ak+1 = ak. Otherwise, f(bk+1) and f(bk) have opposite signs, so the new contrapoint becomes ak+1 = bk. Finally, if |f(ak+1)| < |f(bk+1)|, then ak+1 is probably a better guess for the solution than bk+1, and hence the values of ak+1 and bk+1 are exchanged. This ends the description of a single iteration of Dekker's method. Dekker's method performs well if the function f is reasonably well-behaved. However, there are circumstances in which every iteration employs the secant method, but the iterates bk converge very slowly (in particular, |bk − bk−1| may be arbitrarily small). Dekker's method requires far more iterations than the bisection method in this case. Brent's method 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 , if the previous step used the bisection method, the inequality 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 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 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 is used instead. This modification ensures that at the kth iteration, a bisection step will be performed in at most additional iterations, because the above conditions force consecutive interpolation step sizes to halve every two iterations, and after at most iterations, the step size will be smaller than , which invokes a bisection step. Brent proved that his method requires at most N2 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 superlinearly. Furthermore, Brent's method uses inverse quadratic interpolation instead of linear interpolation (as used by the secant method). If f(bk), f(ak) and f(bk−1) 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