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
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 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 As a consequence, if is a double root and, accordingly, , the convergence criterion (1.11) is no longer valid since is not bounded.11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 more sensitive (or ill‐conditioned) than the simple root 