Enter your function f(x), an initial guess x₀, and the number of iterations to find the root using Newton's Method. The calculator applies the Newton-Raphson formula — xₙ₊₁ = xₙ − f(xₙ)/f′(xₙ) — step by step, showing you the root approximation, function value, and error at each iteration.
Root Approximation
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f(x) at Root
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Iterations Used
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Final Error |xₙ₊₁ − xₙ|
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Newton's Method (also called the Newton-Raphson method) is an iterative numerical technique for finding the roots of a real-valued function. Starting from an initial guess x₀, it repeatedly applies the formula xₙ₊₁ = xₙ − f(xₙ)/f′(xₙ) to produce successively better approximations to the root where f(x) = 0.
Newton's Method converges much faster than simpler methods like bisection — it typically achieves quadratic convergence, meaning the number of correct decimal digits roughly doubles with each iteration. It is especially useful for solving complex polynomial and transcendental equations that cannot be solved algebraically.
The core formula is xₙ₊₁ = xₙ − f(xₙ) / f′(xₙ), where xₙ is the current approximation, f(xₙ) is the function value at that point, and f′(xₙ) is the derivative value. Each application of this formula produces a new, closer approximation to the root.
A good initial guess should be reasonably close to the actual root. You can graph the function to identify where it crosses the x-axis, or use prior knowledge of the problem domain. A poor starting point can cause the method to diverge or converge to an unexpected root.
If f′(xₙ) equals zero at any iteration, the method fails because division by zero is undefined. This usually happens when the current approximation lands at a local maximum or minimum of f(x). In such cases, you should choose a different initial guess.
For well-behaved functions and a reasonably close initial guess, Newton's Method usually converges within 5 to 10 iterations to many decimal places of accuracy. The method's quadratic convergence means errors shrink very rapidly once you are near the root.
The tolerance ε is a stopping criterion. If the absolute difference |xₙ₊₁ − xₙ| falls below ε before all iterations are completed, the method stops early because the approximation has converged sufficiently. Setting ε to 0 forces the calculator to run every requested iteration.
Yes. Newton's Method can fail or diverge if the initial guess is too far from the root, if the function has a zero derivative near the guess, or if the function oscillates in a way that prevents convergence. In those cases, try a different starting point or a more robust method like bisection.