Newton's Method Calculator – Root Finding Algorithm
Find roots of functions using Newton-Raphson iteration.
Table of Contents
How to Use
- Enter the function f(x) using x as the variable
- Enter the derivative f'(x) of the function
- Provide an initial guess close to the expected root
- Set the maximum iterations and tolerance
- Click calculate to see the iterative approximation
What is Newton's Method?
Newton's Method, also known as the Newton-Raphson method, is a powerful numerical technique for finding roots of a function. It uses the idea that a continuous and differentiable function can be approximated by a straight line tangent to it.
The iteration formula is: xₙ₊₁ = xₙ - f(xₙ)/f'(xₙ), where f(x) is the function and f'(x) is its derivative.
How Does It Work?
Starting from an initial guess x₀, the method repeatedly improves the approximation:
- Evaluate the function f(xₙ) at the current point
- Evaluate the derivative f'(xₙ) at the current point
- Calculate the next approximation: xₙ₊₁ = xₙ - f(xₙ)/f'(xₙ)
- Repeat until the change is smaller than the tolerance
Convergence Conditions
Newton's Method converges quadratically when:
- The initial guess is sufficiently close to the root
- The function is continuously differentiable
- The derivative is non-zero at the root
- The function has a simple root (multiplicity 1)
The method may fail or diverge if the derivative is zero, the initial guess is poor, or the function has complex behavior near the root.
Applications
Newton's Method is widely used in:
- Finding square roots and nth roots
- Solving nonlinear equations
- Optimization problems (finding critical points)
- Computer graphics and physics simulations
- Financial calculations and engineering
Frequently Asked Questions
- How do I enter the function?
- Use 'x' as the variable. Supported operations include: +, -, *, /, ^ (power), sqrt, sin, cos, tan, log, exp. For example, 'x^3 - 2*x + 1' or 'sin(x) - x/2'.
- What if the method doesn't converge?
- Try a different initial guess closer to the expected root, increase the number of iterations, or check if the derivative is zero near your guess. Some functions may require special handling.
- Why do I need to enter the derivative?
- Newton's Method requires the derivative to calculate the tangent line at each point. The formula xₙ₊₁ = xₙ - f(xₙ)/f'(xₙ) uses the derivative to determine the direction and step size.
- What tolerance should I use?
- A tolerance of 0.0001 (10⁻⁴) is suitable for most applications. For higher precision, use smaller values like 10⁻⁸. The tolerance determines when the iteration stops based on the change between successive approximations.