What is the Secant method and why would I want to use it versus the Newton-Raphson method?

The Newton-Raphson method of solving the nonlinear equation f(x =0 is given by the recursive formula

(1)

From the above equation, one of the drawbacks of the Newton-Raphson method is that you have to evaluate the derivative of the function. With availability of symbolic manipulators such as Maple, Mathcad, Mathematica and Matlab, this process has become more convenient. However, it is still can be a laborious process. To overcome this drawback, the derivative, f’ (x) of the function, f(x) is approximated as

(2)

Substituting Equation (2) in (1), gives

. (3)

The above equation is called the Secant method. This method now requires two initial guesses, but unlike the bisection method, the two initial guesses do not need to bracket the root of the equation. The Secant method may or may not converge, but when it converges, it converges faster than the Bisection method. However, since the derivative is approximated, it converges slower then Newton-Raphson method.

The Secant method can be also derived from geometry shown in Figure (1). Taking two initial guesses, x_i and x_{i -1}, one draws a straight line between f(x_i) and f(x_i-1) passing through the x -axis at x_{i +1}. ABE and DCE are similar triangles. Hence

On rearranging, it gives the secant method as

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