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relation
(1.21)
or for short, where the prime symbols acting on and must be understood as derivatives with respect to and , respectively. We can now estimate the location of the new root. Since is small,
Since the new root is precisely located at ,
Finally, since , we conclude from the last equation that . Taking the absolute value and recalling that ,
(1.22)
This last expression predicts how far the new root will move in terms of how much we have perturbed . From (1.22), we see that no matter how tiny may be, can potentially be very large if is very small. As an example, take the equation . For , the previous equation has a simple root at and a double root at (see Figure 1.3b, black curve). For and , the simple root exhibits a slight displacement to and , respectively (see dashed and solid gray curves in Figure 1.3b, respectively). However, for the same values of , the double root does experience remarkable changes. In particular, for , the double root disappears, leading to two simple roots located at and . For the effects are even more drastic since the function has no longer any root near .
Figure 1.3 (a) Graph
intercepting ordinates
and
at abscissas
and
, respectively. (b) Roots of the equation
for
(solid black curve),
(dashed gray), and
(solid gray).
From the previous analysis, we can clearly conclude that the double root is more sensitive (or ill‐conditioned) than the simple root . This phenomenon could have been predicted in advance just by evaluating the denominator appearing in (1.22) with and or , since , whereas .
In general, for a given numerical problem, it is common practice to quantify its conditioning by the simple relation
(1.23)
where and are the size of the variations introduced in the input data and their corresponding deviation effect in the outcome solution, respectively, and is a positive constant known as the condition number of the problem. The quantity
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