# Brent method

Numerical methods

Root bracketing methods

Root polishing methods

Brent method combines an interpolation strategy with the bisection algorithm.

On each iteration, Brent method approximates the function using an interpolating curve. On the first iteration, this is a linear interpolation of the two endpoints. For subsequent iterations the algorithm uses an inverse quadratic fit to the last three points, for higher accuracy.

The intercept of the interpolating curve with the x-axis is taken as a guess for the root. If it lies within the bounds of the current interval then the interpolating point is accepted, and used to generate a smaller interval. If the interpolating point is not accepted then the algorithm falls back to an ordinary bisection step. The best estimation of the root is taken from the most recent interpolation or bisection.

If the function is well-behaved, then Brent method will usually proceed by either inverse quadratic or linear interpolation, in which case it will converge superlinearly. On the other hand, Brent method will always fall back on the bisection method if necessary, so it will never do worse than the bisection method; in particular, it will never fail.

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