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Newton's Method Calculator

Find roots of equations using the Newton-Raphson method. Enter any function f(x), set an initial guess, and see step-by-step iterations with convergence analysis.

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What is Newton's Method?

Newton's method (also called the Newton-Raphson method) is a powerful iterative algorithm for finding roots of equations -- values of x where f(x) = 0. Starting from an initial guess x₀, each iteration refines the estimate using the formula:

xn+1 = xn − f(xn) / f'(xn)

Geometrically, each step draws a tangent line to the curve at the current point (xn, f(xn)) and follows it down to the x-axis, where it crosses at xn+1. This new x-intercept becomes the next approximation.

How Does Newton's Method Work?

  • Tangent Line: At each xn, draw the tangent to f(x). Its x-intercept is the next guess.
  • Quadratic Convergence: Correct digits roughly double each iteration for simple roots.
  • Fast Convergence: Typically finds roots in 5-10 iterations to machine precision.
  • Sensitivity: Poor initial guess or flat derivative can cause divergence.

Convergence Properties

Property Description Implication
Order of Convergence Quadratic (order 2) for simple roots Error roughly squares each step: 10⁻² → 10⁻⁴ → 10⁻⁸
Simple Roots f(r) = 0, f'(r) ≠ 0 Fastest convergence, quadratic rate
Multiple Roots f(r) = 0, f'(r) = 0 Convergence drops to linear
Basin of Attraction Set of initial guesses that converge Complex for oscillatory or multi-root functions

For related calculus and equation tools, try the Derivative Calculator, Quadratic Calculator, and Ln Calculator.

Frequently Asked Questions

What is Newton's method (Newton-Raphson method)?

Newton's method is an iterative root-finding algorithm that uses tangent line approximations. Starting from an initial guess x₀, it repeatedly applies the formula xn+1 = xn - f(xn)/f'(xn) to converge toward a root where f(x) = 0. It typically converges quadratically for simple roots, meaning the number of correct digits roughly doubles each iteration.

How do I choose a good initial guess for Newton's method?

Choose an initial guess close to where you expect the root. You can graph the function first or use the intermediate value theorem: if f(a) and f(b) have opposite signs, a root lies between a and b. Avoid starting where f'(x) is zero or near zero, as this causes the method to fail or diverge.

When does Newton's method fail to converge?

Newton's method can fail when the derivative f'(x) is zero or near zero at an iteration point, when the initial guess is too far from the root, when the function has inflection points near the root, or when the method enters a cycle. In these cases, try a different initial guess or use a bracketing method like bisection.

What is the convergence rate of Newton's method?

Newton's method has quadratic convergence for simple roots, meaning the number of correct digits roughly doubles each iteration. For repeated roots (where f'(r) = 0), convergence slows to linear. The quadratic convergence makes it one of the fastest root-finding methods when it converges.

Does this calculator compute the derivative automatically?

Yes. This calculator uses central difference numerical differentiation to compute f'(x) automatically. You only need to enter f(x) and the initial guess. The derivative is approximated as [f(x+h) - f(x-h)] / (2h) with a small step size for high accuracy.

What is the difference between Newton's method and the secant method?

Newton's method requires the derivative f'(x) and converges quadratically, while the secant method approximates the derivative using two previous points and converges superlinearly (order ~1.618). Newton's method is faster per iteration but requires computing the derivative, while the secant method may be more convenient when the derivative is difficult to compute.

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