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99 Pages

### Basic Matlab guide by professor

Course: ECE 8833, Spring 2010
School: Georgia Tech
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Word Count: 14277

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Language MATLAB The of Technical Computing Computation Visualization Programming Getting Started with MATLAB Version 5 How to Contact The MathWorks: 508-647-7000 508-647-7001 The MathWorks, Inc. 24 Prime Park Way Natick, MA 01760-1500 http://www.mathworks.com ftp.mathworks.com comp.soft-sys.matlab support@mathworks.com suggest@mathworks.com bugs@mathworks.com doc@mathworks.com subscribe@mathworks.com...

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Language MATLAB The of Technical Computing Computation Visualization Programming Getting Started with MATLAB Version 5 How to Contact The MathWorks: 508-647-7000 508-647-7001 The MathWorks, Inc. 24 Prime Park Way Natick, MA 01760-1500 http://www.mathworks.com ftp.mathworks.com comp.soft-sys.matlab support@mathworks.com suggest@mathworks.com bugs@mathworks.com doc@mathworks.com subscribe@mathworks.com service@mathworks.com info@mathworks.com Phone Fax Mail Web Anonymous FTP server Newsgroup Technical support Product enhancement suggestions Bug reports Documentation error reports Subscribing user registration Order status, license renewals, passcodes Sales, pricing, and general information Getting Started with MATLAB COPYRIGHT 1984 - 1998 by The MathWorks, Inc. The software described in this document is furnished under a license agreement. The software may be used or copied only under the terms of the license agreement. No part of this manual may be photocopied or reproduced in any form without prior written consent from The MathWorks, Inc. U.S. GOVERNMENT: If Licensee is acquiring the Programs on behalf of any unit or agency of the U.S. Government, the following shall apply: (a) For units of the Department of Defense: the Government shall have only the rights specified in the license under which the commercial computer software or commercial software documentation was obtained, as set forth in subparagraph (a) of the Rights in Commercial Computer Software or Commercial Software Documentation Clause at DFARS 227.7202-3, therefore the rights set forth herein shall apply; and (b) For any other unit or agency: NOTICE: Notwithstanding any other lease or license agreement that may pertain to, or accompany the delivery of, the computer software and accompanying documentation, the rights of the Government regarding its use, reproduction, and disclosure are as set forth in Clause 52.227-19 (c)(2) of the FAR. MATLAB, Simulink, Stateflow, Handle Graphics, and Real-Time Workshop are registered trademarks, and Target Language Compiler is a trademark of The MathWorks, Inc. Other product or brand names are trademarks or registered trademarks of their respective holders. Printing History: December 1996 May 1997 September 1998 First printing (for MATLAB 5) Second printing (for MATLAB 5.1) Third printing (for MATLAB 5.3) Contents Getting Started Starting MATLAB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Matrices and Magic Squares . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Entering Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . sum, transpose, and diag . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Subscripts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Colon Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The magic Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Expressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Expressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Working with Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Generating Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . M-Files . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Concatenation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Deleting Rows and Columns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Command Window . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The format Command . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Suppressing Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Long Command Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Command Line Editing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 4 5 7 8 9 11 11 11 12 12 14 15 15 16 16 17 17 19 19 21 21 21 Graphics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 Creating a Plot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 Figure Windows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 i Adding Plots to an Existing Graph . . . . . . . . . . . . . . . . . . . . . . . . Subplots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Imaginary and Complex Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . Controlling Axes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Axis Labels and Titles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mesh and Surface Plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Visualizing Functions of Two Variables . . . . . . . . . . . . . . . . . . . . Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Printing Graphics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Help and Online Documentation . . . . . . . . . . . . . . . . . . . . . . . . The help Command . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Help Window . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The lookfor Command . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Help Desk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The doc Command . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Printing Online Reference Pages . . . . . . . . . . . . . . . . . . . . . . . . . Link to the MathWorks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The MATLAB Environment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Workspace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . save Commands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Search Path . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Disk File Manipulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The diary Command . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Running External Programs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . More About Matrices and Arrays . . . . . . . . . . . . . . . . . . . . . . . . Linear Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Arrays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Multivariate Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Scalar Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Logical Subscripting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The find Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 27 28 29 30 31 31 32 33 34 34 35 36 37 37 37 37 38 38 39 39 40 40 41 42 42 45 47 48 49 50 ii Contents Flow Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . if . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . switch and case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . for . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . while . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . break . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Other Data Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Multidimensional Arrays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cell Arrays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Characters and Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Scripts and Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Scripts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Global Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Command/Function Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The eval Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Vectorization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Preallocation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Function Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Handle Graphics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Graphics Objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Graphics Objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Object Handles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Object Creation Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Object Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . set and get . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Graphics User Interfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Animations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Movies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 52 53 54 55 55 57 57 59 61 64 67 67 69 70 71 71 72 72 73 76 76 76 77 78 78 79 81 81 83 Learning More . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 iii iv Contents Introduction What Is MATLAB? . . . . . . . . . . . . . . . . . . . vi The MATLAB System . . . . . . . . . . . . . . . . . .vii About Simulink . . . . . . . . . . . . . . . . . . . .vii Introduction What Is MATLAB? MATLAB is a high-performance language for technical computing. It integrates computation, visualization, and programming in an easy-to-use environment where problems and solutions are expressed in familiar mathematical notation. Typical uses include: Math and computation Algorithm development 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, which together represent 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 high-productivity research, development, and analysis. MATLAB features a family of application-specific 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 (M-files) 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. vi The MATLAB System The MATLAB system consists of five main parts: The MATLAB language. This is a high-level matrix/array language with control flow statements, functions, data structures, input/output, and object-oriented programming features. It allows both programming in the small to rapidly create quick and dirty throw-away programs, and programming in the large to create complete large and complex application programs. The MATLAB working environment. This is the set of tools and facilities that you work with as the MATLAB user or programmer. It includes facilities for managing the variables in your workspace and importing and exporting data. It also includes tools for developing, managing, debugging, and profiling M-files, MATLABs applications. Handle Graphics. This is the MATLAB graphics system. It includes high-level commands for two-dimensional and three-dimensional data visualization, image processing, animation, and presentation graphics. It also includes low-level commands 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 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 Application Program Interface (API). This is a library that allows you to write C and Fortran programs that interact with MATLAB. It include facilities for calling routines from MATLAB (dynamic linking), calling MATLAB as a computational engine, and for reading and writing MAT-files. About Simulink Simulink, a companion program to MATLAB, is an interactive system for simulating nonlinear dynamic systems. It is a graphical mouse-driven program that allows you to model a system by drawing a block diagram on the screen and manipulating it dynamically. It can work with linear, nonlinear, continuous-time, discrete-time, multivariable, and multirate systems. vii Introduction Blocksets are add-ins to Simulink that provide additional libraries of blocks for specialized applications like communications, signal processing, and power systems. Real-time Workshop is a program that allows you to generate C code from your block diagrams and to run it on a variety of real-time systems. viii Getting Started Starting MATLAB ..................2 Matrices and Magic Squares . . . . . . . . . . . . . . 3 Expressions . . . . . . . . . . . . . . . . . . . . .11 Working with Matrices . . . . . . . . . . . . . . . .15 The Command Window . . . . . . . . . . . . . . . .19 Graphics . . . . . . . . . . . . . . . . . . . . . . .23 Help and Online Documentation . . . . . . . . . . . .34 The MATLAB Environment . . . . . . . . . . . . . .38 More About Matrices and Arrays . . . . . . . . . . . .42 Flow Control . . . . . . . . . . . . . . . . . . . . .52 Other Data Structures Scripts and Functions . . . . . . . . . . . . . . . .57 . . . . . . . . . . . . . . . .67 Handle Graphics . . . . . . . . . . . . . . . . . . .76 Learning More . . . . . . . . . . . . . . . . . . . .85 Getting Started Starting MATLAB This book is intended to help you start learning MATLAB. It contains a number of examples, so you should run MATLAB and follow along. To run MATLAB on a PC, double-click on the MATLAB icon. To run MATLAB on a UNIX system, type matlab at the operating system prompt. To quit MATLAB at any time, type quit at the MATLAB prompt. If you feel you need more assistance, type help at the MATLAB prompt, or pull down on the Help menu on a PC. We will tell you more about the help and online documentation facilities later. 2 Matrices and Magic Squares Matrices and Magic Squares The best way for you to get started with MATLAB is to learn how to handle matrices. This section shows you how to do that. In MATLAB, a matrix is a rectangular array of numbers. Special meaning is sometimes attached to 1-by-1 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. 3 Getting Started A good example matrix, used throughout this book, appears in the Renaissance engraving Melancholia I by the German artist and amateur mathematician Albrecht Drer. 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 Drers time to have genuinely magical properties. It does turn out to have some fascinating characteristics worth exploring. Entering Matrices 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 built-in functions. Create matrices with your own functions in M-files. Start by entering Drers matrix as a list of its elements. You have only 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, [ ]. To enter Drers matrix, simply type: A = [16 3 2 13; 5 10 11 8; 9 6 7 12; 4 15 14 1] 4 Matrices and Magic Squares 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 exactly 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 Youre 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. Lets verify that using MATLAB. The first statement to try is sum(A) MATLAB replies with ans = 34 34 34 34 When you dont 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. So A' 5 Getting Started 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 easily obtained with the help of the diag function, which picks off that diagonal. diag(A) produces ans = 16 10 7 1 and sum(diag(A)) produces ans = 34 6 Matrices and Magic Squares The other diagonal, the so-called antidiagonal, is not so important mathematically, so MATLAB does not have a ready-made 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 Drers 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 it is possible to compute the sum of the elements in the fourth column of A by typing 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 two-dimensional 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. 7 Getting Started 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 MATLABs most important 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)) 8 Matrices and Magic Squares 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 4-by-4 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 If you have access to the Symbolic Math Toolbox, you can discover that the magic sum for an n-by-n magic square is (n 3 + n)/2. The magic Function MATLAB actually has a built-in 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 Drers A, swap the two middle columns. A = B(:,[1 3 2 4]) 9 Getting Started 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 MATLABs ordering? No doubt he wanted to include the date of the engraving, 1514, at the bottom of his magic square. 10 Expressions 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 Numbers Operators Functions 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 1-by-1 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 power-of-ten 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 11 Getting Started All numbers are stored internally using the long format specified by the IEEE floating-point standard. Floating-point numbers have a finite precision of roughly 16 significant decimal digits and a finite range of roughly 10-308 to 10+308. (The VAX computer uses a different floating-point format, but its precision and range are nearly the same.) Operators Expressions use familiar arithmetic operators and precedence rules. + * / \ Addition Subtraction Multiplication Division Left division (described in the section on Matrices and Linear Algebra in Using MATLAB) 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 12 Expressions For a list of more advanced mathematical and matrix functions, type help specfun help elmat Some of the functions, like sqrt and sin, are built-in. They 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 M-files. 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, -1 Same as i Floating-point relative precision, 2-52 Smallest floating-point number, 2-1022 Largest floating-point number, (2-)21023 Infinity Not-a-number Infinity is generated by dividing a nonzero value by zero, or by evaluating well defined mathematical expressions that overflow, i.e., exceed realmax. Not-a-number 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 13 Getting Started 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 14 Working with Matrices Working with Matrices This section introduces you to other ways of creating matrices. 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 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= 4 9 4 4 R = randn(4,4) R= 1.0668 0.2944 0.0593 1.3362 0.0956 0.7143 0.8323 1.6236 8 5 2 6 8 0 0.6918 0.8580 1.2540 1.5937 1.4410 0.5711 0.3999 0.6900 15 Getting Started load The load command 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 command load magik.dat reads the file and creates a variable, magik, containing our example matrix. M-Files You can create your own matrices using M-files, which are text files containing MATLAB code. Just 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. NOTE To access a text editor on a PC, choose Open or New from the File menu or press the appropriate button on the toolbar. To access a text editor under UNIX, use the ! symbol followed by whatever command you would ordinarily use at your operating system prompt. 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 ]; 16 Working with Matrices 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 4-by-4 magic square, A, and form B = [A A+32; A+48 A+16] The result is an 8-by-8 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 half way to being another magic square. Its elements are a rearrangement of the integers 1:64. Its column sums are the correct value for an 8-by-8 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 8-by-8 magic square. 17 Getting Started 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 isnt 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 18 T he Command Window The Command Window So far, you have been using the MATLAB command line, typing commands and expressions, and seeing the results printed in the command window. This section describes a few ways of altering the appearance of the command window. If your system allows you to select the command window font or typeface, we recommend you use a fixed width font, such as Fixedsys or Courier, to provide proper spacing. The format Command The format command controls the numeric format of the values displayed by MATLAB. The command 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. 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 format long 1.33333333333333 format long e 1.333333333333333e+000 1.234500000000000e006 0.00000123450000 19 Getting Started format long g 1.33333333333333 format bank 1.33 format rat 4/3 format hex 3ff5555555555555 3eb4b6231abfd271 1/810045 0.00 1.2345e006 If the largest element of a matrix is larger than 103 or smaller than 10-3, MATLAB applies a common scale factor for the short and long formats. In addition to the format commands 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. 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); 20 T he Command Window Long Command Lines If a statement does not fit on one line, use 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 commands 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 misspelled command 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. The list of available command line editing keys is different on different computers. Experiment to see which of the following keys is available on your machine. (Many of these keys will be familiar to users of the EMACS editor.) ctrl- ctrl-p Recall previous line Recall next line Move back one character Move forward one character Move right one word ctrl-n ctrl-b ctrl-f ctrl-r 21 Getting Started ctrl- ctrl-l ctrl-a ctrl-e ctrl-u ctrl-d ctrl-h Move left one word Move to beginning of line Move to end of line Clear line Delete character at cursor Delete character before cursor Delete to end of line home end esc del backspace ctrl-k 22 Graphics Graphics MATLAB has extensive facilities for displaying vectors and matrices as graphs, as well as annotating and printing these graphs. This section describes a few of the most important graphics functions and provides examples of some typical applications. 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, to plot the value of the sine function from zero to 2, use t = 0:pi/100:2*pi; y = sin(t); plot(t,y) 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 23 Getting Started Multiple x-y pairs create multiple graphs with a single call to plot. MATLAB automatically cycles through a predefined (but user settable) list of colors to allow discrimination between each set of data. For example, these statements plot three related functions of t, each curve in a separate distinguishing color: y2 = sin(t.25); y3 = sin(t.5); plot(t,y,t,y2,t,y3) 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 It is possible to specify color, linestyle, and markers, such as plus signs or circles, with: plot(x,y,'color_style_marker') 24 Graphics color_style_marker is a 1-, 2-, or 3-character string (delineated by single quotation marks) constructed from a color, a linestyle, 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. Linestyle strings are '' for solid, ' ' for dashed, ':' for dotted, '.' for dash-dot, and 'none' for no line. The most common marker types include '+', 'o', '*', and 'x'. For example, the statement: plot(x,y,'y:+') plots a yellow dotted line and places plus sign markers at each data point. If you specify a marker type but not a linestyle, MATLAB draws only the marker. Figure Windows The plot function automatically opens a new figure window if there are no figure windows already on the screen. If a figure window exists, plot uses that window by default. To open a new figure window and make it the current figure, type figure To make an existing figure window the current figure, type figure(n) where n is the number in the figure title bar. The results of subsequent graphics commands are displayed in this window. Adding Plots to an Existing Graph The hold command allows you to add plots to an existing graph. When you type hold on MATLAB does not remove the existing graph; it adds the new data to the current graph, rescaling if necessary. For example, these statements first 25 Getting Started create a contour plot of the peaks function, then superimpose a pseudocolor plot of the same function: [x,y,z] = peaks; contour(x,y,z,20,'k') hold on pcolor(x,y,z) shading interp The hold on command causes the pcolor plot to be combined with the contour plot in one figure. 26 Graphics Subplots The subplot function allows you to display multiple plots in the same window or print them on the same piece of paper. Typing subplot(m,n,p) breaks the figure window into an m-by-n 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, to 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 0 5 40 20 00 20 5 0 5 40 20 00 20 40 40 1 0.5 0 40 20 00 20 1 0.5 0 5 0 5 5 0 40 5 27 Getting Started Imaginary and Complex Data When the arguments to plot are complex, the imaginary part is ignored except when plot is given a single complex argument. For this special case, the command is a shortcut for a plot 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') draws a 20-sided polygon with little circles at the vertices. 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 28 Graphics Controlling Axes The axis function has a number of options for customizing the scaling, orientation, and aspect ratio of plots. Ordinarily, MATLAB finds the maxima and minima of the data and chooses an appropriate plot box and axes labeling. The axis function overrides the default by setting custom axis limits, axis([xmin xmax ymin ymax]) axis also accepts a number of keywords for axes control. For example axis square makes the entire x-axes and y-axes the same length and axis equal makes the individual tick mark increments on the x- and y-axes the same length. So plot(exp(i*t)) followed by either axis square or axis equal turns the oval into a proper circle. axis auto returns the axis scaling to its default, automatic mode. axis on turns on axis labeling and tick marks. axis off turns off axis labeling and tick marks. The statement grid off turns the grid lines off and grid on turns them back on again. 29 Getting Started Axis Labels and Titles The xlabel, ylabel, and zlabel functions add x-, y-, and z-axis labels. The title function adds a title at the top of the figure and the text function inserts text anywhere in the figure. A subset of Tex notation produces Greek letters, mathematical symbols, and alternate fonts. The following example uses \leq for , \pi for , and \it for italic font. 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,'\it{Note 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 30 Graphics Mesh and Surface Plots MATLAB defines a surface by the z-coordinates of points above a grid in the x-y plane, using straight lines to connect adjacent points. The functions mesh and surf 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. 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. Then use these matrices 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. To evaluate the two-dimensional sinc function, sin(r)/r, between x and y directions: [X,Y] = meshgrid(8:.5:8); R = sqrt(X.^2 + Y.^2) + eps; Z = sin(R)./R; mesh(X,Y,Z) 31 Getting Started 1 0.8 0.6 0.4 0.2 0 0.2 0.4 10 5 0 5 10 10 0 5 5 10 In this example, R is the distance from origin, which is at the center of the matrix. Adding eps avoids the indeterminate 0/0 at the origin. Images Two-dimensional arrays can be displayed as images, where the array elements determine brightness or color of the images. For example, load durer whos shows that file durer.mat in the demo directory contains a 648-by-509 matrix, X, and a 128-by-3 matrix, map. The elements of X are integers between 1 and 128, which serve as indices into the color map, map. Then image(X) colormap(map) axis image 32 Graphics reproduces Drers etching shown at the beginning of this book. A high resolution scan of the magic square in the upper right corner is available in another file. Type load detail and then use the uparrow 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. Printing Graphics The Print option on the File menu and the print command both print MATLAB figures. The Print menu brings up a dialog box that lets you select common standard printing options. The print command provides more flexibility in the type of output and allows you to control printing from M-files. The result can be sent directly to your default printer or stored in a specified file. A wide variety of output formats, including 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: print depsc2 magicsquare.eps Its important to know the capabilities of your printer before using the print command. For example, Level 2 Postscript files are generally smaller and render more quickly when printing than Level 1 Postscript. However, not all PostScript printers support Level 2, so you need to know what your output device can handle. MATLAB produces graduated output for surfaces and patches, even for black and white output devices. However, lines and text are printed in black or white. 33 Getting Started Help and Online Documentation There are several different ways to access online information about MATLAB functions. The help command The help window The MATLAB Help Desk Online reference pages Link to The MathWorks, Inc. The help Command The help command is the most basic way to determine the syntax and behavior of a particular function. Information is displayed directly in the command window. For example help magic prints MAGIC Magic square. MAGIC(N) is an NbyN matrix constructed from the integers 1 through N^2 with equal row, column, and diagonal sums. Produces valid magic squares for N = 1,3,4,5.... NOTE MATLAB online help entries use uppercase characters for the function and variable names to make them stand out from the rest of the text. When typing function names, however, always use the corresponding lowercase characters because MATLAB is case sensitive and all function names are actually in lowercase. 34 H elp and Online Documentation All the MATLAB functions are organized into logical groups, and MATLABs directory structure is based on this grouping. For example, all the linear algebra functions reside in the matfun directory. To list the names of all the functions in that directory, with a brief description of each: help matfun Matrix functions numerical linear algebra. Matrix analysis. norm Matrix or vector norm. normest Estimate the matrix 2norm ... The command help by itself lists all the directories, with a description of the function category each represents: matlab/general matlab/ops ... The Help Window The MATLAB help window is available on PCs by selecting the Help Window option under the Help menu, or by clicking the question mark on the menu bar. It is also available on all computers by typing helpwin To use the help window on a particular topic, type helpwin topic The help window gives you access to the same information as the help command, but the window interface provides convenient links to other topics. 35 Getting Started The lookfor Command The lookfor command allows you to search for functions based on a keyword. It searches through the first line of help text, which is known as the H1 line, for each MATLAB function, and returns the H1 lines containing a specified keyword. For example, MATLAB does not have a function named inverse. So the response from help inverse is inverse.m not found. But lookfor inverse finds over a dozen matches. Depending on which toolboxes you have installed, you will find entries like INVHILB ACOSH ERFINV INV PINV IFFT IFFT2 ICCEPS IDCT Inverse Hilbert matrix. Inverse hyperbolic cosine. Inverse of the error function. Matrix inverse. Pseudoinverse. Inverse discrete Fourier transform. Twodimensional inverse discrete Fourier transform. Inverse complex cepstrum. Inverse discrete cosine transform. Adding all to the lookfor command, as in lookfor all searches the entire help entry, not just the H1 line. 36 H elp and Online Documentation The Help Desk The MATLAB Help Desk provides access to a wide range of help and reference information stored on a disk or CD-ROM in your local system. Many of the underlying documents use HyperText Markup Language (HTML) and are accessed with an Internet Web browser such as Netscape or Microsoft Explorer. The Help Desk process can be started on PCs by selecting the Help Desk option under the Help menu, or, on all computers, by typing helpdesk All of MATLABs operators and functions have online reference pages in HTML format, which you can reach from the Help Desk. These pages provide more details and examples than the basic help entries. HTML versions of other documents, including this manual, are also available. A search engine, running on your own machine, can query all the online reference material. The doc Command If you know the name of a specific function, you can view its reference page directly. For example, to get the reference page for the eval function, type doc eval The doc command starts your Web browser, if it is not already running. Printing Online Reference Pages Versions of the online reference pages, as well as the rest of the MATLAB documentation set, are also available in Portable Document Format (PDF) through the Help Desk. These pages are processed by Adobes Acrobat reader. They reproduce the look and feel of the printed page, complete with fonts, graphics, formatting, and images. This is the best way to get printed copies of reference material. Link to the MathWorks If your computer is connected to the Internet, the Help Desk provides a connection to The MathWorks, the home of MATLAB. You can use electronic mail to ask questions, make suggestions, and report possible bugs. You can also use the Solution Search Engine at The MathWorks Web site to query an up-to-date data base of technical support information. 37 Getting Started The MATLAB Environment The MATLAB environment includes both the set of variables built up during a MATLAB session and the set of disk files containing programs and data that persist between sessions. The Workspace The workspace is the area of memory accessible from the MATLAB command line. Two commands, who and whos, show the current contents of the workspace. The who command gives a short list, while whos gives also size and storage information. Here is the output produced by whos on a workspace containing results from some of the examples in this book. It shows several different MATLAB data structures. As an exercise, you might see if you can match each of the variables with the code segment in this book that generates it. whos Name A D M S h n s v Size 4x4 5x3 10x1 1x3 1x11 1x1 1x5 2x5 Bytes 128 120 3816 442 22 8 10 20 Class double array double array cell array struct array char array double array char array char array Grand total is 471 elements using 4566 bytes. To delete all the existing variables from the workspace, enter clear 38 T he MATLAB Environment save Commands The save commands preserve the contents of the workspace in a MAT-file that can be read with the load command in a later MATLAB session. For example save August17th saves the entire workspace contents in the file August17th.mat. If desired, you can save only certain variables by specifying the variable names after the filename. Ordinarily, the variables are saved in a binary format that can be read quickly (and accurately) by MATLAB. If you want to access these files outside of MATLAB, you may want to specify an alternative format. ascii ascii double ascii double tabs v4 append Use 8-digit text format. Use 16-digit text format. Delimit array elements with tabs. Create a file for MATLAB version 4. Append data to an existing MAT-file. When you save workspace contents in text format, you should save only one variable at a time. If you save more than one variable, MATLAB will create the text file, but you will be unable to load it easily back into MATLAB. The Search Path MATLAB uses a search path, an ordered list of directories, to determine how to execute the functions you call. When you call a standard function, MATLAB executes the first M-file function on the path that has the specified name. You can override this behavior using special private directories and subfunctions. The command path shows the search path on any platform. On PCs, choose Set Path from the File menu to view or modify the path. 39 Getting Started Disk File Manipulation The commands dir, type, delete, and cd implement a set of generic operating system commands for manipulating files. This table indicates how these commands map to other operating systems. MATLAB dir type delete cd MS-DOS dir type del or erase chdir UNIX ls cat rm cd VAX/VMS dir type delete set default For most of these commands, you can use pathnames, wildcards, and drive designators in the usual way. The diary Command The diary command creates a diary of your MATLAB session in a disk file. You can view and edit the resulting text file using any word processor. To create a file called diary that contains all the commands you enter, as well as MATLABs printed output (but not the graphics output), enter diary To save the MATLAB session in a file with a particular name, use diary filename To stop recording the session, use diary off 40 T he MATLAB Environment Running External Programs The exclamation point character ! is a shell escape and indicates that the rest of the input line is a command to the operating system. This is quite useful for invoking utilities or running other programs without quitting MATLAB. On VMS, for example, !edt magik.m invokes an editor called edt for a file named magik.m. When you quit the external program, the operating system returns control to MATLAB. 41 Getting Started More About Matrices and Arrays This sections shows you more about working with matrices and arrays, focusing on Linear Algebra Arrays Multivariate Data Linear Algebra Informally, the terms matrix and array are often used interchangeably. More precisely, a matrix is a two-dimensional numeric array that represents a linear transformation. The mathematical operations defined on matrices are the subject of linear algebra. Drers 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. Youve 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 42 More About Matrices and Arrays The multiplication symbol, *, denotes the matrix multiplication involving inner products between rows and columns. Multiplying a matrix by its transpose also produces a symmetric matrix. 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 = 1.175530e017. 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 floating-point relative precision, so the computed inverse is unlikely to be of much use. 43 Getting Started 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. Thats 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 one. 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 44 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 kth power, Pk, 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 4 - 343 - 642 + 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! Arrays When they are taken away from the world of linear algebra, matrices become two dimensional 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. 45 Getting Started The list of operators includes: + .* ./ .\ .^ .' Addition Subtraction Element-by-element multiplication Element-by-element division Element-by-element left division Element-by-element 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 Array operations are useful for building tables. Suppose n is the column vector n = (0:9)'; Then pows = [n n.^2 2.^n] 46 More About Matrices and Arrays builds a table of squares and powers of two. 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 Multivariate Data MATLAB uses column-oriented 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. 47 Getting Started 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 of MATLABs 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 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 48 More About Matrices and Arrays 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 zeros 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 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, 49 Getting Started use finite(x), which is true for all finite numerical values and false for NaN and Inf. x = x(finite(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 Drers magic square by using logical indexing and scalar expansion to set the nonprimes to 0. A(~isprime(A)) = 0 A= 0 5 0 0 3 0 0 0 2 11 7 0 13 0 0 0 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 k = find(isprime(A))' picks out the locations, using one-dimensional indexing, of the primes in the magic square. k= 2 5 9 10 11 13 50 More About Matrices and Arrays 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 left-hand-side 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 51 Getting Started Flow Control MATLAB has five flow control constructs: if statements switch statements for loops while loops break statements 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. MATLABs 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 werent, the first true condition would be executed. 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, ... 52 Flow Control 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 0s and 1s showing element-by-element 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 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. 53 Getting Started 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 for C Programmers Unlike the C language switch statement, MATLABs 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. 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 54 Flow Control 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. break The break statement lets you exit early from a for or while loop. In nested loops, break exits from the innermost loop only. 55 Getting Started 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 56 O ther Data Structures Other Data Structures This section introduces you to some other data structures in MATLAB, including: Multidimensional arrays Cell arrays Characters and text Structures Multidimensional Arrays Multidimensional arrays in MATLAB are arrays with more than two subscripts. They can be created by calling zeros, ones, rand, or randn with more than two arguments. For example R = randn(3,4,5); creates a 3-by-4-by-5 array with a total of 3x4x5 = 60 normally distributed random elements. A three-dimensional array might represent three-dimensional 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 time-dependent 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). MATLABs and Drers 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 57 Getting Started stores the sequence of 24 magic squares in a three-dimensional array, M. The size of M is size(M) ans = 4 4 24 16 5 2 11 7 14 3 10 6 15 13 8 12 1 ... 13 3 13 8 12 1 10 3 6 10 15 6 15 138 12 8 2 121 11 1 7 14 2 11 2 117 16 14 7 5 14 9 4 3 10 16 56 15 9 4 16 5 9 4 9 4 It turns out that the 22nd matrix in the sequence is Drers: M(:,:,22) ans = 16 5 9 4 3 10 6 15 2 11 7 14 13 8 12 1 The statement sum(M,d) computes sums by varying the dth subscript. So sum(M,1) is a 1-by-4-by-24 array containing 24 copies of the row vector 34 34 34 34 58 O ther Data Structures and sum(M,2) is a 4-by-1-by-24 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 4-by-4-by-1, so it looks like a 4-by-4 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 1-by-3 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] 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. 59 Getting Started 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. Three-dimensional arrays can be used to store a sequence of matrices of the same size. Cell arrays can be used to store a sequences of matrices of different sizes. For example, M = cells(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] 60 O ther Data Structures 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 we have been dealing with up to now. It is a 1-by-5 character array. .. . 13 8 12 1 61 Getting Started Internally, the characters are stored as numbers, but not in floating-point format. The statement a = double(s) converts the character array to a numeric matrix containing floating-point 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 6-by-16 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 MATLAB command window. On a PC, select Preferences under the File menu. Be sure to try the Symbol and 62 O ther Data Structures Wingdings fonts, if you have them on your computer. Here is one example of the kind of output you might obtain. !"#$%&'()*+,-./ 0123456789:;<=>? @ABCDEFGHIJKLMNO PQRSTUVWXYZ[\]^_ abcdefghijklmno pqrstuvwxyz{|}~ __ i 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 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 1-by-11 and v is 2-by-5. 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. The char function accepts any number of lines, adds blanks to each line to make them all the 63 Getting Started same length, and forms a character array with each line in a separate row. For example S = char('A','rolling','stone','gathers','momentum.') produces a 5-by-9 character array S= A rolling stone gathers momentum. There are enough blanks in each of the first four rows of S to make all the rows the same length. Alternatively, you can store the text in a cell array. For example C = {'A';'rolling';'stone';'gathers';'momentum.'} is a 5-by-1 cell array C= 'A' 'rolling' 'stone' 'gathers' 'momentum.' 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+' 64 O ther Data Structures 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 There are several ways to reassemble the various fields into other MATLAB arrays. They are all based on the notation of a comma separated list. If you type S.score it is the same as typing S(1).score, S(2).score, S(3).score This is a comma separated 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] 65 Getting Started it is the same as [S(1).score, S(2).score, S(3).score] which produces a numeric row vector containing all of the scores. ans = 83 91 70 Similarly, typing S.name just assigns the names, one at time, to ans. But enclosing the expression in curly braces, {S.name} creates a 1-by-3 cell array containing the three names. ans = 'Ed Plum' 'Toni Miller' 'Jerry Garcia' And char(S.name) calls the char function with three arguments to create a character array from the name fields, ans = Ed Plum Toni Miller Jerry Garcia 66 Scripts and Functions Scripts and Functions MATLAB is a powerful programming language as well as an interactive computational environment. Files that contain code in the MATLAB language are called M-files. You create M-files using a text editor, then use them as you would any other MATLAB function or command. There are two kinds of M-files: 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 youre a new MATLAB programmer, just create the M-files that you want to try out in the current directory. As you develop more of your own M-files, you will want to organize them into other directories and personal toolboxes that you can add to MATLABs search path. If you duplicate function names, MATLAB executes the one that occurs first in the search path. To view the contents of an M-file, for example, myfunction.m, use type myfunction 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. 67 Getting Started 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) 35 30 25 20 15 10 5 0 0 5 10 15 20 25 30 35 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. 68 Scripts and Functions Functions Functions are M-files that can accept input arguments and return output arguments. The name of the M-file 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 M-file rank.m is available in the directory toolbox/matlab/matfun You can see the file with type rank 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 M-file 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. 69 Getting Started 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) Many M-files 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. 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 M-file 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)'); 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. 70 Scripts and Functions Command/Function Duality MATLAB commands are statements like load help Many commands accept modifiers that specify operands. load August17.dat help magic type rank An alternate method of supplying the command modifiers makes them string arguments of functions. load('August17.dat') help('magic') type('rank') This is MATLABs command/function duality. Any command of the form command argument can also be written in the functional form command('argument') The advantage of the functional approach comes when the string argument is constructed from other pieces. 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 string, to help build the file name. for d = 1:31 s = ['August' int2str(d) '.dat'] load(s) % Process the contents of the d-th file end The eval Function The eval function works with text variables to implement a powerful text macro facility. The expression or statement eval(s) 71 Getting Started 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. for d = 1:31 s = ['load August' int2char(d) '.dat'] eval(s) % Process the contents of the dth file end Vectorization To obtain the most speed out of MATLAB, its important to vectorize the algorithms in your M-files. Where other programming languages might use for or DO loops, MATLAB can use vector or matrix operations. A simple example involves creating a table of logarithms. x = 0; for k = 1:1001 y(k) = log10(x); x = x + .01; end Experienced MATLAB users like to say Life is too short to spend writing for loops. A vectorized version of the same code is x = 0:.01:10; y = log10(x); For more complicated code, vectorization options are not always so obvious. When speed is important, however, you should always look for ways to vectorize your algorithms. Preallocation If you cant 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 72 Scripts and Functions 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. 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 Quadrature Ordinary differential equations MATLAB represents the nonlinear function by a function M-file. 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) 73 Getting Started 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 The graph shows that the function has a local minimum near x = 0.6. The function fmins finds the minimizer, the value of x where the function takes on this minimum. The first argument to fmins is the name of the function being minimized and the second argument is a rough guess at the location of the minimum. p = fmins('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 74 Scripts and Functions integration of ordinary differential equations. MATLABs quadrature routines are quad and quad8. The statement Q = quad8('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 of the interval z= 0.1316 75 Getting Started Handle Graphics MATLAB provides a set of low-level functions that allows you to create and manipulate lines, surfaces, and other graphics objects. This system is called Handle Graphics. Graphics Objects Graphics objects are the basic drawing primitives of MATLABs Handle Graphics system. The objects are organized in a tree structured hierarchy. This reflects the interdependence of the graphics objects. For example, Line objects require Axes objects as a frame of reference. In turn, Axes objects exist only within Figure objects. Graphics Objects There are eleven kinds of Handle Graphics objects: The Root object is the at top of the hierarchy. It corresponds to the computer screen. MATLAB automatically creates the Root object at the beginning of a session. Figure objects are the windows on the root screen other than the Command window. Uicontrol objects are user interface controls that execute a function when users activate the object. These include pushbuttons, radio buttons, and sliders. Axes objects define a region in a figure window and orient their children within this region. Uimenu objects are user interface menus that reside at the top of the figure window. Image objects are two-dimensional objects that MATLAB displays using the elements of a rectangular array as indices into a colormap. Line objects are the basic graphics primitive for most two-dimensional plots. Patch objects are filled polygons with edges. A single Patch can contain multiple faces, each colored independently with solid or interpolated colors. Surface objects are three-dimensional representations of matrix data created by plotting the value of the data as heights above the x-y plane. 76 Handle Graphics Text objects are character strings. Light objects define light sources that affect all objects within the Axes. Object Handles Every individual graphics object has a unique identifier, called a handle, that MATLAB assigns to the object when it is created. Some graphs, such as multiple line plots, are composed of multiple objects, each of which has its own handle. Rather than attempting to read handles off the screen and retype them, you will find that it is always better to store the value in a variable and pass that variable whenever a handle is required. The handle of the root object is always zero. The handle of a figure is an integer that, by default, is displayed in the window title. Other object handles are floating-point numbers that contain information used by MATLAB. For example, if A is the Drer magic square, then h = plot(A) creates a line plot with four lines, one for each column of A. It also returns a vector of handles such as h= 9.00024414062500 6.00048828125000 7.00036621093750 8.00036621093750 The actual numerical values are irrelevant and may vary from system to system. Whatever the numbers are, the important fact is that h(1) is the handle for the first line in the plot, h(2) is the handle for the second, and so on. MATLAB provides several functions to access frequently used object handles: gcf gca gco You can use these functions as input arguments to other functions that require figure and axes handles. Obtain the handle of other objects you create at the time of creation. All MATLAB functions that create objects return the handle (or a vector of handles) of the object created. 77 Getting Started Remove an object using the delete function, passing the objects handle as an argument. For example, delete the current axes (and all of its children) with the statement: delete(gca) Object Creation Functions Calling the function named after any object creates one of those objects. For example, the text function creates text objects, the figure function creates figure objects, and so on. MATLABs high-level graphics functions (like plot and surf) call the appropriate low-level function to draw their respective graphics. Low-level functions simply create one of the eleven graphics objects defined by MATLAB with the exception of the root object, which only MATLAB can create. For example: line([1 3 6], [8 2 0], 'Color', 'red') Object Properties All objects have properties that control how they are displayed. MATLAB provides two mechanisms for setting the values of properties. Object properties can be set by the object creation function, or can be changed with the set function after the object already exists. For example, these statements create three objects and override some of their default properties. days = ['Su';'Mo';'Tu';'We';'Th';'Fr';'Sa'] temp = [21.1 22.2 19.4 23.3 23.9 21.1 20.0]; f = figure a = axes('YLim',[16 26],'Xtick',1:7,'XTickLabel',days) h = line(1:7,temp) 78 Handle Graphics Temperature 26 25 24 23 22 21 20 19 18 17 16 Su Mo Tu We Th Fr Sa days is a character array containing abbreviations for the days of the week and temp is a numeric array of typical temperatures. The figure window is created by calling figure with no arguments, so it has the default properties. The axes exists within the figure and has a specified range for the scaling of the y-axis and specified labels for the tick marks on the x-axis. The line exists within the axes and has specified values for the x and y data. The three object handles f, a, and h, are saved for later use. set and get Object properties are specified by referencing the object after its creation. To do this, use the handle returned by the creating function. The set function allows you to set any objects property by specifying the objects handle and any number of property name/property value pairs. For instance, to change the color and width of the line from the previous example, set(h,'Color',[0 .8 .8],'LineWidth',3) 79 Getting Started To see a list of all settable properties for a particular object, call set with the objects handle: set(h) Color EraseMode: [ {normal} | background | xor | none ] LineStyle: [ {} | | : | . | none ] LineWidth Marker: MarkerSize ... XData YData ZData To see a list of all current settings properties for a particular object, call get with the objects handles: get(h) Color = [0 0.8 0.8] EraseMode = normal LineStyle = LineWidth = [3] Marker = none MarkerSize = [6] ... XData = [ (1 by 7) double array] YData = [ (1 by 7) double array] ZData = [] To query the value of a property, use get with the property name: get(h,'Color') ans = 0 0.8000 0.8000 80 Handle Graphics The axes object carries many of the detailed properties of the overall graphic. For example, the title is another child of the axes. The statements: t = get(a,'title'); set(t,'String','Temperature','FontAngle','oblique') specify a particular title. The title function provides another interface to the same properties. Graphics User Interfaces Here is a simple example illustrating how to use Handle Graphics to build user interfaces. The statement b = uicontrol('Style','pushbutton', ... 'Units','normalized', ... 'Position',[.5 .5 .2 .1], ... 'String','click here'); creates a pushbutton in the center of a figure window and returns a handle to the new object. But, so far, clicking on the button does nothing. The statement s = 'set(b,''Position'',[.8*rand .9*rand .2 .1])'; creates a string containing a command that alters the pushbuttons position. Repeated execution of eval(s) moves the button to random positions. Finally, set(b,'Callback',s) installs s as the buttons callback action, so every time you click on the button, it moves to a new position. Animations MATLAB provides several ways of generating moving, animated, graphics. 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, like n = 20 81 Getting Started 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) Now begin the animation. Here is an infinite while loop, which you will eventually break out of 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 How long does it take for one of the points to get outside of the square? How long before all of the points are outside the square? 82 Handle Graphics 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 Movies If you increase the number of points in the Brownian motion example to something like 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, say nframes = 50; Next, set up the first plot as before, except do not use EraseMode. 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 83 Getting Started Now, allocate enough memory to save the complete movie, M = moviein(nframes); This sets aside a large matrix with nframes columns. Each column is long enough to save one frame. The total amount of memory required is proportional to the number of frames, and to the area of the current axes; it is independent of the complexity of the particular plot. For 50 frames and the default axes, over 7.5 megabytes of memory is required. This example is using square axes, which is slightly smaller, so only about 6 megabytes is required. 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) 84 Learning More Learning More To see more MATLAB examples, select Examples and Demos under the menu, or type demo at the MATLAB prompt. From the menu displayed, run the demos that interest you, and follow the instructions on the screen. For a more detailed explanation of any of the topics covered in this book, see The MATLAB Installation Guide describes how to install MATLAB on your platform. Using MATLAB provides in depth material on the MATLAB language, working environment, and mathematical topics. Using MATLAB Graphics describes how to use MATLABs graphics and visualization tools. The MATLAB Application Program Interface Guide explains how to write C or Fortran programs that interact with MATLAB. MATLAB Product Family New Features provides information on what is new in this release and information that is useful in making the transition from previous releases of MATLAB to this release. MATLAB Toolboxes are collections of M-files that extend MATLAB's capabilities to a number of technical fields. Separate guides are available for each of the Toolboxes. Some of the topics they cover are Communications Control Systems Financial Computation Frequency-Domain System Identification Fuzzy Logic Higher-Order Spectral Analysis Image Processing Linear Matrix Inequalities 85 Getting Started Model Predictive Control Mu-Analysis and Synthesis Numerical Algorithms Neural Networks Optimization Partial Differential Equations Quantitative Feedback Theory Robust Control Signal Processing Simulation (Simulink) Splines Statistics Symbolic Mathematics System Identification Wavelets For the very latest information about MATLAB and other MathWorks products, point your Web browser to: http://www.mathworks.com and use your Internet News reader to access the newsgroup comp.soft-sys.matlab A number of MATLAB-related books available from many different publishers. A booklet entitled MATLAB-Based Books is available from The MathWorks and an up-to-date list is available on the Web site. If you have read this entire book and run all the examples, congratulations, you are off to a great start with MATLAB. If you have skipped to this last section or havent tried any of the examples, may we suggest you spend a little more time Getting Started. In either case, welcome to the world of MATLAB! 86 Index A animation 81 API vii arrays 42, 45 cell 59 columnwise organization 47 concatenating 17 creating in M-files 16 deleting rows or columns 17 elements 11 generating with functions and operators 15 listing contents 11 loading from external data files 16 multidimensional 57 notation for elements 11 variable names 11 axes 29 entering matrices 4 eval 71 expressions 11, 14 evaluating 72 external programs running 41 F file manipulation 40 flow control 52 for 54 format of output display 19 function functions 73 functions 12, 69 C case 53 G global variables 70 graphics 2-D 23 handle graphics 76 objects 76 properties 78 printing 33 cell arrays 59 colon operator 8 command line editing 21 concatenating arrays 17 continuing statements on multiple lines 21 D deleting arrays elements 17 diag 6 doc 37 H Handle Graphics vii, 76 help 34 E editing command lines 21 elements of arrays 11 I if 52 images 32 I-1 Index L library mathematical function vii loading arrays 16 logical vectors 49 lookfor 36 printing 37 Reference Guide 34 operators 12 output controlling format 19 suppressing 21 M magic 9 P plots 23 mesh 31 surface 31 plotting complex data 28 matrices 28 preallocation 72 printing graphics 33 online reference pages 37 mathematical function library vii MATLAB Application Programmers Interface vii Handle Graphics vii history vi language vii mathematical function library vii overview vi starting 2 working environment vii matrices 3, 42 entering 4 matrix 3 M-file for creating arrays 16 movies 83 multidimensional arrays 57 multivariate data, organizing 47 R Reference Guide 34 running external programs 41 S save 39 scripts 67 search path 39 searching online help 36, 37 semicolon to suppress output 21 Simulink vii starting MATLAB 2 statements continuing on multiple lines 21 executing 72 N numbers 11 O objects, graphics 76 online help I-2 Index structures 64 subplot 27 subscripting with logical vectors 49 subscripts 7 sum 5 suppressing output 21 switch 53 T text 61 transpose 5 U user interfaces building 81 V variables 11 global 70 vector logical 49 vectorization 72 W while 55 who 38 whos 38 working environment MATLAB vii workspace listing contents 38 listing storage information 38 I-3 Find millions of documents on Course Hero - Study Guides, Lecture Notes, Reference Materials, Practice Exams and more. Course Hero has millions of course specific materials providing students with the best way to expand their education. Below is a small sample set of documents: Princeton - ECO - 310 ECO 310, Fall 2008 Problem Set 2: Consumer Theory Due in class on October 7Question 1You have just emerged from medical school with a debt service burden of$25,000 per year, and have set up practice. You have to decide how hard to work. For each hour o
Princeton - ECO - 310
ECO 310, Fall 2008 Problem Set 5: Equilibrium Analysis Due in class on November 18 Question 1In this problem we will consider the sh industry on the island of San Serife. For this purpose we will aggregate all the other goods into one composite, and meas
Princeton - ECO - 310
ECO 310, Fall 2008 Problem Set 6: General Equilibrium, Monopoly Due in class on November 25 Question 1Consider a small country with two rms. Firm 1 produces x = min(kx , lx ) units of good x; rm 2 produces y = ky ly units of good y . This country has 24
Princeton - ECO - 310
ECO 310, Fall 2008 Problem Set 7: Oligopoly Due in class on December 2 Question 1Consider a Cournot duopoly with homogeneous products. The inverse demand function is given by P = (q1 + q2 )3/2 . Each rms marginal cost is constant and equal to c &gt; 0. (a)
Princeton - ECO - 310
ECO 310, Fall 2008 Problem Set 8: Uncertainty Due in class on December 9 Question 1Satoru is an expected utility maximizer with von Neumann-Morgenstern index U (W ). He is oered the following gamble: gain $h with probability 1/2+ , and lose$h with proba
Princeton - ECO - 310
Econ 310 Problem Set 1Lucas Manuelli (rst question being slightly altered) October 13, 2008Question 1 (b) For x &gt; 0 we have f 0 (x) = 80 20x so f 0 (4) = 0: In addition for x &gt; 0 we have f 00 (x) = 20 so the function is concave for positive x so that x
Princeton - ECO - 310
Econ 310 Problem Set 2Lucas Manuelli October 5, 2008Exercise 1 (a) I = 50H (b) We simply need to maximize U (I; H ) = U (50H = ln(50H 25000; H ) 25000) + 2 ln(5000 H) 25000taking the derivative and setting equal to zero we see that U0 = 50 50H 25000 2
Princeton - ECO - 310
Economics 310 Microeconomic Theory: A Mathematical Approach Fall 2008 Solution to Problem Set 3 Question 1:For each of the following utility functions over two goods, nd the Marshallian demand, the Hicksian demand and expenditure function The Marshallian
Princeton - ECO - 310
ECO 310 - Fall 2007 Microeconomic Theory - A Mathematical Approach Problem Set 4 - Answer KeyQuestion 1:(a) By Shepherds Lema E/px = X C so X C =M1 py/2 U2px1/2and Y C =1 px/2 U2py1/2.By Roys Identity X = X = (V /px )/(V /I ). Inverting the ex
Princeton - ECO - 310
ECO 310 Fall 2007 Microeconomic Theory A Mathematical Approach Problem Set 5 Answer KeyQuestion 1:(a) Each consumer maximizes U (x; y ) = y + 10 x subject to the budget constraint px + y = M : Substituting out y , the objective is F (x) Now F 0 (x) = 10
Princeton - ECO - 310
Economics 310 Microeconomic Theory: A Mathematical Approach Fall 2008 Solutions to Problem Set 6Question 1(a) The production possibility frontier shows the ecient combination of goods x and y with this countrys endowment of labor and capital. By drawing
Princeton - ECO - 310
ECO 310 - Fall 2008 Microeconomic Theory - A Mathematical Approach Problem Set 7 - Answer KeyQuestion 1:(a) There were two hard parts to this question: nasty algebra, and the intuition. Firm 1 prot is s1= (pc) q1 = f [q1 + q2 ]3 =2c g q1 :So its C
Princeton - ECO - 310
ECO 310 - Fall 2008 Microeconomic Theory - A Mathematical Approach Problem Set 8 - Answer KeyQuestion 1:1 1 U (W0 ) = ( + )U (W0 + h) + ( )U (W0 h) 2 2 1 1 1 1 U (W0 ) ( + )(U (W0 ) + hU (W0 ) + h2 U (W0 ) + ( )(U (W0 ) hU (W0 ) + h2 U (W0 ) 2 2 2 2 h 2
Princeton - ECO - 310
Examination Cover SheetPrinceton University Undergraduate Honor Committee May 2008Course Number: ECONOMICS 310 Course Name: Microeconomic Theory cfw_ A Mathematical Approach Professor: Stephen Morris Date: May 19 Time: 1:30pm This examination is adminis
Princeton - ECO - 310
ECO 310 Microeconomic Theory: A Mathematical Approach Spring 2008 Solutions to the Final ExaminationSection BQuestion 4(i.a) The cost of producing Q units is minimized by setting 10K = L = Q so LRT C (Q) = r(Q/10)+ wQ, LRAC (Q) = LRT C (Q)/Q = r/10+ w
Princeton - ECO - 310
ECO 310 - Fall 2008 Microeconomic Theory - A Mathematical Approach Final Exam 01/14/09 - Answer Key0.1Question 1: P 1, eI/P I + P (ln(1/P ) 1); : P 1, eI/P I + P (ln(1/P ) 1); : P 1, : P 1, eI/P I + P (ln(1/P ) 1); : P 1, eI/P I + P (ln(1/P ) 1); : P 1
Princeton - ECO - 310
1Question 5Suppose Dd and Rd are asset positions for Daisy of democratic and republican assets, while Dr and Rr are those of Robert. Then, a) U d = 0.6 ln Dd + 0.4 ln Rd and U r = 0.2 ln Dr + 0.8 ln Rr 5 points were assigned for a completely correct ans
Princeton - ECO - 310
Examination Cover SheetPrinceton University Undergraduate Honor CommitteeCourse Number: ECO 310 Course Name: Microeconomic Theory: A Mathematical Approach Professor: Satoru Takahashi Date: January 22, 2008 Time: 1:30pmThis examination is administered u
Princeton - ECO - 310
Examination Cover SheetPrinceton University Undergraduate Honor CommitteeCourse Number: ECO 310 Course Name: Microeconomic Theory: A Mathematical Approach Professor: Satoru Takahashi Date: January 22, 2008 Time: 1:30pm This examination is administered
Princeton - ECO - 310
Examination Cover SheetPrinceton University Undergraduate Honor CommitteeCourse Number: ECO 310 Course Name: Microeconomic Theory: A Mathematical Approach Professor: Satoru Takahashi Date: Wednesday, January 14, 2009 Time: 1:30 p.m. This examination is
Princeton - ECO - 310
ECO 310, Fall 2007 Midterm Examination Solutions October 25Question 1 [50 points]If necessary, you may use log 2 0.69, log 3 1.10, log 4 1.39, log 5 1.61, log 6 1.79, and log(1 + z ) z when z is close to 0. Consider the following quasi-linear utility fu
Princeton - ECO - 310
ECO 310 cfw_ Spring 2008 Microeconomic Theory cfw_ A Mathematical Approach Stephen MorrisMidterm Examination Important instructions:1. This is a closed-book examination. Put away your books, handouts, notes, calculators, palmtops, cellular phones, : : :
Princeton - ECO - 310
ECO 310, Fall 2008 Midterm Examination October 24 Important instructions1. This is a closed-book examination. Put away your books, handouts, notes, calculators, palmtops, cellular phones, . . . now. 2. Print your name and your preceptors name clearly on
Princeton - ECO - 310
ECO 310, Fall 2008 Midterm Examination October 23 Important instructions1. This is a closed-book examination. Put away your books, handouts, notes, calculators, palmtops, cellular phones, . . . now. 2. Print your name and your preceptors name clearly on
Princeton - ECO - 310
ECO 310 - Fall 2008 Microeconomic Theory - A Mathematical Approach midterm 10/23 - Answer KeyQuestion 1:(a)(10 points) FONCS: L(x; y; ) = (x x0 )(y y x This implies thaty y0 x x0y0 ) + (IPx xPy y )y0 = P x x0 = Px Py (x Py x0 ) into the budget cons
Princeton - ECO - 310
ECO 310 - Fall 2008 Microeconomic Theory - A Mathematical Approach Midterm 10/24/08 - Answer KeyQuestion 1:(a)(10 points) L(x, y, ) = xy + y + (I Px x Py y ) FONCS: y = Px x + 1 = Py This implies y =Px Py ( x+ 1). Substitute y into the budget constrai
Princeton - ECO - 310
ECO 310 - Fall 2008 Microeconomic Theory - A Mathematical Approach midterm 10/23 - Answer KeyQuestion 1:(a)(10 points) L(x, y, ) = (x x0 )(y y0 ) + (I Px x Py y ) FONCS: y y0 = Px x x0 = Py This implies thaty y0 xx0=Px Py .Substitute y = y0 + x=Px
Princeton - ECO - 310
Princeton University - Department of Economics Fall Term 2008 ECO 310 - MICROECONOMIC THEORY: A MATHEMATICAL APPROACH Lectures: Tuesdays and Thursdays 11am-12:20pm, Friend Center 101 Professor: Satoru Takahashi, 201 Fisher Hall, 8-7919, satorut@princeton.
Princeton - ECO - 362
Princeton University Department of EconomicsEconomics 362 Problem Set 8 Multi-Factor ModelsFall Term 2009-2010Use the following to answer questions 1-2: Consider the multifactor APT. There are two independent economic factors, F1 and F2. The risk-free
Princeton - ECO - 362
Princeton University Department of Economics Economics 362 Answers to Problem Set 9 1. The percentage bond price change will be: Duration y .005 = 7.194 = .0327 or a 3.27% decline. 1+ y 1.10 Fall Term 20092. Computation of duration: a) YTM = 6% (1) Time
Princeton - ECO - 362
Princeton University Department of Economics Economics 362 Answers to Problem Set 10 1. a) To replicate a one-year pure discount bond with a face value of $100, buy a share of stock, and a European put with an exercise price of$100, and sell a European c
Princeton - ECO - 362
Princeton University Department of Economics Economics 362 Financial Investments Fall Term 2009-2010 Lectures: M&amp;W 10-10:50 Location: McCosh Hall 50 Professor Harrison Hong Bendheim Center for Finance 210 Office Hours: Monday 4-5:30pm Emails: Harrison Hon
Princeton - ECO - 362
ECO 362: Financial Markets Solutions for Fall 2008 Final ExamINSTRUCTIONS You have 3 hours to complete this exam. This exam is closed book and closed notes, except for one 8x11 cheat sheet. Calculators are allowed. There are 21 questions for a total of 1
Princeton - ECO - 362
ECO 362: Financial Markets Solutions for Fall 2008 Final ExamINSTRUCTIONS You have 3 hours to complete this exam. This exam is closed book and closed notes, except for one 8x11 cheat sheet. Calculators are allowed. There are 21 questions for a total of 1
Princeton - ECO - 362
Princeton University Department of Economics Economics 362 Problem Set 9 Bond Market Term Structure 1. A nine-year bond has a yield of 10% and a duration of 7.194 years. If the market yield rises by 50 basis points, what is the percentage change in the bo
Concordia Canada - ECE - 616
ASSIGNMENT 1(1) In a specied 6-AM-to-6-AM 24-hour period, a student wakes up at time t1 and goes to sleep at some later time t2 . (a) Find the sample space and sketch in on the x y plane if the outcome of this experiment consists of the pair(t1 , t2 ). (
Concordia Canada - ECE - 616
Concordia Canada - ECE - 616
assignment 21. Show that n k = n nk2. Show that if P [A B C ] = P [A | B C ]P [B | C ]P [C ]. 3. A nonsymmetric binary communications channel is shown in the gure below. Assume the input is 0 with probability p and 1 with probability 1 p. (a) Find the p
Concordia Canada - ECE - 616
assignment 31. Let X be the maximum of the number of heads obtained when Carlos and Michael each ip a fair coin twice. (a) Describe the underlying space S of this random experiment and specify the probabilities of its elementary events. (b) Show the mapp
Concordia Canada - ECE - 616
Assignment 41. Eight number are selected at random from the unit interval. (a) Find the probability that the rst four numbers are less than 0.25 and the last four are greater than 0.25. (b) Find the probability that four numbers are less than 0.25 and fo
Concordia Canada - ECE - 616
Assignment 51. Let Y = A cos(t) + c where A has mean m and variance 2 and and c are constants. Find the mean and variance of Y . compare the results to those obtained in following example. Example: Expected Values of a Sinusoid with Random Phase Let Y =
Concordia Canada - ECE - 616
Assignment 61. Let X and Y have joint pdf: fX,Y (x, y ) = k (x + y ) (a) Find k . (b) Find the joint cdf of (X,Y). (c) Find the marginal pdf of X and of Y . (d) Find P [X &lt; Y ], P [Y &lt; X 2 ], P [X + Y &gt; 0.5]. 2. The random vector (X, Y ) is uniformly dis
Concordia Canada - ECE - 616
Assignment 71. Let X, Y, Z have joint pdf fX,Y,Z (x, y, z ) = k (x + y + z ) (a) Find k . (b) Find fX (x|y, z ) and fZ (z |x, y ). (c) Find fX (x), fY (y ), and fZ (z ). 2. Show that fX,Y,Z (x, y, z ) = fZ (z |x, y )fY (y |x)fX (x). 3. Let U1 ,U2 and U3
Concordia Canada - ECE - 616
Assignment 81. Let X1 , ., Xn be random variables with the same mean and with covariance function: 2 if i = j, COV (Xi , Xj ) = 2 if|i j | = 1, 0 otherwise. Where | &lt; 1. Find the mean and variance of Sn = X1 + + Xn . 2. Let X1 , ., Xn be random variables
Concordia Canada - ECE - 616
Assignment 91. A random process is dened by Y (t) = G(t T ) where g (t) is the rectangular pulse of following gure, and T is a uniformly distributed random variable in the interval (0,1).101t(a) Find the pmf of Y (t). (b) Find mY (t) and CY (t1 , t2
Concordia Canada - ECE - 616
Assignment 101. Let Y (t) = X (t + d) X (t), where X (t) is a Gaussian random process. (a) Find the mean and autocovariance of Y (t). (b) Find the pdf of Y (t). (c) Find the joint pdf of Y (t) and Y (t + s). (d) Show that Y (t) is a Gaussian random proce
Concordia Canada - ECE - 616
Assignment 111. (a) Find the autocorrelation function corresponding to the power spectral density shown in the following gure. (b) Find the total average power. (c) Plot the power in the range |f | &gt; f0 as a function of f0 &gt; 0.B A Af2f10f1f2f2. S
Concordia Canada - ECE - 616
Assignment 121. Let Mn denote the sequence of sample means form an iid random process Xn : Mn = (a) Is Mn a Markov process? (b) If the answer to part a is yes, nd the following state transition pdf: fMn (X |Mn1 = y ). 2. (a) Show that the following autor
Concordia Canada - ECE - 616
Concordia Canada - ECE - 616
Concordia Canada - ECE - 616
Concordia Canada - ECE - 616
Concordia Canada - ECE - 616
Concordia Canada - ECE - 616
Concordia Canada - ECE - 616
Concordia Canada - ECE - 616
Concordia Canada - ECE - 616
Concordia Canada - ECE - 616
Concordia Canada - ECE - 616
Concordia Canada - ECE - 616
Chapter 5, 6 Multiple Random VariablesENCS6161 - Probability and Stochastic ProcessesConcordia UniversityVector Random VariablesA vector r.v. X is a function X : S Rn , where S is the sample space of a random experiment. Example: randomly pick up a st
Concordia Canada - ECE - 616
Chapter 2 Basic Concepts of Probability TheoryENCS6161 - Probability and Stochastic ProcessesConcordia UniversitySpecifying Random ExperimentsExamples of random experiments: tossing a coin, rolling a dice, the lifetime of a harddisk. Sample space: the