Euler Integration

Euler Integration

A single Euler step gives one point. To approximate an entire trajectory, we repeat the step $n$ times from $t_0$ to $t_{\text{end}}$.

Algorithm

def euler(f, t0, y0, t_end, n):
    h = (t_end - t0) / n
    t, y = t0, y0
    ys = [y0]
    for _ in range(n):
        y = y + h * f(t, y)
        t += h
        ys.append(y)
    return ys

The step size is $h = (t_{\text{end}} - t_0) / n$. Smaller $h$ (more steps) gives higher accuracy, at the cost of more computation.

Accuracy vs. Step Size

Euler's method has first-order accuracy: if you halve $h$, the global error halves too. For the ODE $\frac{dy}{dt} = y$ with $y(0) = 1$:

Your Task

Implement euler(f, t0, y0, t_end, n) that returns a list of $n+1$ values (including the initial value).