Introduction to Matlab

by Kristian Sandberg, Department of Applied Mathematics, University of Colorado

Goal

The goal with this Lab is to give a brief introduction to the mathematical software
Matlab. After completing the worksheet you should know how to start Matlab, how to
use the elementary functions in Matlab and how to use Matlab to plot functions.

What is Matlab?

Matlab is widely used in all areas of applied mathematics in education and research at
universities and in the industry. Matlab stands for MATrix LABoratory and the
software is built up around vectors and matrices. This makes the software particularly
useful for linear algebra which we will see later in the course but Matlab is also a
great tool for solving algebraic and differential equations and for numerical
integration. Matlab has powerful graphic tools and can produce nice pictures in both
2D and 3D. It is also a programming language (similar to C) and is one of the easiest
programming languages for writing mathematical programs. Matlab also has some
tool boxes useful for signal processing, image processing, etc.

How to start Matlab

Double-click on the icon for Matlab.

The Matlab environment

Note: From now on an instruction to press a certain key will be denoted by < >, e.g.,
pressing the enter key will be denoted as <enter>. Commands that should be typed at
the prompt, will be written in courier font.

The Matlab environment (on most computer systems) consists of menus, buttons and
a writing area similar to an ordinary word processor. There are plenty of help
functions that you are encouraged to use. The writing area that you will see when you
start Matlab, is called the command window. In this window you give the commands
to Matlab. For example, when you want to run a program you have written for Matlab
you start the program in the command window by typing its name at the prompt. The
command window is also useful if you just want to use Matlab as a scientific
calculator or as a graphing tool. If you write longer programs, you will find it more
convenient to write the program code in a separate window, and then run it in the
command window In the command window you will see a prompt, on Macs and PCs
this prompt is >>. You type your commands immediately after this prompt. Once you
have typed the command you wish Matlab to perform, press <enter>. If you want to
interupt a command that Matlab is running, type <ctrl> + <c>.

Useful functions and operations in Matlab

Using Matlab as a calculator is easy.

Example: Compute 5 sin(2.53-pi)+1/75. In Matlab this is done by simply typing

5*sin(2.5^(3-pi))+1/75

at the prompt. Be careful with parantheses and don't forget to type * whenever you
multiply!

Note that Matlab is case sensitive. This means that Matlab knows a difference
between letters written as lower and upper case letters. For example, Matlab will
understand sin(2) but will not understand Sin(2).

Here is a table of useful operations, functions and constants in Matlab.

Operation, function or constant Matlab command

+ (addition) +

- (subtraction) -

X (multiplication) *

/ (division) /

| x | (absolute value of x) abs(x)

square root of x sqrt(x)

ex exp(x)

ln x (natural logaritm of x) log(x)

log10 x (base 10 logaritm of x) log10(x)

sin x sin(x)

cos x cos(x)
tan x tan(x)

cot x cot(x)

arcsin x asin(x)

arccos x acos(x)

arctan x atan(x)

arccot x acot(x)

n! (n factorial) gamma(n+1)

e exp(1)

pi
pi

i (the imaginary number) i

Exercises

Compute the following expressions using Matlab:

 3cos(pi)
 1+1+1/2+1/6+1/24-e
 ln (1000+2pi-2)
 e i pi
 The number of combinations in which 12 persons can stand in line. (Hint: Use
factorials.)

Variables in Matlab

We can easily define our own variables in Matlab. Let's say we need to use the value
of 3.5sin(2.9) repeatedly. Instead of typing 3.5*sin(2.9)over and over again, we can
denote this variable as x by typing the following:

x=3.5*sin(2.9)

(Please try this in Matlab.) Now type

x+1
and observe what happens. If we do not want to see the result of a calculation we can
use semi-colon to surpress the print-out on the screen even though Matlab still
performs the command in "the background". If you defined x as above, now type

y=2*x;

and observe what happened. In many cases we want to know what variables we have
declared. We can do this by typing whos. If you want to erase the variable from the
Matlab memory, type clear.

Vectors and matrices in Matlab

We create a vector in Matlab by putting the elements within [] brackets.

Example: x=[ 1 2 3 4 5 6 7 8 9 10]

We can also create this vector by typing x=1:10. The vector (1 1.1 1.2 1.3 1.4
1.5) can be created by typing x=[ 1 1.1 1.2 1.3 1.4 1.5 ] or by typing
x=1:0.1:1.5.

Matrices can be created according to the following example. The matrix A=

is created by typing

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

i.e., rows are separated with semi-colons. If we want to use a specific element in a
vector or a matrix, study the following example:

Example:

x=[10 20 30]

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

x(2)

A(3,1)

Here we could reach the second element of the vector by typing the variable and the
position within parantheses. The same principle holds for matrices; the first number
specifies the row of the matrix, and the second number specifies the column of the
matrix.
If the matrices (or vectors which are special cases of a matrices) are of the same
dimensions then matrix addition, matrix subtraction and scalar multiplication works
just like we are used to.

Example: Type

x=[ 1 2 3]

y =[4 5 6]

a=2

x+y

x-y

a*x

and observe what happens.

If want to apply an operation such as squaring each element in a matrix we have do

x=1:10

x.^2

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

A.^2

A^2

and observe the result. Note the dot and what it accomplishes! All built-in functions
such as sin, cos, exp and so on automatically act elementwise on a matrix. Type

y=[0 1/4 1/2 3/4 1]

y=pi*y

sin(y)

and observe the result.

How to plot with Matlab

There are different ways of plotting in Matlab. The following technique is probably
the most useful one.
Example: Plot sin(x2) on the interval [-5,5]. To do this, type the following:

x=-5:0.01:5

y=sin(x.^2)

plot(x,y)

and observe what happens. The following commands are useful when plotting:

Graphing functions Matlab command

Label the horizontal axis. xlabel('text')

Label the vertical axis. ylabel('text')

Attach a title to the plot. title('text')

"Keep plotting in the same window." hold on

Turn off the "keep-plotting-in-the-same-window- hold off

Note that all text must be put within ' '. The last two commands (hold on and hold
off) are best explained by trying them next time you plot.

Exercises

 Plot sin(x) on the interval [-pi,pi] using spacing 0.5, 0.1 and 0.01 between the
points where you will sample the function. (This will change the resolution).
Experiment with the hold on command.
 Attach labels to the axis of the previous plot and give a title to the graph.
 Plot 5 cos(x2+1) on [-2pi,2pi]. Note that the squaring operation will require
you to use the dot . in order for the squaring operation to act on each element
individually. However, the addition operation (+) automatically acts on
elements individually.
 Implement anything related to Physics in Matlab. You can visit
https://www.mathworks.com/academia/courseware/teaching-physics-with-
matlab.html and implement anything of your interest.