The title says it all:

What is the equation for the error of the Newton-Raphson method?

Also… an explanation for each of the terms would be nice… I'm a comp sci guy, not typically a math guy.

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# [Math] the equation for the error of the Newton-Raphson method

calculusnumerical methods

The title says it all:

What is the equation for the error of the Newton-Raphson method?

Also… an explanation for each of the terms would be nice… I'm a comp sci guy, not typically a math guy.

## Best Answer

Suppose you're using Newton-Raphson to solve $f(x)=0$ where $f$ is a twice differentiable function, so $x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}$, and $f(r) = 0$. Then $$r - x_{n+1} = - \frac{f''(c) (r - x_n)^2}{2 f'(x_n)}$$ where $c$ is some point between $r$ and $x_n$. If $f''$ is continuous, $f'(r) \ne 0$ and $x_n$ is close to $r$, $f''(c)/f'(x_n)$ will be close to $f''(r)/f'(r)$, so this says the error in $x_{n+1}$ is approximately a constant times the square of the error in $x_n$.