of the function humps from the matlab/demos directory.
function y = humps(x) y = 1./((x.3).^2 + .01) + 1./((x.9).^2 + .04)  6; Evaluate this function at a set of points in the interval 0 x 1 with
x = 0:.002:1; y = humps(x); Then plot the function with
plot(x,y) 100 90 80 70 60 50 40 30 20 10 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 428 Scripts and Functions The graph shows that the function has a local minimum near x = 0.6. The function fminsearch finds the minimizer, the value of x where the function takes on this minimum. The first argument to fminsearch is a function handle to the function being minimized and the second argument is a rough guess at the location of the minimum.
p = fminsearch(@humps,.5) p = 0.6370 To evaluate the function at the minimizer,
humps(p) ans = 11.2528 Numerical analysts use the terms quadrature and integration to distinguish between numerical approximation of definite integrals and numerical integration of ordinary differential equations. MATLAB quadrature routines are quad and quadl. The statement
Q = quadl(@humps,0,1) computes the area under the curve in the graph and produces
Q = 29.8583 Finally, the graph shows that the function is never zero on this interval. So, if you search for a zero with
z = fzero(@humps,.5) you will find one outside the interval
z = 0.1316 429 4 Programming Vectorization
One way to make your MATLAB programs run faster is to vectorize the algorithms you use in constructing the programs. Where other programming languages might use for loops or DO loops, MATLAB can use vector or matrix operations. A simple example involves creating a table of logarithms.
x = .01; for k = 1:1001 y(k) = log10(x); x = x + .01; end A vectorized version of the same code is
x = .01:.01:10; y = log10(x); For more complicated code, vectorization options are not always so obvious. For More Information See "Maximizing MATLAB Performance" for other techniques that you can use. Preallocation
If you can't vectorize a piece of code, you can make your for loops go faster by preallocating any vectors or arrays in which output results are stored. For example, this code uses the function zeros to preallocate the vector created in the for loop. This makes the for loop execute significantly faster.
r = zeros(32,1); for n = 1:32 r(n) = rank(magic(n)); end Without the preallocation in the previous example, the MATLAB interpreter enlarges the r vector by one element each time through the loop. Vector preallocation eliminates this step and results in faster execution. 430 5
Creating Graphical User Interfaces What Is GUIDE? (p. 52) Laying Out a GUI (p. 53) Programming a GUI (p. 56) Introduces GUIDE, the MATLAB graphical user interface design environment Briefly describes the GUIDE Layout Editor Introduces callbacks to define behavior of the GUI components 5 Creating Graphical User Interfaces What Is GUIDE?
GUIDE, the MATLAB graphical user interface development environment, provides a set of tools for creating graphical user interfaces (GUIs). These tools greatly simplify the process of designing and building GUIs. You can use the GUIDE tools to Lay out the GUI. Using the GUIDE Layout Editor, you can lay out a GUI easily by clicking and dragging GUI components  such as panels, buttons, text fields, sliders, menus, and so on  into the layout area. GUIDE stores the GUI layout in a FIGfile. Program the GUI. GUIDE automatically generates an Mfile that controls how the GUI operates. The Mfile initializes the GUI and contains a framework for the most commonly used callbacks for each component  the commands that execute when a user clicks a GUI component. Using the Mfile editor, you can add code to the callbacks to perform the functions you want. 52 Laying Out a GUI Laying Out a GUI
Starting GUIDE
Start GUIDE by typing guide at the MATLAB command prompt. This displays the GUIDE Quick Start dialog, as shown in the following figure. From the Quick Start dialog, you can Create a new GUI from one of the GUIDE templates  prebuilt GUIs that you can modify for your own purposes. Open an existing GUI. 53 5 Creating Graphical User Interfaces The Layout Editor
When you open a GUI in GUIDE, it is displayed in the Layout Editor, which is the control panel for all of the GUIDE tools. The following figure shows the Layout Editor with a blank GUI template. Layout Area
Component Palette You can lay out your GUI by dragging components, such as panels, push buttons, popup menus, or axes, from the component palette, at the left side of the Layout Editor, into the layout area. For example, if you drag a push button into the layout area, it appears as in the following figure. 54 Laying Out a GUI You can also use the Layout Editor to create menus and set basic properties of the GUI components. To get started using the Layout Editor and setting property values, see "Creating a GUI" in the MATLAB documentation. 55 5 Creating Graphical User Interfaces Programming a GUI
After laying out the GUI and setting component properties, the next step is to program the GUI. You program the GUI by coding one or more callbacks for each of its components. Callbacks are functions that execute in response to some action by the user. A typical action is clicking a push button. A GUI's callbacks are found in an Mfile that GUIDE generates automatically. GUIDE adds templates for the most commonly used calbacks to this Mfile, but you may want to add others. Use the Mfile Editor to edit this file. The following figure shows the Callback template for a push button. To learn more about programming a GUI, see "Creating a GUI" in the MATLAB documentation. 56 6
Desktop Tools and Development Environment Desktop Overview (p. 62) Command Window and Command History (p. 65) Help Browser (p. 67) Access tools, arrange the desktop, and set preferences. Run functions and enter variables. Find and view documentation and demos. Current Directory Browser and Search Manage and use Mfiles with MATLAB. Path (p. 610) Workspace Browser and Array Editor (p. 612) Editor/Debugger (p. 614) MLint Code Check and Profiler Reports (p. 616) Other Development Environment Features (p. 619) Work with variables in MATLAB. Create and debug Mfiles (MATLAB programs). Improve and tune your Mfiles. Interface with source control systems, and publish Mfile results. 6 Desktop Tools and Development Environment Desktop Overview
Use desktop tools to manage your work in MATLAB. You can also use MATLAB functions to perform the equivalent of most of the features found in the desktop tools. The following illustration shows the default configuration of the MATLAB desktop. You can modify the setup to meet your needs. Menus change, depending on the tool you are currently using. View or change the current directory. Click to move Command Window outside of desktop (undock). For More Information For an overview of the desktop tools, watch the video tutorials, accessible by typing demo matlab desktop. For complete details, see the MATLAB "Desktop Tools and Development Environment" documentation. 62 Desktop Overview Arranging the Desktop
These are some common ways to customize the desktop: Show or hide desktop tools via the Desktop menu. Resize any tool by dragging one of its edges. Move a tool outside of the desktop by clicking the undock button tool's title bar. in the Reposition a tool within the desktop by dragging its title bar to the new location. Tools can occupy the same position, as shown for the Current Directory and Workspace browser in the preceding illustration, in which case, you access a tool via its tab. Change fonts and other options by using File > Preferences. Start Button
The MATLAB Start button provides easy access to tools, demos, shortcuts, and documentation. Click the Start button to see the options. 63 6 Desktop Tools and Development Environment For More Information See "Desktop" in the MATLAB Desktop Tools and Development Environment documentation. 64 Command Window and Command History Command Window and Command History
Command Window
Use the Command Window to enter variables and to run functions and Mfile scripts. Run functions and enter variables at the MATLAB prompt. MATLAB displays the results. Press the up arrow key to recall a statement you previously typed. Edit the statement as needed and then press Enter to run it. For more information about entering statements in the Command Window, see "Controlling Command Window Input and Output" on page 228. For More Information See "Running Functions" in the MATLAB Desktop Tools and Development Environment documentation for complete details. 65 6 Desktop Tools and Development Environment Command History
Statements you enter in the Command Window are logged in the Command History. From the Command History, you can view previously run statements, as well as copy and execute selected statements. You can also create an Mfile from selected statements. Timestamp marks the start of each session. Select one or more entries and rightclick to copy, evaluate, or create an Mfile from the selection. To save the input and output from a MATLAB session to a file, use the diary function. For More Information See "Command History" in the MATLAB Desktop Tools and Development Environment documentation, and the reference page for the diary function. 66 Help Browser Help Browser
Use the Help browser to search for and view documentation and demos for all your MathWorks products. The Help browser is an HTML viewer integrated into the MATLAB desktop. To open the Help browser, click the help button in the desktop toolbar. The Help browser consists of two panes, the Help Navigator, which you use to find information, and the display pane, where you view the information. These are the key features: Contents tab  View the titles and tables of contents of the documentation. Index tab  Find specific index entries (selected keywords) in the documentation. Search tab  Look for specific words in the documentation. Demos tab  View and run demonstrations for your MathWorks products. While viewing the documentation, you can Browse to other pages  Use the arrows at the tops and bottoms of the pages to move through the document, or use the back and forward buttons in the toolbar to go to previously viewed pages. Bookmark pages  Use the Favorites menu. Print a page  Click the print button in the toolbar. Find a term in the page  Click the find icon ( ) in the toolbar. Copy or evaluate a selection  Select text, such as code from an example, then rightclick and use a context menu item to copy the selection or evaluate (run) it. 67 6 Desktop Tools and Development Environment Tabs in the Help Navigator pane provide different ways to find information. Use the close box to hide the pane. Drag the separator bar to adjust the width of the panes. View documentation in the display pane. 68 Help Browser Other Forms of Help
In addition to the Help browser, you can use help functions. To get help for a specific function, use the doc function. For example, doc format displays documentation for the format function in the Help browser. If you type help followed by the function name, a briefer form of the documentation for the function appears in the Command Window. Other means for getting help include contacting Technical Support (http://www.mathworks.com/support) and participating in the newsgroup for MATLAB users, comp.softsys.matlab. For More Information See "Help for Using MATLAB" in the MATLAB Desktop Tools and Development Environment documentation, and the reference pages for the doc and help functions. 69 6 Desktop Tools and Development Environment Current Directory Browser and Search Path
MATLAB file operations use the current directory and the search path as reference points. Any file you want to run must either be in the current directory or on the search path. Current Directory
A quick way to view or change the current directory is by using the current directory field in the desktop toolbar, shown here. To search for, view, open, and make changes to MATLAB related directories and files, use the MATLAB Current Directory browser. Alternatively, you can use the functions dir, cd, and delete. Use the Visual Directory and Directory Reports to help you manage Mfiles.
Change the directory here. This field only appears here when the Current Directory browser is undocked from the desktop. Search for files and content within text files. For Visual Directory and Directory Reports Doubleclick a file to open it in an appropriate tool. 610 Current Directory Browser and Search Path For More Information See "File Management Operations" in the MATLAB Desktop Tools and Development Environment documentation, and the reference pages for the dir, cd, and delete functions. Search Path
MATLAB uses a search path to find Mfiles and other MATLAB related files, which are organized in directories on your file system. Any file you want to run in MATLAB must reside in the current directory or in a directory that is on the search path. When you create Mfiles and related files for MATLAB, add the directories in which they are located to the MATLAB search path. By default, the files supplied with MATLAB and other MathWorks products are included in the search path. To see which directories are on the search path or to change the search path, select File > Set Path and use the resulting Set Path dialog box. Alternatively, you can use the path function to view the search path, addpath to add directories to the path, and rmpath to remove directories from the path. For More Information See "Search Path" in the MATLAB Desktop Tools and Development Environment documentation, and the reference pages for the path, addpath, and rmpath functions. 611 6 Desktop Tools and Development Environment Workspace Browser and Array Editor
The MATLAB workspace consists of the set of variables (named arrays) built up during a MATLAB session and stored in memory. You add variables to the workspace by using functions, running Mfiles, and loading saved workspaces. Workspace Browser
To view the workspace and information about each variable, use the Workspace browser, or use the functions who and whos. To delete variables from the workspace, select the variables and select Edit >
Delete. Alternatively, use the clear function. The workspace is not maintained after you end the MATLAB session. To save the workspace to a file that can be read during a later MATLAB session, select File > Save, or use the save function. This saves the workspace to a binary file called a MATfile, which has a .mat extension. You can use options to save to different formats. To read in a MATfile, select File > Import Data, or use the load function. For More Information See "MATLAB Workspace" in the MATLAB Desktop Tools and Development Environment documentation, and the reference pages for the who, clear, save, and load functions. 612 Workspace Browser and Array Editor Array Editor
Doubleclick a variable in the Workspace browser, or use openvar variablename, to see it in the Array Editor. Use the Array Editor to view and edit a visual representation of variables in the workspace.
View and change values of array elements. Arrange the display of array documents. Use document bar to view other variables you have open in the Array Editor. For More Information See "Viewing and Editing Workspace Variables with the Array Editor" in the MATLAB Desktop Tools and Development Environment documentation, and the reference page for the openvar function. 613 6 Desktop Tools and Development Environment Editor/Debugger
Use the Editor/Debugger to create and debug Mfiles, which are programs you write to run MATLAB functions. The Editor/Debugger provides a graphical user interface for text editing, as well as for Mfile debugging. To create or edit an Mfile use File > New or File > Open, or use the edit function. Comment selected lines and specify indenting style using the Text menu. Find and replace text. Arrange the display of documents in the Editor. Set breakpoints where you want execution to pause so you can examine variables. Hold the cursor over a variable and its current value appears (known as a data tip). Use the document bar to access other documents open in the Editor/Debugger. 614 Editor/Debugger You can use any text editor to create Mfiles, such as Emacs. Use preferences (accessible from the desktop File menu) to specify that editor as the default. If you use another editor, you can still use the MATLAB Editor/Debugger for debugging, or you can use debugging functions, such as dbstop, which sets a breakpoint. If you just need to view the contents of an Mfile, you can display the contents in the Command Window using the type function. For More Information See "Editing and Debugging MFiles" in the MATLAB Desktop Tools and Development Environment documentation, and the function reference pages for edit, type, and debug. 615 6 Desktop Tools and Development Environment MLint Code Check and Profiler Reports
MATLAB provides tools to help you manage and improve your Mfiles, including the MLint Code Check and Profiler Reports. MLint Code Check Report
The MLint Code Check Report displays potential errors and problems, as well as opportunities for improvement in your Mfiles. The term "lint" is used by similar tools in other programming languages such as C. Access the MLint Code Check Report and other directory reports from the Current Directory browser. You run a report for all files in the current directory. Alternatively, you can use the mlint function to get results for a single Mfile.
Directory reports In MATLAB, the MLint Code Check Report displays a message for each line of an Mfile it determines might be improved. For example, a common MLint message is that a variable is defined but never used in the Mfile. 616 MLint Code Check and Profiler Reports
The report displays a line number and message for each potential problem or improvement opportunity. Click a line number to open the Mfile in the Editor at that line. For More Information See "Tuning and Managing MFiles" in the MATLAB Desktop Tools and Development Environment documentation, and the reference page for the mlint function. 617 6 Desktop Tools and Development Environment Profiler Report
MATLAB includes the Profiler to help you improve the performance of your Mfiles. Run a MATLAB statement or an Mfile in the Profiler and it produces a report of where the time is being spent. Access the Profiler from the Desktop menu, or use the profile function. For More Information See "Tuning and Managing MFiles" in the MATLAB Desktop Tools and Development Environment documentation, and the reference page for the profile function. 618 Other Development Environment Features Other Development Environment Features
Additional development environment features include Source Control  Access your source control system from within MATLAB. Publishing Results  Use the Editor's cell features to publish Mfiles and results to popular output formats including HTML and Microsoft Word. You can also use MATLAB Notebook to access MATLAB functions from within Microsoft Word. For More Information See "Source Control" and "Publishing Results" in the MATLAB Desktop Tools and Development Environment documentation. 619 6 Desktop Tools and Development Environment 620 Index
Symbols
: operator 27 axis labels 349 titles 349
axis function 348 A
algorithms vectorizing 430 animation 371 annotating plots 316 ans function 24 application program interface (API) 13 Array Editor 613 array operators 222 arrays cell 410 character 412 columnwise organization 224 creating in Mfiles 215 deleting rows and columns 217 elements 210 generating with functions and operators 214 listing contents 210 loading from external data files 215 and matrices 221 multidimensional 48 notation for elements 210 preallocating 430 structure 415 variable names 210 arrow keys for editing commands 230 aspect ratio of axes 348 AVI files 371 axes managing 348 visibility 349 B
bit map 359 break function 46 builtin functions defined 212 C
callbacks 56 case function 44 catch function 47 cell arrays 410 char function 414 character arrays 412 characteristic polynomial 221 colon operator 27 colormap 354 colors lines for plotting 341 Command History 66 command line editing 230 Command Window 65 complex numbers plotting 343 concatenation 216 strings 413 constants special 212 Index1 Index continue function 45 continuing statements on multiple lines 230 control keys for editing commands 230 current directory 610 Current Directory browser 610 F
figure function 346 D
data source 327 debugging Mfiles 614 deleting array elements 217 demos running from the Start button 63 desktop for MATLAB 17 desktop tools 61 determinant of matrix 219 diag function 24 documentation 67 E
editing command lines 230 Editor/Debugger 614 eigenvalue 220 eigenvector 220 elements of arrays 210 entering matrices 23 erase mode 371 eval function 426 executing MATLAB 16 exiting MATLAB 16 exporting graphs 335 expressions evaluating 426 examples 213 using in MATLAB 210 figure tools 37 figure windows 346 with multiple plots 346 find function 227 finding object handles 369 fliplr function 26 floatingpoint numbers 211 flow control 42 for loop 44 format of output display 228 format function 228 function functions 427 function handles defined 427 using 429 function keyword 421 function Mfiles 418 naming 420 function of two variables 352 functions builtin, defined 212 defined 420 how to find 211 running 65 variable number of arguments 421 G
global variables 424 graphical user interface creating 51 laying out 53 programming 56 Index2 Index graphics files 360 Handle Graphics 362 objects 363 printing 360 grids 349 GUIDE 51 line continuation 230 line styles of plots 341 load function 215 loading arrays 215 local variables 421 log of functions used 66 logical vectors 226 H
Handle Graphics 362 defined 13 finding handles 369 Help browser 67 help functions 69 hold function 344 M
magic function 28 I
if function 42 images 358 imaginary numbers 210 K
keys for editing in Command Window 230 L
legend adding to plot 340 legend function 340 library mathematical function 13 lighting 355 limits axes 348 magic square 24 markers 341 MATfile 358 mathematical function library 13 mathematical functions listing advanced 211 listing elementary 211 listing matrix 211 MATLAB application program interface 13 desktop 17 executing 16 exiting 16 history 12 language 13 mathematical function library 13 overview 12 quitting 16 running 16 shutting down 16 starting 16 MATLAB user newsgroup 69 matrices 218 creating 214 entering 23 Index3 Index matrix 22 antidiagonal 26 determinant 219 main diagonal 25 multiplication 218 singular 219 swapping columns 28 symmetric 218 transpose 24 mesh plot 352 Mfiles and toolboxes 12 creating 418 editing 614 for creating arrays 215 function 418 script 418 Microsoft Word and access to MATLAB 619 movies 373 multidimensional arrays 48 multiple data sets plotting 340 multiple plots per figure 346 multivariate data organizing 224 O
object properties 365 objects finding handles 369 graphics 363 online help viewing 67 operators 211 colon 27 output controlling format 228 suppressing 230 overlaying plots 344 P
path 611 plot function 338 plots editing 316 plotting adding legend 340 adding plots 344 basic 338 complex data 343 complex numbers 343 contours 344 editing 316 functions 338 line colors 341 line styles 341 lines and markers 341 mesh and surface 352 multiple data sets 340 multiple plots 346 overview 32 tools 39 N
newsgroup for MATLAB users 69 Notebook 619 numbers 210 floatingpoint 211 Index4 Index PostScript 361 preallocation 430 presentation graphics 329 print function 360 Printing 333 printing example 333 graphics 360 Profiler 618 Property Editor interface 319 Property Inspector 318 Q
quitting MATLAB 16 R
return function 47 revision control systems interfacing to MATLAB 619 running functions 65 running MATLAB 16 source control systems interfacing to MATLAB 619 special constants infinity 212 notanumber 212 specialized graphs 345 Start button 63 starting MATLAB 16 statements continuing on multiple lines 230 executing 426 strings concatenating 413 structures 415 subplot function 346 subscripting with logical vectors 226 subscripts 26 sum function 24 suppressing output 230 surface plot 352 switch function 44 symmetric matrix 218 S
scalar expansion 225 scientific notation 210 script Mfiles 418 scripts 419 search path 611 semicolon to suppress output 230 shutting down MATLAB 16 singular matrix 219 T
text entering in MATLAB 412 TIFF 361 title figure 349 toolboxes 12 tools in the desktop 61 transpose function 24 try function 47 Index5 Index V
variables 210 global 424 local 421 vectorization 430 vectors 22 logical 226 preallocating 430 version control systems interfacing to MATLAB 619 visibility of axes 349 W
while loop 45 windows for plotting 346 windows in MATLAB 17 wireframe surface 352 Word and access to MATLAB 619 word processing access to MATLAB 619 workspace 612 Workspace browser 612 X
xor erase mode 371 Index6 ...
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