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alt="images"/> may represent uncertainties in the parameters, numerical noise or, within the context of this book, numerical inaccuracies due to limited machine precision. The condition number
must be understood as a
noise amplifier, which magnifies small uncertainties. A condition number of order 1 is an indication of
well‐conditioning, whereas a problem with
is definitely
ill‐conditioned.
By comparing (1.23) with (1.22), we can easily identify as the displacement exhibited by the root, as the numerical uncertainty in the evaluation of , and
(1.24)
as the condition number of the root . As we will see in Section 1.6, the performance of Newton's method can be affected if the root we are looking for is ill‐conditioned.
1.7 Local and Global Convergence
Newton's method converges properly only under certain conditions. One required condition is that the initial guess from which the iteration is initiated must be sufficiently close to the root, that is, a local initial guess. In that sense, it is said that Newton's method has only local convergence. Even if the sequence converges to the root, the order may not be always , as in Figure 1.2a.
It is a common misconception that every single method has its associated local order of convergence (Newton's has , secant has , etc.) In actuality, the order also depends on the conditioning of the root to which our sequence approaches. For example, the asymptotic constant from Newton's method appearing in (1.16) is proportional to . As a consequence, if is a double root and, accordingly, , the convergence criterion (1.11) is no longer valid since is not bounded.11
Figure 1.4 (a) Convergence history of Newton's and secant methods when the sequences approach the double root of equation . The ordinates corresponding to the secant method (triangles) have been shifted downwards three units to avoid overlap between the two sets of data and help visualize. (b) Newton's method iterates for the solution of .
Figure 1.4a shows the result of applying Newton's and secant methods to find the double ill‐conditioned root of the equation studied in Section 1.6. Newton's iteration is started from , whereas the secant has been initialized from the interval that contains the root. The reader may check that in this case Newton's and secant methods lose their quadratic and golden ratio orders, respectively, both exhibiting linear convergence, as shown in Figure 1.4a. Double or ill‐conditioned roots appear in physics more frequently than one may expect, particularly in problems where the transcendental equation to be solved is the result of imposing some kind of critical or threshold condition (we refer the reader to Practical 1.2, for example).
In general, root‐finding methods converge to the desired solution only if the initial guess is really close to the sought root, i.e. most of the methods are just locally convergent. In practice, a root‐finding algorithm starting from an initial guess moderately far away from the root could easily lead to a sequence that may wander from one point to another of the real axis, eventually diverging to infinity or converging to a solution (not necessarily the sought one). Figure 1.4b illustrates this phenomenon by showing the result of computing the roots of the function using Newton's method starting from different initial guesses. The first two roots of are located at and (black bullets in Figure 1.4b). In this example, we initialize Newton's method from two initial guesses reasonably close (but not too close) to . To guide the eye, we have indicated the history of each of the two sequences by encircled numbering of their ordinates. The first sequence starts at (gray square) and Newton's first iterate is already very close to , to which
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