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1

Maths for Engineers and Scientists 4: Matlab
Contents
1

Introduction to

2

Matlab

1

1
...


1
...


2

1
...


2

1
...


6

1
...


8

1
...


9

Matlab

1

Problem sets

11

2
...


11

2
...


13

2
...


15

2
...


16

2
...


18

1

Matlab

Introduction to

Matlab
Matlab

These notes mostly refer to

under Windows on PC Caledonia
...
There is a free alternative to

Octave

, which is very similar to

at most minor modications
...
Most of the programs introduced in this document

Matlab
Matlab

Symbolic Math Toolbox for

(apart from those requiring

Matlab
Octave

Matlab
Matlab

) will run under

with

to allow a quick start
...
Griths (downloadable through Vision) provides a

broader introduction to


...
1 What is it?

Matlab

is essentially a programming language and is similar to C, Pascal, Fortran and other

widely-used programming languages
...
)

It also has many built-in functions for nding approximate solutions of

mathematical problems, doing linear algebra etc
...


Matlab
Matlab

runs on most standard computer systems (PCs, Macs, Unix workstations etc
...

it is

is widely used in higher education to teach numerical analysis and other topics because

much

easier to learn and to use than most other programming languages
...


Matlab

is increasingly being used in industry,

both as a quick way to develop and test complex software, and for the facilities provided by the
many add-on toolboxes tailored for dierent applications
...


1

real world = outside university

2

1
...
m (here *** stands for any command name e
...
mean, std, sum etc
...
You may also type doc *** in order to get the same
If you know the name of the command you want to nd out about, type

Matlab

Matlab

Matlab

information in a separate window
...


Matlab

can be found in the Mathematics folder on PC-

1
...

Click on File (at the top left of the Matlab command window) and select New and then

M-file
...
Select

Save As and you should see a dialogue box opening
...
You choose the directory in the Save

in box: pull down the arrow at its right side until you come to something that starts with
your username (it is likely to be there already)
...
m

in the File name box and then click on the Save button to save your

program
...
m

and it is stored in your home directory (corresponding to

in Matlab)
...
m
...
It
is very important that you learn how to save les before you create anything complicated
...
Matrices with only one row or
column can be treated as vectors, and

An Introduction to
Type the lines

Matlab


...
(See Sections 5, 8 and 16 of

3

x
y
z
a
A

=
=
=
=
=

[-2, 4, -8]
[3, 5, 7]
[1; 2; 3]
5
[-1 0; 3 1]

to create two

1×3

(row) vectors (x and

y ),

a column vector (z ), a scalar (a) and a

(A)
...
Matrices are usually entered in for

Matlab

Section 19) or using colon notation
...
Try it out
...
g
...
Note that a
so that B(12,197) is OK, whilst

results in 4, 4 (the rst two commands refer to the same element of
matrix or vector will only accept positive integers as indices,

B(0,1), B(2,-3),

or

B(1
...


Matrix operations
Most of matrix operations are listed in Section 16 of

An Introduction to

many of these can be made to work entry-wise by typing a dot

A*A
in

is the matrix

Matlab

A2 ,

with the

whilst

2×2

A
...
*A
ans =
1
9

0
1

See Section 14 and 16
...



...
Note that

before the operation  e
...


entry equal to

A(i, j)2
...


we type

>> A = [-1 0 ; 3 1]
A =
-1
3

0
1

>> [V,D]=eig(A)
V =
0
1
...
5547
-0
...


A

and the diagonal matrix

D

contains the

Functions
These are introduced in Section 7 and listed in Section 28 of

An Introduction to

Matlab


...
Try out the following
examples (remember that you can nd out what the command

exp(x)
sy = sqrt(y)
abs(A)
floor(sy)
ceil(sy)
max(x)

***

does by typing

help ***)
...
This code:

•

initialises the variable

•

for each integer

The command

i

sum20

to zero;

between 1 and 20 performs the command

i = 1:20

sum20 = sum20 + i
...
It means all the integers between

a:b:c means all
the numbers a, a+b, a+2b,
...
The syntax is first:increment:last
...
g
...

1 and 20, starting with 1 and ending with 20
...


1:2:10
5:-7:-30
14
...
2:8
...
24:9
...
24 and not 9
...
else
The

if

Matlab

Sections 19 and 20 deal with for loops and related things in detail
...


a = 3;
b = 2;
if (aj = -1;
else
j = 0;
end
What is the value of

j

in the above example?

The code below produces a vector

v = zeros(1,10);
for i = 1:10
if (i < 2)
v(i) = 1;
elseif (i > 1 && i <= 5)
v(i) = 2;
else

v = 1 2 2 2 2 3 3 3 3 3
...
2

of

An Introduction to

Matlab

explains the

if

statement in more detail
...
, (xn , yn )
...


The simplest plot command has the form
length,

n

say
...
01:2*pi];
v = sin(t);
plot(t,v)
Section 10 of

An Introduction to

Matlab

describes how to make multi-plots, 3D plots and how

to change the axes scaling and line styles
...
g
...
e
...
Another example is

st = 'my name is Frankenstein'
Note that both quotes are the same
...


which converts a number into a string
...

Note that in the above example using

title, the string has more than one element
...


1
...
You can either type instructions directly at the prompt

or make up a le or les containing the instructions you want
le (called an M-le) to

Matlab

Matlab

to execute and feed this


...
Typing commands at the prompt is ne if you want to execute single commands, but
errors usually creep into the typing of longer expressions and sequences of operations and it is a
pain having to retype things used over and over again
...
3 above
...
Script M-les act as a shorthand for a sequence of

Matlab

commands and are

a bit easier to use than function M-les
...


7

A sample script M-le
Suppose we want to write an M-le to nd the sum of the rst 20 integers, instead of typing the
commands at the
M-les end in


...
The following le which we shall call

S20
...
Follow the instructions in Section 1
...


% S20
%
%
%

Find the sum of the first 20 integers
Computes and outputs the sum 1 + 2 +
...


% Note that this line is ignored
...
The semicolon suppresses the output
...
Typing

help S20

will print out the lines at the top of the le that start with a

up to the rst line that doesn't start with a
anyone else who uses your

Matlab

%
...
It is good style

to always start an M-le with a comment line
...


%,

Sample function M-les
The following le (called

mkvec
...


function v = mkvec(x)
%MKVEC Construct the vector [1, x, x^2, x^3] from x
%
mkvec(x) returns the vector [1, x, x^2, x^3]
v = [1, x, x^2, x^3];

input argument (x), the 4component vector output argument v and a function
statement (the rst line)
...

It contains an

You should get the result

ans =
1

2

4

8

8

You can use the M-le to create named vectors  for example if you type

w = mkvec(6)

then

you obtain

w =
1

6

36

216

instead
...
g
...
m

below:

function v = mkvec2(x,n)
%MKVEC2 Construct the vector [1, x, x^2,
...
, x^n]
for k = 0:n
v(k+1) = x^k;
end
and more than one output argument  e
...
the le

function
%MKVEC3
%
%

mkvec3
...
, x^n] from x and n
[v, len] = mkvec3(x,n) returns the vector
v = [1, x, x^2,
...


for k = 0:n
v(k+1) = x^k;
end
len = length(v);
You can also use dot operations to avoid loops in the previous example, e
...


mkvec3a
...
, x^n] from x and n
[v, len] = mkvec3(x,n) returns the vector
v = [1, x, x^2,
...

not use loops

k = 0:n;
v = x
...


1
...
The syntax is quite similar to

with(inttrans):
f:=t:
laplace(f, t, s);

we can type

Matlab


...
6 The

Matlab ODE solvers

The Matlab ODE solvers are written to solve systems of DE's of the form

dx1
dt
dx2
dt
dx3
dt

=

f1 (t, x1 , x2 , x3 ,
...
)

=

f3 (t, x1 , x2 , x3 ,
...


i
...


dx
= F (t, x)
...
g
...


ode45, ode23, ode113,

(each solver has its advantages and disadvantages)
...
1 Problem set 1
1
...
6, 0
...
, 1
...
5)

(c) Find the transpose of the vector
(d) If

A

and

B

n×n
2×2

are

it out for some

x

using 2 colons and three numbers
...


matrices, what is the dierence between

A∗B

and

A
...


x = (0
...
7,
...
4, 4
...


2
...

(b) Find its length
...

Use

length

and

sum

and see Section 14
...



...


using the line style stars and put appropriate labels on the

graph
...
1, 0
...
, 10)
...
Generate the vector

(b) Plot vector

An Introduction to

An Introduction to

Matlab


...

1/3 1/4 1/5


(b) Enter a vector

b = (3, 1
...
4333)T

(c) Find the solution
Note that the matrix

x

A

to the linear system
is the

simply by using the command
5
...
Create
(a) a

9×9

identity matrix,

(b) a

7×9

zero matrix,

(c) a

4×6

matrix of ones
...


b

is a column vector)
...


(hint: use

\)
...
(See

output ? Try it out by typing

help format)
...
e
...


(a) Use the editor to create a function M-le called
positive integer

and as output the result

N,

p=

N
X

p

anysump
...


j=1
(b) Compute from the Matlab command window, the value of the sum for
8
...
Use

(b) Add labels to the

x

and

y

x = −5 : 0
...


x

to plot

f (x) =

coordinates
...


(c) Annotate the graph to show which curve is being depicted
...


(a) Follow the examples in Section 10 of

x + x2

An Introduction to

− x3 /10 − 10 with a dashed line style and

the same graph
...
05 : 10

and

y

for the

x

Matlab

g(x) = cos(x)

to plot

f (x) =

with a solid line in

coordinates
...


(c) Annotate the graph to show which curve is which and place the text f=g here near
where the curves cross
...
Write a function M-le that takes as input the positive integer

outputs the sum

n
X

n

and number

r,

and

kr
...
Plot an

(x, y)

n (n + 1) (2n + 1)
6

n
...


Use the information from the

nd the root of the equation

sin(x) + 1 − x = 0
...
2 Problem set 2
x ∈ [−π, π] (labelled
Figure window (subplot)
...
Produce 4 separate graphs against
tions (a)-(d) below all in the same

2
...


(a) Write a function M-le using
vector

b

for loops to dene the N × N



β

γ


 α
β



...



...

0 ···

0
...


...







...

0 


β γ 
α β

γ

...


α
0

which will work for any integer
matrix

tridiagonal matrix

given by

N ≥ 3
...
The

is called tridiagonal because all its entries are zero apart from those on or

immediately next to the diagonal
...


in the

of

the case

N = 4,

and

(Use the backslash

\

command
...


(c) Use the command

the eigenvalues and eigenvectors of the matrix

3
...

(a) Write an function M-le where the input is the size

N

and the output is the time

(cputime) in seconds it takes by each of the two methods to compute the solution of

Ax = b

for randomly chosen

A

and

b (rand)
...
, 800
...

4
...


Step 1 Make a guess at the solution (call it
Step 2 calculate the sequence of values

xk+1 = xk −

x1 ),
x2 , x3 ,
...
,

14

|xk+1 − xk | < TOL,
k = 100 without nding

TOL

Step 3 stop when

where

or if we reach

a solution
...
It will make life easier for you if you write the M-le in such a way that
the equation to be solved can be changed easily
...


fzero rst of all, then apply
−6 and try two
your Newton function to the equation cos(x) = x with TOL= 10
dierent starting guesses x1 = 1
...
4 (which doesn't)
...
Is there a pattern in which starting guesses work and which do
not? (for, if, break)
...
3 Problem set 3
1
...


subplot(2,2,2)

and mark

it with a star at its maximum
...
The derivative of a well-behaved function
various

nite dierence approximations:
(F) : f 0 (x) ≈

f (x + h) − f (x)
,
h

f (x)

with respect to

(C) : f 0 (x) ≈

x

can approximated by

f (x + h) − f (x − h)

...
The number
relatively small, and the approximations should, in theory, get better as
(a) Write a function M-le that takes as input
central approximate derivatives of
(b) Set

x = 1/2

x and h,
f (x) = sin−1 (x)
...


and outputs the forward and

and plot the modulus of the error in the results against

h

for

h =

10−10 , 10−9 ,
...
Which gives better results and why do they stop getting better
as

h

decreases? Use the

loglog

function to plot, and label the graph appropriately
...
4 Problem set 4
1
...
To nd

a

f (x) dx,

the interval

a≤x≤b

is split into

N

strips of equal width, and the function is assumed to be constant over the strip making
it easy to integrate
...
The approximation formula that results is

Zb
I≡

f (x) dx ≈ h

N
X

f (xj ) ≡ IN

j=1

a

h = (b − a)/N and the mid points of the strips are
xj = a + h(j − 21 )
...

R2
3
(a) Write an M-le to approximate the integral
1 sin(x ) dx with N = 8, 16, 32,
...

(b) Tabulate the result to full precision and make an estimate of the error in the results
...
One of the best known examples in dynamical systems and chaos theory is the Logistic

xk+1 = λxk (1 − xk ) for k = 1, 2,
...


Map
...


(b) Use this to plot
same gure
...
7, N = 100

x1 ∈ (0, 1)
...
, xN )
...
7, 0
...
7 + 10−6 , 0
...
7 case
...
7, N = 500

with

pattern the solution settles into
...
3, 3
...
739, 3
...
)
3
...


h > 0
...

Step 3 Find an approximation of y step-by-step from the recursion

yn+1 = yn + hf (yn , tn )
with initial value
If

h

y0 = a
...


17

(a) Write an M-function with inputs the parameter

output the two vectors

t

and

y

h

and length of the interval

l,

and

obtained by applying Euler's method to the initial

value problem

dy
= (1 − t)y,
dt
(for

y(0) = 1
...


(b) The exact solution of this ODE can actually be found analytically,
Plot a graph comparing the exact and the approximate solutions for

2

y(t) = et−t /2
...
Use

a blue lled line for the exact solution and a green dashed line for the approximated
one
...
(plot)
...
g
...


18

2
...
Use

Matlab Maple
(or

) to compute Laplace transforms from the Table of Laplace trans-

forms from the lecture notes and from the Tutorial exercises
...
Use

Matlab

(or

Maple

) to nd inverse Laplace transforms of the functions from the

Tutorial exercises
...
Solve the LCR circuit example problem from the Lecture notes using the

Matlab

built-in

ODE solvers and compare the computed solution to the exact solution from the lecture
notes
...
Use the

Matlab

build-in ODE solver to solve the following system of ODEs for

t ∈ (0, 5)

dx1
= x1 + 2x2 ,
dt
dx2
= 2x1 − 2x2 ,
dt
x1 (0) = 2
x2 (0) = 1, ,
Compare the computed solution with the exact solution from Section 3
...

5
...
g
...
01)
...
, Lecture notes
where

Figure 3
...
5 sin (2t − 6)

t < 3,
t ≥ 3

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Title: Learn Matlab
Description: Detailed notes on the basics of Matlab. These notes are for anyone looking to start learning Matble and is most relevant for scientists and engineers.

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