Secant Methods
A secant is a line through two distinct points on a curve. There are many numerical methods which use the secants to obtain new estimates for a root. Below are three common secant methods called (1) the secant method, (2) regula falsi, and (3) the Illinois Algorithm.The secant method requires two approximations to the root, the approximations do not need to bracket the root. In fact, since the secant method is equivalent to the Newton-Raphson method in which the derivative is replaced by a numerical approximation to the derivative, all one needs to do is find an estimate to the root and for the second estimate simply add a small number to the first estimate so as to approximate the derivative. Successive approximations are obtained by finding the root of the secant determined by the immediate two previous approximations. The secant method can have difficulties if the root is located near a local maximum or minimum because the estimate can be quite far from the root.
The regula falsi method requires two initial estimates which not only bracket the root but also for which the function evaluated at the two estimates has opposite signs. In the regula falsi method the secant is chosen from points on the curve corresponding to the most recent estimates which bracket a root. Regula falsi is sometimes a good method for starting the iterative procedure, but it should not be used for final convergence to a root.
The Illinois algorithm is a variant of the regula falsi method in which during the iterative process if a change to the same endpoint occurs twice in succession, then the ordinate corresponding to the other endpoint is halved. This technique avoids the potentially slow convergence of the regula falsi method.
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