Newton's Method Calculator

About Newton's Method Calculator

Newton's method converges quickly to roots. Starting from an initial guess, each iteration improves the approximation quadratically.

How to use this calculator

1. Enter the function f(x) whose root you want to find.
2. Give an initial guess x_0 that is close to the root.
3. Compute the next approximation with Newton's formula, then repeat with the new value.
4. Stop when two successive values are close enough for the accuracy you need.

The formula explained

The formula updates a guess by moving along the tangent line to the graph of f(x). Each new value x_{n+1} is usually closer to a root than x_n.

• \(x_n\) = the current approximation after n iterations
• \(x_{n+1}\) = the next approximation
• \(f(x_n)\) = the value of the function at the current approximation
• \(f'(x_n)\) = the derivative of the function at the current approximation

Step by step method

1. Choose a starting value x_0 near the root.
2. Evaluate f(x_0) and f'(x_0), then compute x_1 = x_0 - f(x_0)/f'(x_0).
3. Repeat the same process with x_1, x_2, and so on until the values stop changing much.

Worked example

Problem. Use Newton's method to approximate a root of f(x)=x^2-2 starting from x_0=1.

1. Compute f(1)=1^2-2=-1 and f'(x)=2x, so f'(1)=2.
2. Apply Newton's formula, x_1 = 1 - (-1)/2 = 1.5.
3. Repeat: f(1.5)=0.25 and f'(1.5)=3, so x_2 = 1.5 - 0.25/3 = 1.416666..., which is already close to sqrt{2}.

Tips and common mistakes

• Your starting guess should be close to the root, otherwise the method may fail or jump away.
• If f'(x_n)=0 or very close to 0, the formula breaks down or becomes unstable.

Frequently asked questions