GNU Octave

GNU Octave is a free, open-source programming language designed primarily for numerical computation. It is largely compatible with MATLAB, making it a popular free alternative in academia, engineering, and science.

What can you do with it?

• Solve linear algebra problems
• Perform statistical analysis
• Plot graphs and visualizations
• Process signals and images
• Simulate mathematical models
• Write reusable scripts and functions

Windows

Go to https://octave.org/download, download the .exe installer, and run it.

macOS

Using Homebrew:

Copy$ brew install octave

Or download the .dmg from the official website.

Linux (Ubuntu / Debian)

Copy$ sudo apt update
$ sudo apt install octave

Verify Installation

Copy$ octave --version

When you launch Octave, you'll see several panels:

The Command Window works like a powerful calculator. Type an expression and press Enter.

Copy>> 2 + 3
ans = 5

>> 10 - 4
ans = 6

>> 3 * 7
ans = 21

>> 15 / 4
ans = 3.7500

>> 2 ^ 8        % Exponentiation
ans = 256

>> mod(17, 5)   % Modulo (remainder)
ans = 2

Suppressing Output

Add a semicolon (;) at the end of a line to suppress its output:

Copy>> x = 42;    % No output shown
>> x          % Now show it
x = 42

Comments

Use % to write comments (Octave ignores everything after it):

Copy>> 5 + 3   % This adds 5 and 3

Variables store data for later use. Octave is case-sensitive — myVar and myvar are different variables.

Copy>> x = 10
x = 10

>> name = "Alice"
name = Alice

>> pi_approx = 3.14159
pi_approx = 3.1416

Naming Rules

• Must start with a letter
• Can contain letters, numbers, and underscores
• Cannot use reserved words like for, if, end

Workspace Commands

Copywho          % List all variables
whos         % List variables with size and type
clear x     % Delete variable x
clear        % Delete ALL variables

Copy>> class(3.14)
ans = double

>> class("hello")
ans = char

>> class(true)
ans = logical

Matrices are the core data structure in Octave. A vector is simply a 1-row or 1-column matrix.

Creating Vectors

Copy% Row vector (spaces or commas)
>> v = [1 2 3 4 5]
v = 1   2   3   4   5

% Column vector (semicolons)
>> c = [10; 20; 30]
c =
   10
   20
   30

Ranges

Copy>> 1:5            % 1 to 5, step 1
ans = 1   2   3   4   5

>> 0:0.5:2        % 0 to 2, step 0.5
ans = 0.0000   0.5000   1.0000   1.5000   2.0000

>> 10:-2:2        % Count down
ans = 10   8   6   4   2

Creating Matrices

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

Useful Matrix Builders

Copyzeros(3, 3)       % 3×3 matrix of zeros
ones(2, 4)        % 2×4 matrix of ones
eye(4)            % 4×4 identity matrix
rand(3)           % 3×3 random values (0–1)
linspace(0,1,5)  % 5 evenly spaced points from 0 to 1

Indexing

Copy>> v = [10 20 30 40 50];
>> v(1)           % First element → 10
>> v(end)         % Last element  → 50
>> v(2:4)         % Elements 2–4  → 20 30 40

>> A(2, 3)        % Row 2, Col 3
>> A(1, :)        % Entire first row
>> A(:, 2)        % Entire second column

Size Commands

Copysize(A)       % [rows, cols]
rows(A)       % Number of rows
columns(A)   % Number of columns
length(v)    % Length of a vector
numel(A)     % Total number of elements

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

>> A * B           % Matrix multiply
ans =
   19   22
   43   50

>> A .* B          % Element-wise multiply
ans =
    5   12
   21   32

Linear Algebra

Copydet(A)       % Determinant
inv(A)       % Inverse
rank(A)      % Rank
eig(A)       % Eigenvalues (and eigenvectors)
A \ b        % Solve Ax = b (better than inv(A)*b)

Math

Copysqrt(16)      % → 4
abs(-5)       % → 5
floor(3.7)   % → 3  (round down)
ceil(3.2)    % → 4  (round up)
round(3.5)   % → 4  (round nearest)
log(exp(1))  % Natural log → 1
log2(8)     % → 3
log10(100)  % → 2
exp(1)       % e¹ ≈ 2.7183

Trigonometry (radians)

Copysin(pi/2)    % → 1
cos(0)       % → 1
tan(pi/4)   % → 1
asin(1)     % → pi/2

Statistics

Copyv = [4 7 2 9 1 5];
sum(v)       % Sum of elements
mean(v)      % Average
median(v)   % Median
std(v)       % Standard deviation
var(v)       % Variance
min(v)       % Minimum
max(v)       % Maximum
sort(v)      % Sort ascending
cumsum(v)   % Cumulative sum

if / elseif / else

Copyx = 15;

if x > 20
  disp("x is greater than 20");
elseif x > 10
  disp("x is between 10 and 20");
else
  disp("x is 10 or less");
end

Comparison & Logical Operators

for Loop

Copyfor i = 1:5
  printf("i = %d\n", i);
end

while Loop

Copyn = 1;
while n <= 5
  disp(n);
  n = n + 1;
end

break and continue

Copyfor i = 1:10
  if i == 4
    continue;    % Skip this iteration
  end
  if i == 7
    break;       % Exit the loop
  end
  disp(i);
end
% Prints: 1, 2, 3, 5, 6

Functions are reusable blocks of code. Save them in their own .m file with the same name as the function.

Basic Function (save as add_numbers.m)

Copyfunction result = add_numbers(a, b)
  result = a + b;
end

Copy>> add_numbers(3, 7)
ans = 10

Multiple Return Values

Copyfunction [mn, mx] = min_max(v)
  mn = min(v);
  mx = max(v);
end

Copy>> [lo, hi] = min_max([3 1 9 4]);
>> lo
lo = 1
>> hi
hi = 9

Anonymous Functions

Quick one-liner functions using the @ symbol:

Copy>> square = @(x) x.^2;
>> square(5)
ans = 25

>> f = @(x, y) x^2 + y^2;
>> f(3, 4)
ans = 25

A script is a plain text file with a .m extension containing a sequence of Octave commands. Unlike functions, scripts share the base workspace.

How to create and run

1. Open the Editor panel (or any text editor)
2. Write your code and save as my_script.m
3. Run from the Command Window: type my_script and press Enter

Example: circle_stats.m

Copy% circle_stats.m — Circle area and circumference
radius = 7;

area = pi * radius^2;
circumference = 2 * pi * radius;

printf("Radius:        %.2f\n", radius);
printf("Area:          %.4f\n", area);
printf("Circumference: %.4f\n", circumference);

Copy>> circle_stats
Radius:        7.00
Area:          153.9380
Circumference: 43.9823

Basic Line Plot

Copyx = 0:0.1:2*pi;
y = sin(x);

plot(x, y);
title("Sine Wave");
xlabel("x (radians)");
ylabel("sin(x)");
grid on;

Line Style Options

Copyplot(x, y, 'r--', 'LineWidth', 2);  % Red dashed, thicker

Multiple Lines

Copyx = linspace(0, 2*pi, 100);
plot(x, sin(x), 'b-', x, cos(x), 'r--');
legend("sin(x)", "cos(x)");
grid on;

Subplots

Copysubplot(2, 1, 1);       % 2 rows, 1 col, plot #1
plot(x, sin(x));  title("Sine");

subplot(2, 1, 2);       % 2 rows, 1 col, plot #2
plot(x, cos(x));  title("Cosine");

Other Plot Types

Copybar([3 7 2 5 9]);               % Bar chart
scatter(rand(1,50), rand(1,50));  % Scatter plot
hist(randn(1, 1000), 30);        % Histogram

% 3D surface
[X, Y] = meshgrid(-3:0.1:3, -3:0.1:3);
Z = sin(X) .* cos(Y);
surf(X, Y, Z);

Saving Figures

Copysaveas(gcf, "my_plot.png");    % Save as PNG
print("my_plot.pdf", "-dpdf"); % Save as PDF

Writing and Reading Text Files

Copy% Write to a file
fid = fopen("data.txt", "w");
fprintf(fid, "Hello, World!\n");
fprintf(fid, "Value: %f\n", 3.14);
fclose(fid);

% Read a file line by line
fid = fopen("data.txt", "r");
line = fgetl(fid);
while ischar(line)
  disp(line);
  line = fgetl(fid);
end
fclose(fid);

CSV Files

Copy% Save matrix to CSV
A = [1 2 3; 4 5 6; 7 8 9];
csvwrite("matrix.csv", A);

% Load CSV
B = csvread("matrix.csv");

Saving & Loading Variables

Copysave("workspace.mat");            % Save all variables
save("results.mat", "x", "y");   % Save specific ones
load("workspace.mat");           % Reload them

Essential Commands

Keyboard Shortcuts

Vectorization — Write Faster Code

Avoid slow loops by letting Octave operate on entire vectors at once:

Copy% Slow (loop)
for i = 1:1000
  y(i) = sin(i * 0.01);
end

% Fast (vectorized)
x = (1:1000) * 0.01;
y = sin(x);

Quick-Reference Cheat Sheet