Multiple Roots

Figure 3.12: 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. 3.12left.
• $f(x) = (x - 1) (e^{(x-1)} - 1)$ has a double root at $x = 1$, as seen in Fig. 3.12right.
• 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.

Table 3.5: Left: Errors when finding a double root. Right: Successive errors with Newton's method for $f(x)=(x+1)^3=0$ (Triple root).

• Table 3.5left gives the errors of successive iterates (Newton's method is applied to a double root) and the convergence is clearly linear with ratio of errors is $\frac{1}{2}$.
• When Newton's method is applied to a triple root, convergence is still linear, as seen in Table 3.5right. The ratio of errors is larger, about $\frac{2}{3}$.