ABSTRACT
This chapter considers the solution of the equation f(x) = 0 for a suitably smooth function. Any non-linear system can be expressed in this form. In some cases the function in question satisfies the conditions for convergence, but the point chosen as starting point is not sufficiently close to the root. In this case other methods such as bisection are used to obtain a better starting point. The bisection method always chooses the mid-point of the current interval. The Bus and Dekker algorithm is a hybrid of the secant and the bisection method. Brent's method combines the bisection method, the secant method, and inverse quadratic interpolation. It is also known as the Wijngaarden-Dekker-Brent method. At every iteration, Brent's method decides which method out of these three is likely to do best, and proceeds by doing a step according to that method.
