Newton's method

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Newton's method uses the derivative of a differentiable function to approximate its real or complex roots. For a function $f(x)$, the approximations are defined recursively by \[x_{i+1} = x_i - \frac{f(x_i)}{f'(x_i)}.\] To begin the recursion, an initial guess $x_0$ must be chosen. Often the choice of $x_0$ determines which of several possible roots is found, and in some cases the initial guess can cause the recursion to cycle or diverge instead of converging to a root.

Derivation

At each step of the recursion, we have $x_i$ and seek a root of $f(x)$. Since all nonconstant linear functions have exactly one root, as long as $f'(x_i) \neq 0$ we can construct a tangent-line approximation to $f(x)$ and find its root as an approximation. The tangent-line approximation of $f(x)$ about $x_i$ is \[f(x_i) + (x - x_i)f'(x_i).\] We seek the value $x = x_{i+1}$ such that the above expression equals $0$; hence, \[f(x_i) + (x_{i+1} - x_i)f'(x_i) = 0.\] Dividing by $f'(x_i)$ (as long as $f'(x_i) \neq 0$), \[\frac{f(x_i)}{f'(x_i)} + x_{i+1} - x_i = 0.\] Therefore, the desired value of $x_{i+1}$ is \[x_{i+1} = x_i - \frac{f(x_i)}{f'(x_{i})}.\]

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