Newton's Method Calculator
Find roots fast using Newton’s Method: enter function, guess, iterations. See steps, convergence, derivative preview instantly, with clear results displayed.
Equation Preview
Helping Notes
- Newton step: xₙ₊₁ = xₙ − f(xₙ)/f′(xₙ). Ensure f′(xₙ) ≠ 0 during iteration.
- Different starting guesses may lead to different roots or divergence.
- Expressions follow math.js syntax (e.g., sin(x), cos(x), exp(x), sqrt(x), x^2).
- This tool auto-derives f′(x) symbolically for you.
Results
Approximate Root
Convergence & Error
Iteration Log
Error
What is Newton’s Method Calculator?
This calculator automates a classic root-finding technique for solving nonlinear equations by iteratively refining an initial guess until it reaches a solution. At each step, it uses the tangent line at the current point to predict where the curve meets the x-axis. The iterative update rule is:
When the function is smooth and the starting value is near a true root with a nonzero derivative, convergence is typically rapid (often quadratic). Practical stopping tests include either a small function value or a small step size:
For well-behaved problems, the error often shrinks approximately quadratically:
About the Newton’s Method Calculator
The tool accepts a function f(x), its derivative f′(x), an initial guess x0, a maximum iteration count, and tolerance settings. It returns a clear, step-by-step table of xn, f(xn), and update size; highlights when a stopping rule is triggered; and flags numerical issues such as extremely small f′(xn) or oscillations. You can explore multiple starting values to study convergence behavior and sensitivity.
How to Use this Newton’s Method Calculator
- Enter the function f(x) (e.g., x^2 - 2).
- Provide the derivative f′(x) (e.g., 2x), or choose a numerical derivative option if available.
- Set an initial guess x0, maximum iterations, and tolerances δ, ε.
- Run the iteration governed by the update rule:
- Review the iteration log and final estimate; adjust inputs if the method stalls or diverges.
Examples
Example 1: Solve x^2 − 2 = 0 (root \u221A2)
Let f(x)=x^2-2, f′(x)=2x, x0=1.
After a few steps, the estimate stabilizes near \u221A2 \u2248 1.41421356.
Example 2: Solve cos(x) − x = 0
Let f(x)=\cos x - x, f′(x)=-\sin x - 1, x0=0.5.
FAQs
Does this method always converge?
No. Poor initial guesses, flat derivatives, or non-smooth functions can cause divergence, cycling, or slow progress.
How should I choose a good starting value?
Graph the function and pick a point near a root; avoid regions where the derivative is near zero or changes sign abruptly.
What happens if the derivative is zero (or extremely small)?
The update can become unstable or huge. Try a different start, damping the step, or switching to a bracketing method to reinitialize.
Is this faster than bisection or secant methods?
Often yes (quadratic convergence) when assumptions hold, though bracketing methods guarantee progress if a sign change interval is known.
Can it find multiple or complex roots?
Different initial guesses may lead to different real roots. Complex roots require extending the arithmetic to complex numbers.