Using Octave

First, follow the installation instructions for:

• macOS
• GNU/Linux and other Unix systems
• Microsoft Windows

or consult the GNU Octave manual to install GNU Octave on your system. Then, start the GNU Octave by clicking the icon in the programs menu or launch the interactive prompt by typing octave in a terminal. See the manual page on running Octave.

The GNU Octave graphical user interface (GUI).

Variable Assignment

Assign values to variables with = (Note: assignment is pass-by-value). Read more about variables.

a = 1;

Comments

# or % start a comment line, that continues to the end of the line. Read more about comments.

% Função e derivada f = @(x) 1 - x^3;  % f(x) = 1 - x^3 f_prime = @(x) -3*x^2; % f'(x) = -3*x^2

% Condição inicial e tolerância x0 = 2;  % ponto inicial tolerancia = 1e-6;  % tolerância max_iter = 100;  % máximo de iterações para evitar loop infinito iter = 0;  % contador de iterações erro = inf;  % inicializando o erro como infinito

% Método de Newton-Raphson while erro > tolerancia && iter < max_iter

   iter = iter + 1;  % incrementar contador de iteração
   x1 = x0 - f(x0) / f_prime(x0);  % fórmula de Newton-Raphson
   erro = abs(x1 - x0);  % erro absoluto entre iterações
   x0 = x1;  % atualizar x0 para o novo valor

end

% Resultado disp(['Raiz aproximada: ', num2str(x1)]); disp(['Número de iterações: ', num2str(iter)]);

Elementary math

Many mathematical operators are available in addition to the standard arithmetic. Operations are floating-point. Read more about elementary math.

x = 3/4 * pi;
y = sin (x)

y =  0.70711

Matrices

Arrays in Octave are called matrices. One-dimensional matrices are referred to as vectors. Use a space or a comma , to separate elements in a row and semicolon ; to start a new row. Read more about matrices.

rowVec = [8 6 4]

rowVec =
   8   6   4

columnVec = [8; 6; 4]

columnVec =
   8
   6
   4

mat = [8 6 4; 2 0 -2]

mat =
   8   6   4
   2   0  -2

size(mat)

ans =
   2   3

length(rowVec)

ans =  3

Linear Algebra

Many common linear algebra operations are simple to program using Octave’s matrix syntax. Read more about linear algebra.

columnVec * rowVec

ans =
   64   48   32
   48   36   24
   32   24   16

rowVec * columnVec

ans =  116

columnVec'

ans =
   8   6   4

Accessing Elements

Octave is 1-indexed. Matrix elements are accessed as matrix(rowNum, columnNum). Read more about accessing elements.

mat(2,3)

ans = -2

Control flow with loops

Octave supports for and while loops, as well as other control flow structures. Read more about control flow.

x = zeros (50,1);
for i = 1:2:100 # iterate from 1 to 100 with step size 2
  x(i) = i^2;
endfor

y = zeros (50,1);
k = 1;
step = 2;
while (k <= 100)
  y(k) = k^2;
  k = k + step;
endwhile

Vectorization

For-loops can often be replaced or simplified using vector syntax. The operators *, /, and ^ all support element-wise operations writing a dot . before the operators. Many other functions operate element-wise by default (sin, +, -, etc.). Read more about vectorization.

i = 1:2:100;      # create an array with 50-elements
x = i.^2;         # each element is squared
y = x + 9;        # add 9 to each element
z = y./i;         # divide each element in y by the corresponding value in i
w = sin (i / 10); # take the sine of each element divided by 10

% Função f(x) = 1 - x^3 f = @(x) 1 - x^3;

% Derivada de f(x) -> f'(x) = -3*x^2 f_prime = @(x) -3*x^2;

% Função para aplicar o Método de Newton-Raphson function [raiz, num_iteracoes, iteracoes_x, iteracoes_y] = newton_raphson(x0, tolerancia)

   % Inicializar variáveis
   iter = 0;                   % Contador de iterações
   erro = inf;                 % Inicializa o erro como infinito
   x = x0;                     % Definir o ponto inicial
   iteracoes_x = x;            % Inicializar vetor de iterações para o gráfico
   iteracoes_y = f(x);         % Inicializar valores de f(x) para o gráfico
   
   % Iterar até que o erro seja menor que a tolerância
   while erro > tolerancia && iter < 100
       iter = iter + 1;                          % Incrementa a iteração
       x_new = x - f(x) / f_prime(x);            % Fórmula de Newton-Raphson
       erro = abs(x_new - x);                    % Calcula o erro absoluto
       x = x_new;                                % Atualiza x
       
       % Armazena os valores de x e f(x) para o gráfico
       iteracoes_x = [iteracoes_x, x];
       iteracoes_y = [iteracoes_y, f(x)];
   end
   
   % Retorna a raiz aproximada, número de iterações e os valores das iterações
   raiz = x;                                    % Raiz aproximada
   num_iteracoes = iter;                        % Número de iterações

end

% Condições de cada caso % Caso 1: x0 = 2 e tolerância = 1e-6 x0_case1 = 2; tolerancia_case1 = 1e-6;

% Caso 2: x0 = 7 e tolerância = 1e-3 x0_case2 = 7; tolerancia_case2 = 1e-3;

% Resultado para o primeiro caso (x0 = 2, tolerância = 1e-6) [raiz_case1, iter_case1, iteracoes_x_case1, iteracoes_y_case1] = newton_raphson(x0_case1, tolerancia_case1); disp(['Caso 1: Raiz aproximada: ', num2str(raiz_case1)]); disp(['Número de iterações: ', num2str(iter_case1)]);

% Resultado para o segundo caso (x0 = 7, tolerância = 1e-3) [raiz_case2, iter_case2, iteracoes_x_case2, iteracoes_y_case2] = newton_raphson(x0_case2, tolerancia_case2); disp(['Caso 2: Raiz aproximada: ', num2str(raiz_case2)]); disp(['Número de iterações: ', num2str(iter_case2)]);

% Plotagem do gráfico figure;

% Caso 1: Plotando as iterações subplot(2,1,1); plot(iteracoes_x_case1, iteracoes_y_case1, '-o', 'LineWidth', 2, 'MarkerSize', 6); title('Método de Newton-Raphson - Caso 1 (x0 = 2)'); xlabel('Iterações (x)'); ylabel('f(x)'); grid on;

% Caso 2: Plotando as iterações subplot(2,1,2); plot(iteracoes_x_case2, iteracoes_y_case2, '-o', 'LineWidth', 2, 'MarkerSize', 6); title('Método de Newton-Raphson - Caso 2 (x0 = 7)'); xlabel('Iterações (x)'); ylabel('f(x)'); grid on;

Strings

Strings are simply arrays of characters. Strings can be composed using C-style formatting with sprintf or fprintf. Read more about strings.

firstString = "hello world";
secondString = "!";
[firstString, secondString] # concatenate both strings

ans = hello world!

fprintf ("%s %.10f \n", "The number is:", 10)

The number is: 10.0000000000

If-else

Conditional statements can be used to create branching logic in your code. Read more in the manual.

# Print 'Foo'      if divisible by 7,
#       'Fizz'     if divisible by 3,
#       'Buzz'     if divisible by 5,
#       'FizzBuzz' if divisible by 3 and 5
for i = 1:1:20
  outputString = "";
  if (rem (i, 3) == 0)  # rem is the remainder function
    outputString = [outputString, "Fizz"];
  endif
  if (rem (i, 5) == 0)
    outputString = [outputString, "Buzz"];
  elseif (rem(i,7) == 0)
    outputString = "Foo";
  else
    outputString = outputString;
  endif
  fprintf("i=%g: %s \n", i, outputString);
endfor

i=1:  
i=2:  
i=3: Fizz 
i=4:  
i=5: Buzz 
i=6: Fizz 
i=7: Foo 
i=8:  
i=9: Fizz 
i=10: Buzz 
i=11:  
i=12: Fizz 
i=13:  
i=14: Foo 
i=15: FizzBuzz 
i=16:  
i=17:  
i=18: Fizz 
i=19:  
i=20: Buzz

Getting Help

The help and doc commands can be invoked at the Octave prompt to print documentation for any function.

help plot
doc plot

Octave packages

Community-developed packages can be added from the Octave Packages website to extend the functionality of Octave’s core library. (Matlab users: Packages act similarly to Matlab’s toolboxes.) The pkg command is used to manage these packages. For example, to use the image processing library visit its page on Octave Packages, copy the install command and run it in octave

pkg install "https://downloads.sourceforge.net/project/octave/Octave%20Forge%20Packages/Individual%20Package%20Releases/image-2.14.0.tar.gz" # install package
pkg load image           # load new functions into workspace

Read more about packages.

Octave User Codes

There are also User Codes available for GNU Octave which are not part of the core program or any of the packages.

See Category User Codes.