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Octave's killer use case is numerical methods. Most classical algorithms ship as one-liners.

Root finding

%% fzero — find a root of f(x) = 0
f = @(x) x^2 - 4;
x = fzero(f, 1)             %% start near x=1, find x=2

fzero works for any continuous function. Newton-style under the hood.

Optimization

%% fminbnd — minimize over a bounded interval
f = @(x) (x - 3).^2 + 1;
x_min = fminbnd(f, 0, 10)   %% finds x = 3

%% fminunc — unconstrained minimization (multidimensional)
f = @(v) v(1)^2 + v(2)^2;
x_min = fminunc(f, [10, 10])    %% finds [0, 0]

%% fminsearch — Nelder-Mead simplex (no derivative needed)

Numerical integration

%% quad — adaptive quadrature
f = @(x) sin(x);
q = quad(f, 0, pi)          %% ≈ 2.0

%% quadgk — adaptive Gauss-Kronrod (more accurate)
%% trapz, simpson — fixed-grid integration

Differential equations

%% Solve dy/dt = -y, y(0) = 1
f = @(t, y) -y;
[t, y] = ode45(f, [0, 5], 1);
printf("%.4f\n", y(end));    %% e^-5 ≈ 0.0067

ode45 is the Runge-Kutta 4(5) integrator. Other solvers: ode23, ode23s (stiff), ode15s.

Polynomial fitting

x = 0:10;
y = 2*x + 3 + randn(1, 11);     %% noisy linear
p = polyfit(x, y, 1)              %% [slope, intercept]
y_fit = polyval(p, x);            %% evaluate the polynomial

Interpolation

x = [0, 1, 2, 3, 4];
y = [1, 2, 4, 7, 11];
x_query = 2.5;
y_interp = interp1(x, y, x_query, "spline")     %% spline interpolation

Methods: "linear", "spline", "pchip", "nearest".

FFT

fs = 1000;                   %% sample rate
t = 0:1/fs:1;
signal = sin(2*pi*50*t) + sin(2*pi*120*t);
Y = fft(signal);
P = abs(Y) / length(signal);
f = (0:length(signal)-1) * fs / length(signal);
%% peaks at 50 Hz and 120 Hz

Why Octave for numerical work

• One-liner access to LAPACK, BLAS, FFTW under the hood
• Compatible MATLAB syntax — algorithms in textbooks copy/paste
• Free, scriptable, embeddable
• Statistics, signal, image packages add specialized algorithms

For research-grade numerical code, Octave is competitive with MATLAB and beats most general-purpose languages by far in expressiveness.