CM Hub: Introduction to Matlab

Introduction to Matlab course for the CM Hub at Imperial College

3 × 2 hour classes

• Part 1: Call-and-response Matlab, basic arithmetic, simple scripts
• Part 2: 2D plots, functions, Collatz conjecture, if, for, while
• Part 3: Data analysis, linear algebra, 3D plots

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On completion of this workshop you will be able to:

• use Matlab to run scripts,
• apply fundamental components of the Matlab language including variables, loops, conditionals and functions,
• create programs designed to solve simple problems,
• interpret common errors and use these to help debug a program.

Prerequisites

• No programming experience is required
• Students are welcome to bring their own fully charged laptops to these sessions although there are computers in situ.
• If on a laptop, please install MATLAB.

YouTube videos for distance learning

Timestamps are in the descriptions and correspond to the numbering in this document.

• Part 1
• Part 2
• Part 3
• Playlist

Part 1. Call-and-response Matlab, basic arithmetic, simple scripts

1. Where Matlab sits among other languages

Pros of Matlab:

• Shallow learning curve for maths
• Comes with a lot of useful baked-in tools
• Interactivity/debugging is quite easy
• Backwards compatible and has been around for a long time

Cons of Matlab:

• Ideosyncratic
• £££££££££££
• Bad for programming in general
• Isn't used outside of universities
• Plays badly with other languages

In short: Matlab is a good plug-and-play language for medium-sized maths problems.

2. Awareness of the Matlab desktop environment

• The command window allows for call-and-response interface

3. Command-line arithmetic

Join in:

• 3+4
• 3*4
• 3/4
• 3^4

Try:

• Does Matlab respect BIDMAS?: 3+4*5

4. Use of variables

Join in:

• x=2
• x
• x=-9
• x
• width = 3 (long names are good)
• area = width^2
• y=12/2+5
• (y+1)^2
• y=sin(x)*cos(x)

5. Getting help

• Find square root of x: Google the function
• Or use F1 on the function

Try:

• Find sin^–1(1)
• Find the remainder when 14 is divided by 3
• Find |–4|

6. The semicolon

Try:

• What's the difference between x=40 and x=40; ?

7. Creating vectors and matrices

The building block of Matlab is the matrix. These can represent lists, data tables, or matrices as mathematical objects.

Join in:

• x = [29 43 13 3.2 -26]
• y = [1 2; 3 4]
• [1:10] (note endpoints)
• [1:2:10]
• [3:-0.1:2]

Try:

• What do apostrophes do? x' y'
• What do commas do?: [29, 43, 13, 3.2, -26]
• How long is [1:0.5:10]?

8. Matrix manipulation

Join in:

• x = [3 1 4 1 5 9]
• x(2)
• x(4) = 50
• x
• y = [3 10; -1 6]
• y(1,2) = 100
• y
• A = [1:10]'*[1:10] (explanation later)
• A(4, [1 2])
• A([8 9],[8 9])
• A(3,4:end)
• A(1:end,4)
• A(:,4)
• sum(A(:,4)): sum sums the columns

Try:

• Create the matrix A = magic(5) (explanation later)
• Get the element in the 1st row, 1st column
• Get the element in the 5th row, 2nd column
• Get all elements in the last row
• Get the element in the 6th row, 4th column (...)
• What does A([2 1 1 1],4) do? Try it.

Now try:

• Show that the sum of the first column of A = the sum of the last column of A
• Find the sum of the diagonal of A (hint: search for the function that gives the diagonal of A... or guess!)
• Harder! Find the sum of the '/'-leaning diagonal of A. Hint: try showing the rows of A in reverse before using diag
• Harder! Produce the elements of A for which both coordinates are odd
• Replace the bottom row of A with zeros

9. Vector arithmetic

Join in:

• x = [1:10]
• y = [11:20]
• x+y
• x-y
• 2*x
• x/2
• x*y... what do you expect to happen? Why is this problematic but x+x isn't? (Hint: think about matrices)
• x.*y
• x.^2
• 2.^x

Try:

• sum the numbers from 1 to 100 (are you faster than Gauss?)
• sum the squares from 1 to 5
• What is the mean of the powers of 2 from the zeroth power to the sixth power? (Google the function to find the mean... or guess!)

10. Writing simple scripts

A script is the simplest type of Matlab program.

Join in:

• Create a script magic_square_test.m
• Let's see if switching the top and bottom row of a magic square keeps it a magic square:

n = 4; % matrix size
M = magic(n);
top_row = M(1,:);
bottom_row = M(end,:);
M(1,:) = bottom_row;
M(end,:) = top_row;
disp(sum(M)); % to display we can use 'disp' or just leave off the semicolon
disp(sum(diag(M)));
disp(n*(n^2+1)/2); % magic constant

• Run the script.
• Breakpoints
• Change the script so that we do it with a matrix of size 3 instead.

Part 2. Plotting, functions, Collatz conjecture, if, for, while

1. Plotting in 2D

Join in, putting this in a script, first_plot.m:

• x = [0:10]
• y = exp(x)
• plot(x,y)
• How do we make this graph smoother?

Try:

• plot sin(x) for x between 0 and 2π
• plot a circle: recall x = cos(θ), y = sin(θ) for θ between 0 and 2π to make a circle with radius 1.

Join in:

• x = [0:0.1:10]
• y = exp(x)
• plot(x,y,'r-')
• Change to: plot(x,y,'go')

Try:

• Plot a magenta, dotted line with large line width and squares as markers. (Look at the F1 help file for plot)

Join in:

• xlabel('x')
• ylabel('exp(x)')
• title('Exponential growth is fast')
• xlim([0 5])
• Now let's try multiple plots
• x = [0:0.1:10]
• y1 = exp(x)
• y2 = exp(0.9*x)
• plot(x,y1,'r-')
• hold on
• plot(x,y2,'k--')
• legend('exp(x)','exp(0.9x)')

Try:

• The same but use loglog, semilogx or semilogy instead of plot (the syntax is the same). What do they do?

2. Writing and calling simple functions

Now we move from simple call-and-response to writing whole programs

• Difference between scripts and functions.
• Let's do functions first.

Join in:

• Create a new file collatz_function.m and inside it write:

function y = collatz_function(n)
  y = 3*n+1;
end

• Save and run collatz_function(5) from the command line

Try:

• Create a function first_and_last which takes a vector v and returns the sum of the first element in the vector and the last element in the vector
• Test it out in the command line, letting test_vector=[1:10] and running first_and_last(test_vector).

Join in:

• Change first_and_last so it outputs the first and last elements separately
• Test it out:
  • [first, last] = first_and_last(v)
  • first
  • last

3. If

Join in:

• Create a function function y = sign_function(x) so that the core functionality reads:

if x > 0
  y = 1;
elseif x == 0
  y = 0;
else
  y = -1;
end

• Note: == is not =

Try:

• Change collatz_function(n) so that if n is even, it returns n/2, otherwise it returns 3n+1.

4. For and while loops

Join in:

• Create a new script squares_up_to.m. Inside let's write

n = 10;
for i=1:n
  disp(i^2); % this squares i and then displays it on the screen
end

• Run the script

Try:

• Change the script to display the first 10 odd cubes.

Join in:

• Create a new function count_up_to.m. Inside let's write

function count_up_to(n)
  i = 1;
  disp(i);
  while i < n
    i = i + 1;
    disp(i);
  end
end

Try:

• The Collatz conjecture:

  • The Collatz conjecture is a famous mathematical conjecture about a sequence which starts with a positive integer n. The next term in the sequence is given by collatz_function(n). The conjecture says that this sequence will always reach the number 1 (where it ends).
  • For example, the sequence for n=5 is 5, 16, 8, 4, 2, 1.
  • Your job is to create a new function, collatz_conjecture(n), which takes a starting number n and displays the terms in the sequence.
  • Take a moment to think about the logic you need!
  • Suggested method: Use a while loop inside the function collatz_conjecture(n). While n does not equal 1, run collatz_function(n) to get the next term in the sequence.
  • "Not equal to" is ~= in Matlab.
  • Hint 1: You have to let the output of collatz_function(n) become the input of the function the next time round.
  • Hint 2: You have to change the value of n within your while loop otherwise n will never equal 1.

• For the keen: Write a script, collatz_trials.m which loops through the numbers 1 to 10, printing out the Collatz path every time.

Part 3. Data analysis, linear algebra

1. Saving and reading data

There are lots of ways of saving and reading data in Matlab. A good question to ask is 'do I want to open the saved data in another program?'

Join in:

• Download the file examples/exchange_rates.csv
• Move the CSV file to your folder
• Have a look at this file in Excel
• Create a new script, exchange_rate_data.m
• data = csvread('exchange_rates.csv',1,0);

Your turn:

• Plot the GBP/USD price (12th column) against the day of the year (1st column)
• On the same graph, plot the EUR/USD price (11th column) against the day of the year
• On a new graph, plot the EUR/GBP price against the day of the year. Can you guess which year this data is from?
• What was the minimum number of euros you could buy with £1 that year?
• On which day of the year was this the case? (Hint: look up min in the help files)

Join in:

• Create another data matrix, data_pounds, which contains the exchange rate of these currencies versus GBP, instead of USD
• Save the EUR/GBP data matrix
• csvwrite('exchange_rates_pounds.csv',data_pounds)
• Look at it in Excel

2. Linear algebra: Inversion of matrices (and when this is a bad idea)

Join in:

• A = [1 0 5; 2 1 6; 3 4 0]
• A
• inv(A)
• Let's solve Ax = (–1 0 1)^T
• b = [-1; 0; 1]
• If Ax = b then x = A^–1b, so inv(A)*b
• A\b

Try:

• Solve the system of equations x+y=2, -x+3y=3.
• Let A = [1 2 3; 4 5 6; 5 7 9] and b = [-1; 0; 1]. Solve Ax=b. What is the determinant of A? (Google!)

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Extra stuff

X1. Linear algebra: Matrix arithmetic

Join in:

• A = [1 0 5; 2 1 6; 3 4 0]
• A.^2
• A^2

Try:

• Create a 2x3 matrix, call it B
• Create a 3x4 matrix, call it C
• Try calculating B*C and C*B
• Multiply B by the transpose of B
• Make one of the elements of B imaginary (i)
• Calculate B'. What does the apostrophe actually do?

X2. Size

Size gives rows × columns

• size(A)
• size(B)
• x = [1:10]
• size(x) ... note it's 1×10 (a row vector), not 10×1 (a column vectors).

Vectors by default are row vectors in Matlab.

X3. Linear algebra: More matrix arithmetic

Join in:

• A
• Let's multiply A by (3 1 4)^T
• x = [3 1 4]' (note apostrophe) or x = [3; 1; 4] (note semicolons)
• A*x

Try:

• Let x = [3 1 4] (without the apostrophe). Will calculating A*x work? Try it.
• Create the 3×3 identity matrix I = eye(3). Multiply I by x.

X4. Eigenvalues and eigenvectors

• A = [-2 -4 2; -2 1 2; 4 2 5]
• eig(A)

Try:

• How to get eigenvectors? (Google or F1)

X5. Plotting in 3D

Join in:

• t = [0:0.1:10];
• x = sin(t);
• y = cos(t);
• z = t;
• plot3(x,y,z)
• grid on

Visualise f(x,y) = sin(x)cos(2y) for 0 ≤ x,y ≤ 2π:

• Create grid
  • x = linspace(0,2*pi,300)
  • y = linspace(0,2*pi,300)
  • [xg,yg] = meshgrid(x,y);
• f = sin(xg).*cos(2*yg);
• contour(xg,yg,f,20);
• surf(x,y,f)
• shading interp

X6. Strings

Single-quote strings are vectors, with each character an element in the vector

Join in:

• greeting = 'hello there' (note: single quotes)
• greeting(3)
• [greeting(1:4) greeting(7)]

Double-quote strings (R2016b upwards only) are individual things. You can create vectors of multiple strings

• greeting = "hello there"
• greeting(3)
• greeting(1)
• conversation = ["hello there", "general kenobi"]
• conversation(2)

Try:

• Let surname equal your surname
• Find the last letter of your surname
• Output the first and last letter of your surname, put together
• If conversation = ['hello there', 'general kenobi'] (with single quotes), what is conversation(2)?

X7. Built-in functions to try

Try:

• sin(x), cos(x), tan(x)
• floor(x), ceil(x)
• max(x), min(x)
• triu(A), rand(n)

X8. Curve Fitting

X9. Numerical Root Finding

Extra challenges

Challenge 1

• Plot the graphs of sin(ax), sin(bx) and sin(cx) for x from –π to π, with these plots in different colours. Include a legend, a title and a label for the x-axis.

Challenge 2

• Create a new function fib.m. Let this function take a number n and output the nth Fibonacci number. Recall the algorithm:
  • F(1) = 1
  • F(2) = 1
  • F(i) = F(i-2) + F(i-1)
• Create a vector of the first 20 Fibonacci numbers
• Save them to a file
• Plot them

Challenge 3

• Create a 10×10 matrix of random numbers (hint: use rand) and call it A
• Find the eigenvalues of A
• Show that the sum of the eigenvalues of A = the trace of A
• Plot the eigenvalues on an Argand diagram, using a circular marker at each eigenvalue
• Now add a row and column of zeros to A, to form B, which is therefore an 11×11 matrix. (Think about how you might want to do this!)
• Now plot the eigenvalues of B on the same graph as the eigenvalues of A, in another colour.
• What can you say about the eigenvalues of B compared to the eigenvalues of A?

Licence

This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International Licence.