### MATLAB is a multi-paradigm numerical computing environment.

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. MATLAB is widely used for for matrix-based computation designed for scientific and engineering use.MATLAB can be used as a tool for simulating various electrical networks. Matlab Programming Language:

MATLAB is a software package for mathematical calculations. It is a very powerful package, but is also very simple to use. One of the attractions of MATLAB is its versatility. You can use it interactively or use it like a programming language. It can handle every think from a simple expression to a set of complex mathematical calculations on large sets of data. There is a massive number of predefine functions to choose from. There is also a large selection of simple to use graphics functions to plot and display data to the screen.

The purpose of this document is to introduce the fundamentals of MATLAB. After reading this document you should have a good idea of the type of problems that MATLAB can cope with and how to solve those problems. The document is designed for people that have never used MATLAB before. It is not the intention of this document to be a complete guide to MATLAB. It covers only a small fraction of the hundreds of commands available within MATLAB. However it does show you how to use the basic MATLAB commands and functions. It also explains how to use the help in MATLAB to find out what is available.

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 non interactive 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.

MATLAB is a fourth-generation programming language and numerical analysis environment. Uses for MATLAB include matrix calculations, developing and running algorithms, creating user interfaces (UI) and data visualization. The multi-paradigm numerical computing environment allows developers to interface with programs developed in different languages. MATLAB is used by engineers and scientists in many fields such as image and signal processing, communications, control systems for industry, smart grid design, robotics as well as computational finance.

Introduction:

The primarily objective is to help you learn quickly the first steps. The emphasis here is “learning by doing”. Therefore, the best way to learn is by trying it yourself. Working through the examples will give you a feel for the way that MATLAB operates. In this introduction we will describe how MATLAB handles simple numerical expressions and mathematical formulas. The name MATLAB stands for MATrix LABoratory. MATLAB was written originally to provide easy access to matrix software developed by the LINPACK (linear system package) and EISPACK (Eigen system package) projects.

MATLAB is a high-performance language for technical computing. It integrates computation, visualization, and programming environment. Furthermore, MATLAB is a modern programming language environment: it has sophisticated data structures, contains built-in editing and debugging tools, and supports object-oriented programming. These factors make MATLAB an excellent tool for teaching and research.

MATLAB has many advantages compared to conventional computer languages (e.g., C, FORTRAN) for solving technical problems. MATLAB is an interactive system whose basic data element is an array that does not require dimensioning. The software package has been commercially available since 1984 and is now considered as a standard tool at most universities and industries worldwide.

It has powerful built-in routines that enable a very wide variety of computations. It also has easy to use graphics commands that make the visualization of results immediately available. Specific applications are collected in packages referred to as toolbox. There are toolboxes for signal processing, symbolic computation, control theory, simulation, optimization, and several other fields of applied science and engineering.

MATLAB (matrix laboratory) is a fourth-generation high-level programming language and interactive environment for numerical computation, visualization and programming. MATLAB is developed by MathWorks. It allows matrix manipulations; plotting of functions and data; implementation of algorithms; creation of user interfaces; interfacing with programs written in other languages, including C, C++, Java, and FORTRAN; analyze data; develop algorithms; and create models and applications.  It has numerous built-in commands and math functions that help you in mathematical calculations, generating plots, and performing numerical methods.

The MATLAB programming language is part of the commercial MATLAB software  that is often employed in research and industry and is an example of a high-level “scripting” or “4th generation” language. The most striking difference to C and other compiled languages is that the code is interpreted when the program is executed (an interpreter program reads the source code line by line and translates it into machine instructions on the fly), i.e. no compilation is required. While this decreases the execution speed, it frees the programmer from memory management, allows dynamic typing and interactive sessions. It is worth mentioning that programs written in scripting languages are usually significantly shorter than equivalent programs written in compiled languages and also take significantly less time to code and debug.

In short, there is a trade-off between the execution time (small for compiled languages) and the development time (small for interpreted languages). An important feature for teaching purposes is the ability of MATLAB (and other interpreted languages) to have interactive sessions. The user can type one or several commands at the command prompt and after pressing return, these commands are executed immediately. This allows interactive testing of small parts of the code (without any delay stemming from compilation) and encourages experimentation. Using the interactive prompt, interpreted languages also tend to be easier to debug than compiled executables. The MATLAB package comes with sophisticated libraries for matrix operations, general numeric methods and plotting of data. Universities may have to acquire licences and this may cost tens of thousands of pounds.

Features of MATLAB

Following are the basic features of MATLAB:

• It is a high-level language for numerical computation, visualization and application development.
• It also provides an interactive environment for iterative exploration, design and problem solving.
• It provides vast library of mathematical functions for linear algebra, statistics, Fourier analysis, filtering, optimization, numerical integration and solving ordinary differential equations.
• It provides built-in graphics for visualizing data and tools for creating custom plots.
• MATLAB's programming interface gives development tools for improving code quality, maintainability, and maximizing performance.
• It provides tools for building applications with custom graphical interfaces.
• It provides functions for integrating MATLAB based algorithms with external applications and languages such as C, Java, .NET and Microsoft Excel.

Uses of MATLAB

MATLAB is widely used as a computational tool in science and engineering encompassing the fields of physics, chemistry, math and all engineering streams. It is used in a range of applications including:

• signal processing and Communications
• image and video Processing
• control systems
• test and measurement
• computational finance
• computational biology

MATLAB System

MATLAB system consists of five main parts:

1. 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.

2. 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, MATLAB's applications.

3. Handle Graphics

This is the MATLAB graphics system. It includes high-level commands for two-dimensional and threedimensional 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.

4. 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.

5. A minimum MATLAB session

The goal of this minimum session (also called starting and exiting sessions) is to learn the first steps:

• How to log on
• Invoke MATLAB
• Do a few simple calculations
• How to quit MATLAB

Starting MATLAB

After logging into your account, you can enter MATLAB by double-clicking on the MATLAB shortcut icon (MATLAB 7.0.4) on your Windows desktop. When you start MATLAB, a special window called the MATLAB desktop appears. The desktop is a window that contains other windows. The major tools within or accessible from the desktop are:

• The Command Window
• The Command History
• The Workspace
• The Current Directory
• The Help Browser
• The Start button

When MATLAB is started for the first time, You can customize the arrangement of tools and documents to suit your needs. Now, we are interested in doing some simple calculations. We will assume that you have sufficient understanding of your computer under which MATLAB is being run. You are now faced with the MATLAB desktop on your computer, which contains the prompt

• (>>) in the Command Window. Usually, there are 2 types of prompt:
• >> for full version
• EDU> for educational version

Note: To simplify the notation, we will use this prompt, >>, as a standard prompt sign, though our MATLAB version is for educational purpose.

Using MATLAB as a calculator

As an example of a simple interactive calculation, just type the expression you want to evaluate. Let’s start at the very beginning. For example, let’s suppose you want to calculate the expression, 1 + 2 × 3. You type it at the prompt command (>>) as follows,

>> 1+2*3

ans =

7

You will have noticed that if you do not specify an output variable, MATLAB uses a default variable ans, short for answer, to store the results of the current calculation. Note that the variable ans is created (or overwritten, if it is already existed). To avoid this, you may assign a value to a variable or output argument name. For example,

>> x = 1+2*3

x =

7

will result in x being given the value 1 + 2 × 3 = 7. This variable name can always be used to refer to the results of the previous computations. Therefore, computing 4x will result in

>> 4*x

ans =

28.0000

Quitting MATLAB

To end your MATLAB session, type quit in the Command Window, or select File −→ Exit MATLAB in the desktop main menu.

Getting started

After learning the minimum MATLAB session, we will now learn to use some additional operations.

1. Creating MATLAB variables

MATLAB variables are created with an assignment statement. The syntax of variable assignment is

variable name = a value (or an expression)

For example,

>> x = expression

where expression is a combination of numerical values, mathematical operators, variables, and function calls. On other words, expression can involve:

• manual entry
• built-in functions
• user-defined functions

Overwriting variable

Once a variable has been created, it can be reassigned. In addition, if you do not wish to see the intermediate results, you can suppress the numerical output by putting a semicolon (;) at the end of the line. Then the sequence of commands looks like this:

>> t = 5;

>> t = t+1

t = 6

Error messages

If we enter an expression incorrectly, MATLAB will return an error message. For example, in the following, we left out the multiplication sign, *, in the following expression

>> x = 10;

>> 5x

??? 5x

Error: Unexpected MATLAB expression.

Making corrections

To make corrections, we can, of course retype the expressions. But if the expression is lengthy, we make more mistakes by typing a second time. A previously typed command can be recalled with the up-arrow key ↑. When the command is displayed at the command prompt, it can be modified if needed and executed.

Controlling the hierarchy of operations or precedence

Let’s consider the previous arithmetic operation, but now we will include parentheses. For example, 1 + 2 × 3 will become (1 + 2) × 3

>> (1+2)*3

ans = 9

and, from previous example

>> 1+2*3

ans = 7

By adding parentheses, these two expressions give different results: 9 and 7. The order in which MATLAB performs arithmetic operations is exactly that taught in high school algebra courses. Exponentiations are done first, followed by multiplications and divisions, and finally by additions and subtractions. However, the standard order of precedence of arithmetic operations can be changed by inserting parentheses. For example, the result of 1+2×3 is quite different than the similar expression with parentheses (1+2)×3. The results are 7 and 9 respectively. Parentheses can always be used to overrule priority, and their use is recommended in some complex expressions to avoid ambiguity.

Controlling the appearance of floating point number

MATLAB by default displays only 4 decimals in the result of the calculations, for example −163.6667, as shown in above examples. However, MATLAB does numerical calculations in double precision, which is 15 digits. The command format controls how the results of computations are displayed. Here are some examples of the different formats together with the resulting outputs.

>> format short

>> x=-163.6667

If we want to see all 15 digits, we use the command format long

>> format long

>> x= -1.636666666666667e+002

To return to the standard format, enter format short, or simply format. There are several other formats. For more details, see the MATLAB documentation, or type help format.

Note - Up to now, we have let MATLAB repeat everything that we enter at the prompt (>>). Sometimes this is not quite useful, in particular when the output is pages en length. To prevent MATLAB from echoing what we type, simply enter a semicolon (;) at the end of the command. For example,

>> x=-163.6667;

>> x

x =

-163.6667

Managing the workspace

The contents of the workspace persist between the executions of separate commands. Therefore, it is possible for the results of one problem to have an effect on the next one. To avoid this possibility, it is a good idea to issue a clear command at the start of each new independent calculation.

>> clear

The command clear or clear all removes all variables from the workspace. This frees up system memory. In order to display a list of the variables currently in the memory,

type

>> who

while, whos will give more details which include size, space allocation, and class of the variables

Keeping track of your work session

It is possible to keep track of everything done during a MATLAB session with the diary command.

>> diary

or give a name to a created file,

>> diary FileName

where FileName could be any arbitrary name you choose. The function diary is useful if you want to save a complete MATLAB session. They save all input and output as they appear in the MATLAB window. When you want to stop the recording, enter diary off. If you want to start recording again, enter diary on. The file that is created is a simple text file. It can be opened by an editor or a word processing program and edited to remove extraneous material, or to add your comments. You can use the function type to view the diary file or you can edit in a text editor or print. This command is useful, for example in the process of preparing a homework or lab submission.

Entering multiple statements per line

It is possible to enter multiple statements per line. Use commas (,) or semicolons (;) to enter more than one statement at once. Commas (,) allow multiple statements per line without suppressing output.

>> a=7; b=cos(a), c=cosh(a)

b = 0.6570

c = 548.3170

Miscellaneous commands

Here are few additional useful commands:

• To clear the Command Window, type clc
• To abort a MATLAB computation, type ctrl-c
• To continue a line, type . . .

Getting help

To view the online documentation, select MATLAB Help from Help menu or MATLAB Help directly in the Command Window. The preferred method is to use the Help Browser. The Help Browser can be started by selecting the ? icon from the desktop toolbar. On the other hand, information about any command is available by typing

>> help Command

Another way to get help is to use the lookfor command. The lookfor command differs from the help command. The help command searches for an exact function name match, while the lookfor command searches the quick summary information in each function for a match. For example, suppose that we were looking for a function to take the inverse of a matrix. Since MATLAB does not have a function named inverse, the command help inverse will produce nothing. On the other hand, the command lookfor inverse will produce detailed information, which includes the function of interest, inv.

>> lookfor inverse

Note - At this particular time of our study, it is important to emphasize one main point. Because MATLAB is a huge program; it is impossible to cover all the details of each function one by one. However, we will give you information how to get help. Here are some examples:

• Use on-line help to request info on a specific function

>> help sqrt

• In the current version (MATLAB version 7), the doc function opens the on-line version of the help manual. This is very helpful for more complex commands.

>> doc plot

Mathematical functions

MATLAB offers many predefined mathematical functions for technical computing which contains a large set of mathematical functions. Typing help elfun and help specfun calls up full lists of elementary and special functions respectively. There is a long list of mathematical functions that are built into MATLAB. These functions are called built-ins. Many standard mathematical functions, such as sin(x), cos(x), tan(x), e x , ln(x), are evaluated by the functions sin, cos, tan, exp, and log respectively in MATLAB.

Examples

We illustrate here some typical examples which related to the elementary functions previously defined.

As a first example, the value of the expression y = e−a

sin(x) + 10√y, for a = 5, x = 2, and

y = 8 is computed by

>> a = 5; x = 2; y = 8;

>> y = exp(-a)*sin(x)+10*sqrt(y)

y = 28.2904

The subsequent examples are

>> log(142)

ans = 4.9558

>> log10(142)

ans = 2.1523

Note the difference between the natural logarithm log(x) and the decimal logarithm (base

10) log10(x).

To calculate sin(π/4) and e 10, we enter the following commands in MATLAB,

>> sin(pi/4)

ans = 0.7071

>> exp(10)

ans = 2.2026e+004

Basic plotting

1. overview

MATLAB has an excellent set of graphic tools. Plotting a given data set or the results of computation is possible with very few commands. You are highly encouraged to plot mathematical functions and results of analysis as often as possible. Trying to understand mathematical equations with graphics is an enjoyable and very efficient way of learning mathematics. Being able to plot mathematical functions and data freely is the most important step, and this section is written to assist you to do just that.

Creating simple plots

The basic MATLAB graphing procedure, for example in 2D, is to take a vector of xcoordinates, x = (x1, . . . , xN ), and a vector of y-coordinates, y = (y1, . . . , yN ), locate thepoints (xi , yi), with i = 1, 2, . . . , n and then join them by straight lines. You need to prepare x and y in an identical array form; namely, x and y are both row arrays or column arrays of the same length.

The MATLAB command to plot a graph is plot(x,y). The vectors x = (1, 2, 3, 4, 5, 6) and y = (3, −1, 2, 4, 5, 1) produce

>> x = [1 2 3 4 5 6];

>> y = [3 -1 2 4 5 1];

>> plot(x,y)

Note: The plot functions 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 we specify two vectors, as mentioned above, plot(x,y) produces a graph of y versus x.

For example, to plot the function sin (x) on the interval [0, 2π], we first create a vector of x values ranging from 0 to 2π, then compute the sine of these values, and finally plot the result:

>> x = 0:pi/100:2*pi;

>> y = sin(x);

>> plot(x,y)

Adding titles, axis labels, and annotations

MATLAB enables you to add axis labels and titles. For example, using the graph from the previous example, add an x- and y-axis labels. Now label the axes and add a title. The character \pi creates the symbol π.

>> xlabel(’x = 0:2\pi’)

>> ylabel(’Sine of x’)

>> title(’Plot of the Sine function’)

The color of a single curve is, by default, blue, but other colors are possible. The desired color is indicated by a third argument. For example, red is selected by plot(x,y,’r’). Note the single quotes, ’ ’, around r.

Multiple data sets in one plot

Multiple (x, y) pairs arguments create multiple graphs with a single call to plot. For example, these statements plot three related functions of x: y1 = 2 cos(x), y2 = cos(x), and y3 = 0.5 ∗ cos(x), in the interval 0 ≤ x ≤ 2π.

>> x = 0:pi/100:2*pi;

>> y1 = 2*cos(x);

>> y2 = cos(x);

>> y3 = 0.5*cos(x);

>> plot(x,y1,’--’,x,y2,’-’,x,y3,’:’)

>> xlabel(’0 \leq x \leq 2\pi’)

>> ylabel(’Cosine functions’)

>> legend(’2*cos(x)’,’cos(x)’,’0.5*cos(x)’)

>> title(’Typical example of multiple plots’)

>> axis([0 2*pi -3 3])

By default, MATLAB uses line style and color to distinguish the data sets plotted in the graph. However, you can change the appearance of these graphic components or add annotations to the graph to help explain your data for presentation.

Specifying line styles and colors

It is possible to specify line styles, colors, and markers (e.g., circles, plus signs, . . . ) using the plot command:

plot(x,y,’style_color_marker’)

where style_color_marker is a triplet of values. To find additional information, type help plot or doc plot.

Introduction

Matrices are the basic elements of the MATLAB environment. A matrix is a two-dimensional array consisting of m rows and n columns. Special cases are column vectors (n = 1) and row vectors (m = 1).

In this blog we will illustrate how to apply different operations on matrices. The following topics are discussed: vectors and matrices in MATLAB, the inverse of a matrix, determinants, and matrix manipulation. MATLAB supports two types of operations, known as matrix operations and array operations. Matrix operations will be discussed first.

Matrix generation

Matrices are fundamental to MATLAB. Therefore, we need to become familiar with matrix generation and manipulation. Matrices can be generated in several ways

Entering a vector

A vector is a special case of a matrix. The purpose of this section is to show how to create vectors and matrices in MATLAB. As discussed earlier, an array of dimension 1 ×n is called a row vector, whereas an array of dimension m × 1 is called a column vector. The elements of vectors in MATLAB are enclosed by square brackets and are separated by spaces or by commas. For example, to enter a row vector, v, type

>> v = [1 4 7 10 13]

v = 1 4 7 10 13

Column vectors are created in a similar way, however, semicolon (;) must separate the components of a column vector,

>> w = [1;4;7;10;13]

w =

1

4

7

10

13

On the other hand, a row vector is converted to a column vector using the transpose operator. The transpose operation is denoted by an apostrophe or a single quote (’).

>> w = v’

w =

1

4

7

10

13

Thus, v(1) is the first element of vector v, v(2) its second element, and so forth. Furthermore, to access blocks of elements, we use MATLAB’s colon notation (:). For example, to access the first three elements of v, we write,

>> v(1:3)

ans =

1 4 7

Or, all elements from the third through the last elements,

>> v(3,end)

ans =

7 10 13

where end signifies the last element in the vector. If v is a vector, writing

>> v(:)

produces a column vector, whereas writing

>> v(1:end)

produces a row vector.

Entering a matrix

A matrix is an array of numbers. To type a matrix into MATLAB you must

• begin with a square bracket, [
• separate elements in a row with spaces or commas (,)
• use a semicolon (;) to separate rows
• end the matrix with another square bracket, ].

Here is a typical example. To enter a matrix A, such as,

A =

1 2 3

4 5 6

7 8 9

(2.1)

type,

>> A = [1 2 3; 4 5 6; 7 8 9]

MATLAB then displays the 3 × 3 matrix as follows,

A =

1 2 3

4 5 6

7 8 9

Note that the use of semicolons (;) here is different from their use mentioned earlier to suppress output or to write multiple commands in a single line. Once we have entered the matrix, it is automatically stored and remembered in the Workspace. We can refer to it simply as matrix A. We can then view a particular element in a matrix by specifying its location. We write,

>> A(2,1)

ans = 4

A(2,1) is an element located in the second row and first column. Its value is 4.

Matrix indexing

We select elements in a matrix just as we did for vectors, but now we need two indices. The element of row i and column j of the matrix A is denoted by A(i,j). Thus, A(i,j) in MATLAB refers to the element Aij of matrix A. The first index is the row number and the second index is the column number. For example, A(1,3) is an element of first row and third column. Here, A(1,3)=3. Correcting any entry is easy through indexing. Here we substitute A(3,3)=9 by A(3,3)=0. The result is

>> A(3,3) = 0

A =

1 2 3

4 5 6

7 8 0

Single elements of a matrix are accessed as A(i,j), where i ≥ 1 and j ≥ 1. Zero or negative subscripts are not supported in MATLAB.

Colon operator

The colon operator will prove very useful and understanding how it works is the key to efficient and convenient usage of MATLAB. It occurs in several different forms. Often we must deal with matrices or vectors that are too large to enter one element at a time. For example, suppose we want to enter a vector x consisting of points (0, 0.1, 0.2, 0.3, · · · , 5). We can use the command

>> x = 0:0.1:5;

The row vector has 51 elements.

Linear spacing

On the other hand, there is a command to generate linearly spaced vectors: linspace. It is similar to the colon operator (:), but gives direct control over the number of points. For example,

y = linspace(a,b)

generates a row vector y of 100 points linearly spaced between and including a and b.

y = linspace(a,b,n)

generates a row vector y of n points linearly spaced between and including a and b. This is useful when we want to divide an interval into a number of subintervals of the same length.

For example,

>> theta = linspace(0,2*pi,101)

divides the interval [0, 2π] into 100 equal subintervals, then creating a vector of 101 elements.

Colon operator in a matrix

The colon operator can also be used to pick out a certain row or column. For example, the statement A(m:n,k:l specifies rows m to n and column k to l. Subscript expressions refer to portions of a matrix. For example,

>> A(2,:)

ans =

4 5 6

is the second row elements of A.

The colon operator can also be used to extract a sub-matrix from a matrix A.

>> A(:,2:3)

ans =

2 3

5 6

8 0

A(:,2:3) is a sub-matrix with the last two columns of A.

A row or a column of a matrix can be deleted by setting it to a null vector, [ ].

>> A(:,2)=[]

ans =

1 3

4 6

7 0

Creating a sub-matrix

To extract a submatrix B consisting of rows 2 and 3 and columns 1 and 2 of the matrix A, do the following

>> B = A([2 3],[1 2])

B =

4 5

7 8

To interchange rows 1 and 2 of A, use the vector of row indices together with the colon operator.

>> C = A([2 1 3],:)

C =

4 5 6

1 2 3

7 8 0

It is important to note that the colon operator (:) stands for all columns or all rows. To create a vector version of matrix A, do the following

>> A(:)

ans =

1

2

3

4

5

6

7

8

0

The submatrix comprising the intersection of rows p to q and columns r to s is denoted by A(p:q,r:s).

As a special case, a colon (:) as the row or column specifier covers all entries in that row or column; thus

• A(:,j) is the jth column of A, while
• A(i,:) is the ith row, and
• A(end,:) picks out the last row of A.

The keyword end, used in A(end,:), denotes the last index in the specified dimension. Here are some examples.

>> A

A =

1 2 3

4 5 6

7 8 9

>> A(2:3,2:3)

ans =

5 6

8 9

>> A(end:-1:1,end)

ans =

9

6

3

>> A([1 3],[2 3])

ans =

2 3

8 9

Deleting row or column

To delete a row or column of a matrix, use the empty vector operator, [ ].

>> A(3,:) = []

A =

1 2 3

4 5 6

Third row of matrix A is now deleted. To restore the third row, we use a technique for creating a matrix

>> A = [A(1,:);A(2,:);[7 8 0]]

A =

1 2 3

4 5 6

7 8 0

Matrix A is now restored to its original form.

Dimension

To determine the dimensions of a matrix or vector, use the command size. For example,

>> size(A)

ans =

3 3

means 3 rows and 3 columns. Or more explicitly with,

>> [m,n]=size(A)

Continuation

If it is not possible to type the entire input on the same line, use consecutive periods, called an ellipsis . . ., to signal continuation, then continue the input on the next line.

B = [4/5 7.23*tan(x) sqrt(6); ...

1/x^2 0 3/(x*log(x)); ...

x-7 sqrt(3) x*sin(x)];

Note that blank spaces around +, −, = signs are optional, but they improve readability

Transposing a matrix

The transpose operation is denoted by an apostrophe or a single quote (’). It flips a matrix about its main diagonal and it turns a row vector into a column vector. Thus,

>> A’

ans =

1 4 7

2 5 8

3 6 0

By using linear algebra notation, the transpose of m × n real matrix A is the n × m matrix that results from interchanging the rows and columns of A. The transpose matrix is denoted AT.

Concatenating matrices

Matrices can be made up of sub-matrices. Here is an example. First, let’s recall our previous matrix A.

A =

1 2 3

4 5 6

7 8 9

The new matrix B will be,

>> B = [A 10*A; -A [1 0 0; 0 1 0; 0 0 1]]

B =

1 2 3 10 20 30

4 5 6 40 50 60

7 8 9 70 80 90

-1 -2 -3 1 0 0

-4 -5 -6 0 1 0

-7 -8 -9 0 0 1

Matrix generators

MATLAB provides functions that generates elementary matrices. The matrix of zeros, the matrix of ones, and the identity matrix are returned by the functions zeros, ones, and eye, respectively.

Array operations and Linear equations

Array operations

MATLAB has two different types of arithmetic operations: matrix arithmetic operations and array arithmetic operations. We have seen matrix arithmetic operations in the previous lab. Now, we are interested in array operations.

Matrix arithmetic operations

As we mentioned earlier, MATLAB allows arithmetic operations: +, −, ∗, and ˆ to be carried out on matrices. Thus,

• A+B or B+A is valid if A and B are of the same size
• A*B is valid if A’s number of column equals B’s number of rows
• A^2 is valid if A is square and equals A*A
• α*A or A*α multiplies each element of A by α

Array arithmetic operations

On the other hand, array arithmetic operations or array operations for short, are done element-by-element. The period character, ., distinguishes the array operations from the matrix operations. However, since the matrix and array operations are the same for addition (+) and subtraction (−), the character pairs (.+) and (.−) are not used. The list of array operators is shown below in Table 3.2. If A and B are two matrices of the same size with elements A = [aij ] and B = [bij ], then the command

>> C = A.*B

produces another matrix C of the same size with elements cij = aij bij . For example, using the same 3 × 3 matrices,

A =

1 2 3

4 5 6

7 8 9

, B =

10 20 30

40 50 60

70 80 90

we have,

>> C = A.*B

C =

10 40 90

160 250 360

490 640 810

To raise a scalar to a power, we use for example the command 10^2. If we want the operation to be applied to each element of a matrix, we use .^2. For example, if we want to produce new matrix whose elements are the square of the elements of the matrix A, we enter

>> A.^2

ans =

1 4 9

16 25 36

49 64 81

Solving linear equations

One of the problems encountered most frequently in scientific computation is the solution of systems of simultaneous linear equations. With matrix notation, a system of simultaneous linear equations is written

Ax = b (3.1)

where there are as many equations as unknown. A is a given square matrix of order n, b is a given column vector of n components, and x is an unknown column vector of n components. In linear algebra we learn that the solution to Ax = b can be written as x = A−1

b, where

A−1

is the inverse of A.

For example, consider the following system of linear equations

x + 2y + 3z = 1

4x + 5y + 6z = 1

7x + 8y = 1

The coefficient matrix A is

A =

1 2 3

4 5 6

7 8 9

and the vector b =

1

1

1

With matrix notation, a system of simultaneous linear equations is written

Ax = b (3.2)

This equation can be solved for x using linear algebra. The result is x = A−1

b.

There are typically two ways to solve for x in MATLAB:

1. The first one is to use the matrix inverse, inv

>> A = [1 2 3; 4 5 6; 7 8 0];

>> b = [1; 1; 1];

>> x = inv(A)*b

x =

-1.0000

1.0000

-0.0000

2. The second one is to use the backslash (\)operator. The numerical algorithm behind this operator is computationally efficient. This is a numerically reliable way of solving system of linear equations by using a well-known process of Gaussian elimination.

>> A = [1 2 3; 4 5 6; 7 8 0];

>> b = [1; 1; 1];

>> x = A\b

x =

-1.0000

1.0000

-0.0000

This problem is at the heart of many problems in scientific computation. Hence it is important that we know how to solve this type of problem efficiently. Now, we know how to solve a system of linear equations

Introduction to programming in MATLAB

So far in these lab sessions, all the commands were executed in the Command Window. The problem is that the commands entered in the Command Window cannot be saved and executed again for several times. Therefore, a different way of executing repeatedly commands with MATLAB is:

1. to create a file with a list of commands,

2. save the file, and

3. run the file.

If needed, corrections or changes can be made to the commands in the file. The files that are used for this purpose are called script files or scripts for short. This section covers the following topics:

• M-File Scripts
• M-File Functions

M-File Scripts

A script file is an external file that contains a sequence of MATLAB statements. Script files have a filename extension .m and are often called M-files. M-files can be scripts that simply execute a series of MATLAB statements, or they can be functions that can accept arguments and can produce one or more outputs.

Script side-effects

All variables created in a script file are added to the workspace. This may have undesirable effects, because:

• Variables already existing in the workspace may be overwritten.
• The execution of the script can be affected by the state variables in the workspace.

As a result, because scripts have some undesirable side-effects, it is better to code any complicated applications using rather function M-file.

M-File functions

As mentioned earlier, functions are programs (or routines) that accept input arguments and return output arguments. Each M-file function (or function or M-file for short) has its own area of workspace, separated from the MATLAB base workspace.

Anatomy of a M-File function

This simple function shows the basic parts of an M-file.

function f = factorial(n)

% FACTORIAL(N) returns the factorial of N.

% Compute a factorial value.

f = prod(1:n);

The first line of a function M-file starts with the keyword function. It gives the function name and order of arguments. In the case of function factorial, there are up to one output argument and one input argument. Table 4.1 summarizes the M-file function.

As an example, for n = 5, the result is,

>> f = factorial(5)

f = 120

Input and output arguments

As mentioned above, the input arguments are listed inside parentheses following the function name. The output arguments are listed inside the brackets on the left side. They are used to transfer the output from the function file. The general form looks like this

function [outputs] = function_name(inputs)

Function file can have none, one, or several output arguments.

Input to a script file

When a script file is executed, the variables that are used in the calculations within the file must have assigned values. The assignment of a value to a variable can be done in three ways.

1. The variable is defined in the script file.

2. The variable is defined in the command prompt.

3. The variable is entered when the script is executed.

We have already seen the two first cases. Here, we will focus our attention on the third one. In this case, the variable is defined in the script file. When the file is executed, the user is prompted to assign a value to the variable in the command prompt. This is done by using the input command. Here is an example.

% This script file calculates the average of points

% scored in three games.

% The point from each game are assigned to a variable

% by using the ‘input’ command.

game1 = input(’Enter the points scored in the first game ’);

game2 = input(’Enter the points scored in the second game ’);

game3 = input(’Enter the points scored in the third game ’);

average = (game1+game2+game3)/3

The following shows the command prompt when this script file (saved as example3) is executed.

>> example3

>> Enter the points scored in the first game 15

>> Enter the points scored in the second game 23

>> Enter the points scored in the third game 10

average = 16

The input command can also be used to assign string to a variable. A typical example of M-file function programming can be found in a recent paper which related to the solution of the ordinary differential equation (ODE)

Output commands

As discussed before, MATLAB automatically generates a display when commands are executed. In addition to this automatic display, MATLAB has several commands that can be used to generate displays or outputs. Two commands that are frequently used to generate output are: disp and fprintf.

Control flow and operators

MATLAB is also a programming language. Like other computer programming languages, MATLAB has some decision making structures for control of command execution. These decision making or control flow structures include for loops, while loops, and if-else-end constructions. Control flow structures are often used in script M-files and function M-files. By creating a file with the extension .m, we can easily write and run programs. We do not need to compile the program since MATLAB is an interpretative (not compiled) language. MATLAB has thousand of functions, and you can add your own using m-files. MATLAB provides several tools that can be used to control the flow of a program (script or function). In a simple program as shown in the previous Chapter, the commands are executed one after the other. Here we introduce the flow control structure that make possible to skip commands or to execute specific group of commands.

Control flow

MATLAB has four control flow structures: the if statement, the for loop, the while loop, and the switch statement.

The ‘‘if...end’’ structure

MATLAB supports the variants of “if” construct.

• if ... end
• if ... else ... end
• if ... elseif ... else ... end

The simplest form of the if statement is

if expression

statements

end

Here are some examples based on the familiar quadratic formula.

1. discr = b*b - 4*a*c;

if discr < 0

disp(’Warning: discriminant is negative, roots are

imaginary’);

end

2. discr = b*b - 4*a*c;

if discr < 0

disp(’Warning: discriminant is negative, roots are

imaginary’);

else

disp(’Roots are real, but may be repeated’)

end

3. discr = b*b - 4*a*c;

if discr < 0

disp(’Warning: discriminant is negative, roots are

imaginary’);

elseif discr == 0

disp(’Discriminant is zero, roots are repeated’)

else

disp(’Roots are real’)

end

It should be noted that:

• elseif has no space between else and if (one word)
• no semicolon (;) is needed at the end of lines containing if, else, end
• indentation of if block is not required, but facilitate the reading.
• the end statement is required

The ‘‘for...end’’ loop

In the for ... end loop, the execution of a command is repeated at a fixed and predetermined number of times. The syntax is

for variable = expression

statements

end

Usually, expression is a vector of the form i:s:j. A simple example of for loop is

for ii=1:5

x=ii*ii

end

It is a good idea to indent the loops for readability, especially when they are nested. Note that MATLAB editor does it automatically. Multiple for loops can be nested, in which case indentation helps to improve the readability. The following statements form the 5-by-5 symmetric matrix A with (i, j) element i/j for j ≥ i:

n = 5; A = eye(n);

for j=2:n

for i=1:j-1

A(i,j)=i/j;

A(j,i)=i/j;

end

end

The ‘‘while...end’’ loop

This loop is used when the number of passes is not specified. The looping continues until a stated condition is satisfied. The while loop has the form:

while expression

statements

end

The statements are executed as long as expression is true.

x = 1

while x <= 10

x = 3*x

end

It is important to note that if the condition inside the looping is not well defined, the looping will continue indefinitely. If this happens, we can stop the execution by pressing Ctrl-C.

Other flow structures

• The break statement. A while loop can be terminated with the break statement, which passes control to the first statement after the corresponding end. The break statement can also be used to exit a for loop.
• The continue statement can also be used to exit a for loop to pass immediately to the next iteration of the loop, skipping the remaining statements in the loop.
• Other control statements include return, continue, switch, etc. For more detail about these commands, consul MATLAB documentation.

Operator precedence

We can build expressions that use any combination of arithmetic, relational, and logical operators. Precedence rules determine the order in which MATLAB evaluates an expression. We have already seen this in the “Tutorial Lessons”. ordered from highest (1) to lowest (9) precedence level. Operators are evaluated from left to right.

Saving output to a file

In addition to displaying output on the screen, the command fprintf can be used for writing the output to a file. The saved data can subsequently be used by MATLAB or other softwares.

To save the results of some computation to a file in a text format requires the following steps:

1. Open a file using fopen

2. Write the output using fprintf

3. Close the file using fclose

Here is an example (script) of its use

% write some variable length strings to a file

op = fopen(’weekdays.txt’,’wt’);

fprintf(op,’Sunday\nMonday\nTuesday\nWednesday\n’);

fprintf(op,’Thursday\nFriday\nSaturday\n’);

fclose(op);

This file (weekdays.txt) can be opened with any program that can read .txt file.

Debugging M-files

This section introduces general techniques for finding errors in M-files. Debugging is the process by which you isolate and fix errors in your program or code. Debugging helps to correct two kind of errors:

• Syntax errors - For example omitting a parenthesis or misspelling a function name.
• Run-time errors - Run-time errors are usually apparent and difficult to track down. They produce unexpected results.

Debugging process

We can debug the M-files using the Editor/Debugger as well as using debugging functions from the Command Window. The debugging process consists of

• Preparing for debugging
• Setting breakpoints
• Running an M-file with breakpoints
• Stepping through an M-file
• Examining values
• Correcting problems
• Ending debugging

Preparing for debugging

Here we use the Editor/Debugger for debugging. Do the following to prepare for debugging:

• Open the file
• Save changes
• Be sure the file you run and any files it calls are in the directories that are on the search path.

Setting breakpoints

Set breakpoints to pause execution of the function, so we can examine where the problem might be. There are three basic types of breakpoints:

• A standard breakpoint, which stops at a specified line.
• A conditional breakpoint, which stops at a specified line and under specified conditions.
• An error breakpoint that stops when it produces the specified type of warning, error, NaN, or infinite value. You cannot set breakpoints while MATLAB is busy, for example, running an M-file.

Running with breakpoints

After setting breakpoints, run the M-file from the Editor/Debugger or from the Command Window. Running the M-file results in the following:

• The prompt in the Command Window changes to K>> indicating that MATLAB is in debug mode.
• The program pauses at the first breakpoint. This means that line will be executed when you continue. The pause is indicated by the green arrow.
• In breakpoint, we can examine variable, step through programs, and run other calling functions.

Examining values

While the program is paused, we can view the value of any variable currently in the workspace. Examine values when we want to see whether a line of code has produced the expected result or not. If the result is as expected, step to the next line, and continue running. If the result is not as expected, then that line, or the previous line, contains an error. When we run a program, the current workspace is shown in the Stack field. Use who or whos to list the variables in the current workspace.

Viewing values as datatips

First, we position the cursor to the left of a variable on that line. Its current value appears. This is called a datatip, which is like a tooltip for data. If you have trouble getting the datatip to appear, click in the line and then move the cursor next to the variable.

Correcting and ending debugging

While debugging, we can change the value of a variable to see if the new value produces expected results. While the program is paused, assign a new value to the variable in the Command Window, Workspace browser, or Array Editor. Then continue running and stepping through the program.

Ending debugging

After identifying a problem, end the debugging session. It is best to quit debug mode before editing an M-file. Otherwise, you can get unexpected results when you run the file. To end debugging, select Exit Debug Mode from the Debug menu.

The program can be used as a scratchpad to evaluate expressions typed at the command line, or it can be used to execute large prewritten programs. Applications may be written and changed with the built-in integrated development environment and debugged with the MATLAB debugger. Because the language is so simple to use, it is optimal for the fast prototyping of new applications.

Many program development tools are supported to make the program easy to use. They contain an integrated editor/debugger, on-line documentation and manuals, a workspace browser, and extensive demos.

Platform Independence

MATLAB is supported on different computer systems, providing a considerable measure of platform independence. The language is provided on Windows 2000/XP/Vista, Linux, various versions of UNIX, and the Macintosh. Applications written on any platform will run on the other entire platform, and information files written on any platform may be read apparently on any other platform. As a result, programs written in MATLAB can shift to new platforms when the needs of the user change.

Predefined Functions

MATLAB comes complete with a huge library of predefined functions that provides tested and prepackaged solutions to many primary technical tasks. For example, suppose that we are writing a program that must evaluate the statistics associated with an input data set. In most languages, we would need to write our subroutines or functions to implement calculations such as the arithmetic mean, standard deviation, median, and so on. These and hundreds of other services are built right into the MATLAB language, making your job much more comfortable.

In addition to the vast libraries of services built into the basic MATLAB language, there are many special-purpose toolboxes applicable to help solve complex problems in particular areas. For example, a user can buy standard toolkits to solve problems in signal processing, control systems, communications, image processing, and neural networks, etc. There is also a broad compilation of free user-contributed MATLAB programs that are shared through the MATLAB Web site.

Device-Independent Plotting

MATLAB has many basic plotting and imaging commands. The plots and pictures can be displayed on any graphical output device provided by the computer on which MATLAB is running. This facility makes MATLAB an outstanding tool for visualizing technical information.

Graphical User Interface

MATLAB contains a tool that allows a programmer to interactively design a Graphical User Interface (GUI) for his program. With this capability, the programmer can design refined data-analysis programs that can be operated by relatively inexperienced users.

MATLAB Compiler

MATLAB's adaptability and platform independence are produced by compiling MATLAB applications into a machine-independent p-code and then interpreting the p-code instruction at runtime. This method is equivalent to that used by Microsoft's Visual Basic language. Unfortunately, the resulting applications can sometimes execute slowly because the MATLAB code is interpreted rather than compiled.

A separate MATLAB compiler is available. This compiler can compile MATLAB programs into a real executable that runs faster than the interpreted code. It is a great technique to convert a prototype MATLAB program into an executable suitable for sale and distribution to users.

There is two major disadvantage of MATLAB programming language:

Interpreted language

The first disadvantage is that it is an interpreted language and, therefore, may execute more slowly than compiled language. This problem can be check by properly structuring the MATLAB program.

Cost

A full copy of MATLAB is five to ten times more costly than a conventional C or FORTRAN compiler. This comparatively high cost is more than offset by the decreased time necessary for an engineer or scientist to create a working program, so MATLAB is cost-effective for businesses. However, it is too expensive for most individuals to consider purchasing. Fortunately, there is also an inexpensive Student Edition of MATLAB, which is an excellent tool for students wishing to learn the language. The Student Edition of MATLAB is virtually identical to the full edition.