Figure 3.10:
Left: The curve on the left has a triple root at [the function is ]. The curve on the right has a double root at [the function is ].Right: Plot of
.
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- A function can have more than one root of the same value. See Fig. 3.10left.
- 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.
has a double root at , as seen in Fig. 3.10right.
- Table 3.6left 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 3.6right. With a triple root, the ratio of errors is larger, about
, compared to
for the double root of Table 3.6left.
Table 3.6:
Right: Errors when finding a double root. Left: Successive errors with Newton's method, for
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2004-12-28