1550078481-Ordinary_Differential_Equations__Roberts_

194 Ordinary Differential Equations

line. See Figure 4.1. The equation of the tangent line at (x1, f(x1))- that is,
the equation of the line through the point (x1,f(x1)) with slope f'(x1)- is

y - f(x1) = J'(x 1)(x - x1).

Provided f' (xi) =f. 0, the tangent line intersects the x-axis at some point, say,

(x 2 , 0). Substituting x = x 2 and y = 0 into the above equation, we find x2

satisfies

Or solving for x 2 , we get

The value x 2 is a new approximation to a root off. The process is repeated.

Thus, Newton-Raphson's method for finding a root of a differentiable function

f is as follows:

Make an initial guess, x1, of a root.

Then provided f' (xn) =f. 0 iteratively compute

Xn+i = Xn -j(xn)/J'(xn)

until Xn+i is as good an approximation of a root as desired or until a specified

number of iterations have been computed without the partial sequence x 1 , x2,

... , Xn+i appearing to converge to a root.

If the partial sequence does not appear to be converging, then the process
can be started over again with a new guess for x 1.

y

Figure 4.1 The Newton-Raphson Method

Tangent Line

at (x 1 , f(x 1 ))

x