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Newton's method

Newton's method finds a root by following the tangent of $f$ down to where it crosses the x-axis, repeatedly. Near a simple root it converges quadratically — the number of correct digits roughly doubles each step — but it needs the derivative and can diverge with a poor initial guess.

Symbols

Example

$f(x) = x^3 - 5$, $x_0 = 3$:

$x_1 = 3 - \frac{27-5}{27} \approx 2.1852$.

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Part of Numerical Methods: Solving equations.