Newton-Raphson Method
The fastest root-finding method for smooth functions — converges quadratically when it works.
Try Newton-Raphson — No Sign-in NeededWhat is the Newton-Raphson Method?
The Newton-Raphson method is an iterative root-finding algorithm that uses the derivative of a function to rapidly zero in on a root. Starting from an initial guess x₀, the method constructs a tangent line to the curve at that point and finds where the tangent crosses the x-axis — that crossing becomes the next guess. Because the tangent line is a linear approximation of the curve, each step places the new estimate much closer to the true root.
The method was independently developed by Isaac Newton in the 1660s and Joseph Raphson in 1690. It remains one of the most widely used numerical methods in science and engineering today, powering everything from iterative solvers in finite-element software to the square-root instruction in modern CPUs.
The key strength of Newton-Raphson is its convergence rate. Once the iterates are close enough to the root, the number of correct decimal digits roughly doubles with every step. This is called quadratic convergence, and it means the method can reach machine precision in as few as five or six iterations — a dramatic advantage over simpler methods like bisection.
The Formula
Each new estimate subtracts the ratio of the function value to its derivative. When f(xₙ) is small and f′(xₙ) is large, the correction is tiny and the method has nearly converged.
Step-by-Step Algorithm
- 1
Choose a starting point x₀ near the expected root.
- 2
Evaluate f(x₀) and f′(x₀).
- 3
Compute x₁ = x₀ − f(x₀) / f′(x₀).
- 4
Check convergence: if |f(x₁)| < tolerance or |x₁ − x₀| < tolerance, stop.
- 5
Otherwise set x₀ ← x₁ and return to step 2.
Convergence & Behaviour
Newton-Raphson converges quadratically near a simple root — meaning the error at step n+1 is proportional to the square of the error at step n. In practice this means the method doubles the number of correct digits each iteration. However, this guarantee only holds once the iterates are in the "basin of attraction" of the root. Far from the root, the method can oscillate, overshoot, or diverge entirely. Near a point where f′(x) ≈ 0 (a local extremum or inflection point), the correction term explodes and the method shoots far away. For roots of multiplicity greater than 1 (where both f and f′ vanish), convergence degrades to linear.
Use Newton-Raphson when the function is smooth and differentiable, you can compute or express f′(x) analytically, your starting point is reasonably close to the root, and speed matters — it's the best available option for well-behaved functions.
Avoid it when f′(x) cannot be computed analytically (use the Secant method instead), when the function has vertical tangents or flat regions near the root, or when you need a guaranteed result without a good initial guess (use Bisection to bracket first).
Worked Example
Find √2 by solving f(x) = x² − 2 = 0, starting from x₀ = 1.
- 1.f(x) = x² − 2, f′(x) = 2x
- 2.x₁ = 1 − (1 − 2)/(2·1) = 1 + 0.5 = 1.5
- 3.x₂ = 1.5 − (2.25 − 2)/(3) = 1.5 − 0.08333 = 1.41667
- 4.x₃ = 1.41667 − (2.00694 − 2)/(2·1.41667) ≈ 1.41422
- 5.x₄ ≈ 1.41421356 — correct to 8 decimal places
Result: √2 ≈ 1.41421356 in just 4 iterations. Each step doubled the correct digits, demonstrating quadratic convergence.
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