Editing Brent's method (section)

== Brent's method ==
{{Harvtxt|Brent|1973}} proposed a small modification to avoid the problem with Dekker's method. He inserts an additional test which must be satisfied before the result of the secant method is accepted as the next iterate. Two inequalities must be simultaneously satisfied:

Given a specific numerical tolerance <math>\delta</math>, if the previous step used the bisection method, the inequality <math display=inline> |\delta| < |b_k - b_{k-1}| </math> must hold to perform interpolation, otherwise the bisection method is performed and its result used for the next iteration.

If the previous step performed interpolation, then the inequality <math display=inline>|\delta| < |b_{k-1} - b_{k-2}|</math> is used instead to perform the next  action (to choose) interpolation (when inequality is true) or bisection method (when inequality is not true).

Also, if the previous step used the bisection method, the inequality <math display=inline>|s-b_k| < \begin{matrix} \frac12 \end{matrix} |b_k - b_{k-1}|</math>
must hold, otherwise the bisection method is performed and its result used for the next iteration. If the previous step performed interpolation, then the  inequality <math display=inline>|s-b_k| < \begin{matrix} \frac12 \end{matrix} |b_{k-1} - b_{k-2}|</math>
is used instead.

This modification ensures that at the ''k''th iteration, a bisection step will be performed in at most <math>2\log_2(|b_{k-1}-b_{k-2}|/\delta)</math> additional iterations, because the above conditions force consecutive interpolation step sizes to halve every two iterations, and after at most <math>2\log_2(|b_{k-1}-b_{k-2}|/\delta)</math> iterations, the step size will be smaller than <math>\delta</math>, which invokes a bisection step. Brent proved that his method requires at most ''N''<sup>2</sup> iterations, where ''N'' denotes the number of iterations for the bisection method. If the function ''f'' is well-behaved, then Brent's method will usually proceed by either inverse quadratic or linear interpolation, in which case it will converge [[rate of convergence|superlinearly]].

Furthermore, Brent's method uses [[inverse quadratic interpolation]] instead of [[linear interpolation]] (as used by the secant method). If ''f''(''b''<sub>''k''</sub>), ''f''(''a''<sub>''k''</sub>) and ''f''(''b''<sub>''k''&minus;1</sub>) are distinct, it slightly increases the efficiency. As a consequence, the condition for accepting ''s'' (the value proposed by either linear interpolation or inverse quadratic interpolation) has to be changed: ''s'' has to lie between (3''a''<sub>''k''</sub> + ''b''<sub>''k''</sub>) / 4 and ''b''<sub>''k''</sub>.