MATLAB/Octave Scientific Computing

MATLAB/​Octave Scientific Computing

MATLAB is a numerical computing environment optimized for matrix operations and scientific computing. GNU Octave is a free, open-source alternative with high MATLAB compatibility.

Quick Start

Running MATLAB scripts:

# MATLAB (commercial)
matlab -nodisplay -nosplash -r "run('script.m'); exit;"

# GNU Octave (free, open-source)
octave script.m

Install GNU Octave:

# macOS
brew install octave

# Ubuntu/​Debian
sudo apt install octave

# Windows - download from https://octave.org/​download

Core Capabilities

1. Matrix Operations

MATLAB operates fundamentally on matrices and arrays:

% Create matrices
A = [1 2 3; 4 5 6; 7 8 9];  % 3x3 matrix
v = 1:10;                     % Row vector 1 to 10
v = linspace(0, 1, 100);      % 100 points from 0 to 1

% Special matrices
I = eye(3);          % Identity matrix
Z = zeros(3, 4);     % 3x4 zero matrix
O = ones(2, 3);      % 2x3 ones matrix
R = rand(3, 3);      % Random uniform
N = randn(3, 3);     % Random normal

% Matrix operations
B = A';              % Transpose
C = A * B;           % Matrix multiplication
D = A .* B;          % Element-wise multiplication
E = A \ b;           % Solve linear system Ax = b
F = inv(A);          % Matrix inverse

For complete matrix operations, see references/​matrices-arrays.md.

2. Linear Algebra

% Eigenvalues and eigenvectors
[V, D] = eig(A);     % V: eigenvectors, D: diagonal eigenvalues

% Singular value decomposition
[U, S, V] = svd(A);

% Matrix decompositions
[L, U] = lu(A);      % LU decomposition
[Q, R] = qr(A);      % QR decomposition
R = chol(A);         % Cholesky (symmetric positive definite)

% Solve linear systems
x = A \ b;           % Preferred method
x = linsolve(A, b);  % With options
x = inv(A) * b;      % Less efficient

For comprehensive linear algebra, see references/​mathematics.md.

3. Plotting and Visualization

% 2D Plots
x = 0:0.1:2*pi;
y = sin(x);
plot(x, y, 'b-', 'LineWidth', 2);
xlabel('x'); ylabel('sin(x)');
title('Sine Wave');
grid on;

% Multiple plots
hold on;
plot(x, cos(x), 'r--');
legend('sin', 'cos');
hold off;

% 3D Surface
[X, Y] = meshgrid(-2:0.1:2, -2:0.1:2);
Z = X.^2 + Y.^2;
surf(X, Y, Z);
colorbar;

% Save figures
saveas(gcf, 'plot.png');
print('-dpdf', 'plot.pdf');

For complete visualization guide, see references/​graphics-visualization.md.

4. Data Import/​Export

% Read tabular data
T = readtable('data.csv');
M = readmatrix('data.csv');

% Write data
writetable(T, 'output.csv');
writematrix(M, 'output.csv');

% MAT files (MATLAB native)
save('data.mat', 'A', 'B', 'C');  % Save variables
load('data.mat');                   % Load all
S = load('data.mat', 'A');         % Load specific

% Images
img = imread('image.png');
imwrite(img, 'output.jpg');

For complete I/​O guide, see references/​data-import-export.md.

5. Control Flow and Functions

% Conditionals
if x > 0
    disp('positive');
elseif x < 0
    disp('negative');
else
    disp('zero');
end

% Loops
for i = 1:10
    disp(i);
end

while x > 0
    x = x - 1;
end

% Functions (in separate .m file or same file)
function y = myfunction(x, n)
    y = x.^n;
end

% Anonymous functions
f = @(x) x.^2 + 2*x + 1;
result = f(5);  % 36

For complete programming guide, see references/​programming.md.

6. Statistics and Data Analysis

% Descriptive statistics
m = mean(data);
s = std(data);
v = var(data);
med = median(data);
[minVal, minIdx] = min(data);
[maxVal, maxIdx] = max(data);

% Correlation
R = corrcoef(X, Y);
C = cov(X, Y);

% Linear regression
p = polyfit(x, y, 1);  % Linear fit
y_fit = polyval(p, x);

% Moving statistics
y_smooth = movmean(y, 5);  % 5-point moving average

For statistics reference, see references/​mathematics.md.

7. Differential Equations

% ODE solving
% dy/​dt = -2y, y(0) = 1
f = @(t, y) -2*y;
[t, y] = ode45(f, [0 5], 1);
plot(t, y);

% Higher-order: y'' + 2y' + y = 0
% Convert to system: y1' = y2, y2' = -2*y2 - y1
f = @(t, y) [y(2); -2*y(2) - y(1)];
[t, y] = ode45(f, [0 10], [1; 0]);

For ODE solvers guide, see references/​mathematics.md.

8. Signal Processing

% FFT
Y = fft(signal);
f = (0:length(Y)-1) * fs / length(Y);
plot(f, abs(Y));

% Filtering
b = fir1(50, 0.3);           % FIR filter design
y_filtered = filter(b, 1, signal);

% Convolution
y = conv(x, h, 'same');

For signal processing, see references/​mathematics.md.

Common Patterns

Pattern 1: Data Analysis Pipeline

% Load data
data = readtable('experiment.csv');

% Clean data
data = rmmissing(data);  % Remove missing values

% Analyze
grouped = groupsummary(data, 'Category', 'mean', 'Value');

% Visualize
figure;
bar(grouped.Category, grouped.mean_Value);
xlabel('Category'); ylabel('Mean Value');
title('Results by Category');

% Save
writetable(grouped, 'results.csv');
saveas(gcf, 'results.png');

Pattern 2: Numerical Simulation

% Parameters
L = 1; N = 100; T = 10; dt = 0.01;
x = linspace(0, L, N);
dx = x(2) - x(1);

% Initial condition
u = sin(pi * x);

% Time stepping (heat equation)
for t = 0:dt:T
    u_new = u;
    for i = 2:N-1
        u_new(i) = u(i) + dt/(dx^2) * (u(i+1) - 2*u(i) + u(i-1));
    end
    u = u_new;
end

plot(x, u);

Pattern 3: Batch Processing

% Process multiple files
files = dir('data/​*.csv');
results = cell(length(files), 1);

for i = 1:length(files)
    data = readtable(fullfile(files(i).folder, files(i).name));
    results{i} = analyze(data);  % Custom analysis function
end

% Combine results
all_results = vertcat(results{:});

Reference Files

• matrices-arrays.md - Matrix creation, indexing, manipulation, and operations
• mathematics.md - Linear algebra, calculus, ODEs, optimization, statistics
• graphics-visualization.md - 2D/​3D plotting, customization, export
• data-import-export.md - File I/​O, tables, data formats
• programming.md - Functions, scripts, control flow, OOP
• python-integration.md - Calling Python from MATLAB and vice versa
• octave-compatibility.md - Differences between MATLAB and GNU Octave
• executing-scripts.md - Executing generated scripts and for testing

GNU Octave Compatibility

GNU Octave is highly compatible with MATLAB. Most scripts work without modification. Key differences:

• Use # or % for comments (MATLAB only %)
• Octave allows ++, --, += operators
• Some toolbox functions unavailable in Octave
• Use pkg load for Octave packages

For complete compatibility guide, see references/​octave-compatibility.md.

Best Practices

1. Vectorize operations - Avoid loops when possible:

   % Slow
   for i = 1:1000
       y(i) = sin(x(i));
   end
   
   % Fast
   y = sin(x);
   

2. Preallocate arrays - Avoid growing arrays in loops:

   % Slow
   for i = 1:1000
       y(i) = i^2;
   end
   
   % Fast
   y = zeros(1, 1000);
   for i = 1:1000
       y(i) = i^2;
   end
   

3. Use appropriate data types - Tables for mixed data, matrices for numeric:

   % Numeric data
   M = readmatrix('numbers.csv');
   
   % Mixed data with headers
   T = readtable('mixed.csv');
   

4. Comment and document - Use function help:

   function y = myfunction(x)
   %MYFUNCTION Brief description
   %   Y = MYFUNCTION(X) detailed description
   %
   %   Example:
   %       y = myfunction(5);
       y = x.^2;
   end
   

Additional Resources

• MATLAB Documentation: https://www.mathworks.com/help/matlab/
• GNU Octave Manual: https://docs.octave.org/latest/
• MATLAB Onramp (free course): https://www.mathworks.com/learn/tutorials/matlab-onramp.html
• File Exchange: https://www.mathworks.com/matlabcentral/fileexchange/

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