SecantMethodofsolvingNonlinearequationsGeneralEngineering.docVIP

Chapter 03.05 Secant Method of Solving Nonlinear Equations After reading this chapter, you should be able to: derive the secant method to solve for the roots of a nonlinear equation, use the secant method to numerically solve a nonlinear equation. What is the secant method and why would I want to use it instead of the Newton-Raphson method? The Newton-Raphson method of solving a nonlinear equation is given by the iterative formula (1) 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 still can be a laborious process, and even intractable if the function is derived as part of a numerical scheme. To overcome these drawbacks, the derivative of the function, is approximated as (2) Substituting Equation (2) in Equation (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 is an open method and may or may not converge. However, when secant method converges, it will typically converge faster than the bisection method. However, since the derivative is approximated as given by Equation (2), it typically converges slower than the Newton-Raphson method. The secant method can also be derived from geometry, as shown in Figure 1. Taking two initial guesses, and , one draws a straight line between and passing through the -axis at . ABE and DCE are similar triangles. Hence On rearranging, the secant method is given as Figure 1 Geometrical representation of the secant method. Example 1 You are working for ‘DOWN THE TOILET COMPANY’ that makes floats (Figure 2) for ABC commodes. The floating ba