Secant method
In numerical analysis, the secant method is a root-finding algorithm that uses a succession of roots of secant lines to better approximate a root of a function f. The secant method can be thought of as a finite difference approximation of Newton's method. However, the method was developed independently of Newton's method, and predates it by over 3,000 years.
The Newton-Raphson algorithm requires the evaluation of two functions (the function and its derivative) per each iteration. If they are complicated expressions it will take considerable amount of effort to do hand calculations or large amount of CPU time for machine calculations. Hence it is desirable to have a method that converges (please see the section order of the numerical methods for theoretical details) as fast as Newton's method yet involves only the evaluation of the function.
Let x0 and x1 are two initial approximations for the root 's' of f(x) = 0 and f(x0) & f(x1) respectively, are their function values. If x 2 is the point of intersection of x-axis and the line-joining the points (x0, f(x0)) and (x1, f(x1)) then x2 is closer to 's' than x0 and x1. The equation relating x0, x1 and x2 is found by considering the slope 'm'.
\[m = \frac{f(x_{1})-f(x_{0})}{x_{1}-x_{2}} = \frac{f(x_{2})-f(x_{1})}{x_{2}-x_{1}} =\frac{0-f(x_{1})}{x_{2}-x_{1}} \]
\[x_{2}-x_{1} = \frac{-f(x_{1})(x_{1}-x_{0})}{f(x_{1})-f(x_{0})} \]
\[x_{2} = x_{1} - \frac{-f(x_{1})(x_{1}-x_{0})}{f(x_{1})-f(x_{0})} \]
in general the iterative process can be written as
\[x_{i+1} = x_{i}-\frac{f(x_{i})(x_{i}-x_{i-1})}{f(x_{i})-f(x_{i-1})} \] i=1,2,3...........
Algorithm - Secant Method
Given an equation f(x) = 0
Let the initial guesses be x0 and x1
Do
\[x_{i+1} = x_{i}-\frac{f(x_{i})(x_{i}-x_{i-1})}{f(x_{i})-f(x_{i-1})} \] i=1,2,3........
while (none of the convergence criterion C1 or C2 is met)
C1. Fixing apriori the total number of iterations N.
C2. By testing the condition | xi+1 - xi | (where i is the iteration number) less than some tolerance limit, say epsilon, fixed apriori.
