Multiple Roots

Figure 4: Left: The curve on the left has a triple root at $x = -1$ [the function is $(x + 1)^3$]. The curve on the right has a double root at $x = 2$ [the function is $(x - 2)^2$].Right: Plot of $(x - 1) (e^{(x-1)} - 1)$.

• A function can have more than one root of the same value. See Fig. 4left.
• The methods we have described do not work well for multiple roots. For example, Newton's method is only linearly convergent at a double root. $f(x) = (x - 1) (e^{(x-1)} - 1)$ has a double root at $x = 1$, as seen in Fig. 4right.
• Table 2left gives the errors of successive iterates and the convergence is clearly linear.
• When Newton's method is applied to a triple root, convergence is still linear, as seen in Table 2right. With a triple root, the ratio of errors is larger, about $\frac{2}{3}$, compared to $\frac{1}{2}$ for the double root of Table 2left.

Table 2: Right: Errors when finding a double root. Left: Successive errors with Newton's method, for $f(x)=(x+1)^3=0$.