Newton-Raphson Method

An iterative method that finds roots of a function using its derivative, converging quadratically near a solution.

Definition

The Newton-Raphson method finds a root of a function f(x) by iterating x := x - f(x) / f'(x). Each step uses the tangent line at the current guess to predict where the function crosses zero.

For systems of equations, each step solves a linear system involving the Jacobian matrix, so the method's cost per iteration can be significant. Quasi-Newton methods approximate the Jacobian to reduce that cost while keeping much of Newton's fast convergence.

Its quadratic convergence makes it the method of choice when a good initial guess and a computable derivative are available, but it can diverge or oscillate from a poor start. Robust solvers therefore combine Newton's speed with the guaranteed convergence of bisection, switching to the safe method when Newton misbehaves. For systems of equations, each step solves a linear system with the Jacobian, and quasi-Newton methods approximate it to cut that cost.

def newton(f, df, x0, tol=1e-10, itmax=50):
    x = x0
    for _ in range(itmax):
        fx = f(x)
        if abs(fx) < tol:
            return x
        x = x - fx / df(x)
    return x

Behavior

• Converges quadratically near a simple root.
• Requires the derivative and a reasonable initial guess.
• Can diverge or cycle if started poorly.
• Bisection is a slower but more robust fallback.

Why it matters

Newton's method is the fastest general root-finder when it works and generalizes to systems of equations, where it underlies many nonlinear solvers. Its speed makes it central to implicit time-stepping and optimization.

Fusion connection

Newton iteration solves the nonlinear equations of plasma equilibrium, converging rapidly to a consistent magnetic configuration for a given set of Hyperion parameters.