Numerical Methods: Newton-Raphson, Euler, and Runge-Kutta

Newton-Raphson finds a root of an equation; Euler and Runge-Kutta step a differential equation forward. That single split decides which method you reach for, and the rest is matching the method to the conditions of the problem.

Whether a method works well depends on a few conditions: a sensible starting guess, a usable derivative, or a step size $h$ that is small enough for the problem.

The three methods at a glance

Newton-Raphson updates a guess for $x$

If you want $x$ such that $f(x)=0$, Newton-Raphson follows the tangent line:

The intuition: near a smooth root, the tangent line is a local linear model, and its intercept is usually a better guess than the current point. With $f(x)=x^2-2$ and $x_0=1.5$,

and one more step gives about $1.4142$, already close to $\sqrt{2}$.

Euler uses one slope, one step

For an initial-value problem $y'=f(t,y),\ y(t_0)=y_0$, Euler steps forward with the slope it already knows:

It is the simplest approximation, so it is easy to learn and implement, but its error grows quickly if $h$ is too large or the solution changes rapidly.

Runge-Kutta samples several slopes per step

RK4, the classical fourth-order method, samples slope information four times inside one step:

The weighted average of those slope estimates tracks the curve much better than Euler for the same step size.

When to reach for each one

Use Newton-Raphson to solve a nonlinear equation when you can compute or approximate the derivative. Use Euler for the basic idea of stepping through an ODE or a quick baseline. Use Runge-Kutta (RK4) for a practical accuracy upgrade without changing the problem setup. If the ODE is stiff, neither Euler nor classical RK4 is reliable; the method has to match the equation.

Side-by-side: Euler vs. RK4 on the same ODE

Take $y' = y,\ y(0)=1$ and use one step of size $h=0.1$ to estimate $y(0.1)$.

Euler step. At $t=0$, $y_0=1$, so the slope is $f(0,1)=1$ and

RK4 step. Same problem:

The exact value is $e^{0.1} \approx 1.105170918$, so the RK4 step is far closer. Euler uses the slope only at the left endpoint; RK4 samples how the slope changes during the step.

Confusion points that cost the most

Mixing up the problem types. Newton-Raphson is for roots of equations; Euler and Runge-Kutta are for differential equations. Choose the wrong family and the setup is wrong before any arithmetic.

Assuming the method always converges. Newton-Raphson can fail with a poor starting guess or when $f'(x)$ is very small near the iterate. Euler and RK methods misbehave when the step size is too large.

Treating the step size as a minor detail. For ODE methods, $h$ is part of the method. A smaller $h$ often improves accuracy but costs more, and stiff problems may need methods built for stiffness rather than just a smaller step.

Forgetting the answer is approximate. Many digits do not mean trustworthy. Ask whether the approximation is stable, converging, and accurate enough for the purpose.

These methods appear whenever a model is clear but an exact symbolic answer is inconvenient or unavailable: physics, engineering, optimization, finance, and scientific computing. The practical question is always whether the answer is accurate enough for the decision at hand. To feel the difference, rerun the ODE example with $h=0.05$ and watch the RK4 answer pull ahead of Euler even further.