Brent's method

In numerical analysis, Brent's method is a root-finding algorithm combining the bisection method, the secant method and inverse quadratic interpolation. It has the reliability of bisection but it can be as quick as some of the less-reliable methods. The algorithm tries to use the potentially fast-converging secant method or inverse quadratic interpolation if possible, but it falls back to the more robust bisection method if necessary. Brent's method is due to Richard Brent and builds on an earlier algorithm by Theodorus Dekker. Consequently, the method is also known as the Brent–Dekker method.

The idea to combine the bisection method with the secant method goes back to Dekker (1969).

Suppose that we want to solve the equation f(x) = 0. As with the bisection method, we need to initialize Dekker's method with two points, say a₀ and b₀, such that f(a₀) and f(b₀) have opposite signs. If f is continuous on [a₀, b₀], the intermediate value theorem guarantees the existence of a solution between a₀ and b₀.

Three points are involved in every iteration:

Two provisional values for the next iterate are computed. The first one is given by linear interpolation, also known as the secant method:

and the second one is given by the bisection method

If the result of the secant method, s, lies strictly between b_k and m, then it becomes the next iterate (b_k+1 = s), otherwise the midpoint is used (b_k+1 = m).

Then, the value of the new contrapoint is chosen such that f(a_k+1) and f(b_k+1) have opposite signs. If f(a_k) and f(b_k+1) have opposite signs, then the contrapoint remains the same: a_k+1 = a_k. Otherwise, f(b_k+1) and f(b_k) have opposite signs, so the new contrapoint becomes a_k+1 = b_k.

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