Newton Raphson Method

TL;DR

• Iterative procedure that refines an initial guess to approximate a root of an equation.
• Each iteration uses the function value and its derivative to compute a better approximation.
• Repeat the update until the approximation is sufficiently close to the actual root.

Definition

The Newton–Raphson method is a widely used iterative method to find the roots of a given equation. It is based on the idea that if we can approximate the root of a given equation, we can use that approximation to find an even better approximation.

Explanation

• Start by choosing an initial guess for the root; this can be any value believed to be near the actual root.
• Use the chosen approximation together with the function and its derivative to compute an improved approximation.
• The method applies the update formula as presented in the source:

$$f(x) = x - \frac{f(x)}{f'(x)}$$

• Repeat the update iteratively until the result is close enough to the true root.

Examples

Example 1 — quadratic: $x^2 - 6x + 9 = 0$

Choose initial guess: (x = 3).

Apply the update:

$$x - \frac{x^2 - 6x + 9}{2x - 6} = 3 - \frac{3^2 - 63 + 9}{23 - 6} = 3 - \frac{9 - 18 + 9}{6 - 6} = 3 - 0 = 3$$

The improved approximation is (x = 3).

Example 2 — cubic: $x^3 - x^2 - x - 1 = 0$

Initial guess: (x = 1).

First improved approximation:

$$x = 1 - \frac{1^3 - 1^2 - 1 - 1}{31^2 - 21 - 1} = 1 - \frac{-1}{1} = 1 - (-1) = 2$$

Second improved approximation:

$$x = 2 - \frac{2^3 - 2^2 - 2 - 1}{32^2 - 22 - 1} = 2 - \frac{8 - 4 - 2 - 1}{12 - 4 - 1} = 2 - \frac{1}{7} = 2 - 0.14 = 1.86$$

As shown, each improved approximation moves closer to the actual root; the process is continued until the approximation is sufficiently accurate.

Related terms

• Derivative
• Root (of an equation)
• Iterative method