Page 177 - Excel for Scientists and Engineers: Numerical Methods

154 EXCEL: NUMERICAL METHODS

This formula divides the value of y2 by 2 if there has been no change in x2 in the
preceding two iteration cycles (this has occurred in rows 5,6 and 7, for example).
Otherwise the function is calculated by means of the usual formula.
A nested IF could be used to handle the case where either x1 or x2 is "stuck."

Figure 8-9. Modifying the Regula FaIsi method to handle slow convergence.
(folder 'Chapter 08 Examples', workbook 'Roots of Equations', worksheet 'Regula Falsi (3)')

The Newton-Raphson Method

The preceding methods require manual selection of a pair of starting values
with opposite signs. The Newton-Raphson method (sometimes referred to
simply as Newton's method) requires only a single function value as the starting
value, and is therefore self-starting. The Newton-Raphson method is a classic
exercise from freshman calculus-it uses the first derivative of the function (the
slope of the curve) at the initial estimate, XI, and extrapolates this tangent line to
the x axis to obtain an improved value, x2. The process is repeated to obtain
further approximations to the root, as illustrated in Figure 8-10, until the desired
convergence level is reached.