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Newton's method

Many equations cannot be solved exactly. Newton's method starts from a guess and follows the tangent down to the x-axis to get a better guess. The method converges very fast when you start close to the solution.

x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}

one Newton step

Symbols

$x_n$	current guess
$f'(x_n)$	slope at the guess

Example

Solve $x^2 - 2 = 0$ from $x_0 = 1$ :

$x_1 = 1 - (-1)/2 = 1.5$ and $x_2 = 1.5 - 0.25/3 \approx 1.4167$ .

If f'(x) is close to zero, the step can be huge. Then choose another starting point.

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