an:00178996
Zbl 0828.65046
Donovan, George C.; Miller, Arnold R.; Moreland, Timothy J.
Pathological functions for Newton's method
EN
Am. Math. Mon. 100, No. 1, 53-58 (1993).
0002-9890
1993
j
65H05
false convergence phenomenon; stopping criterion; Newton's method
In the solution of equations by numerical methods, a commonly used stopping criterion is
(1) \(|x_{n + 1} - x_n|< \varepsilon\), where \(x_n\) is the \(n\)th term of the sequence generated by the method, and \(\varepsilon > 0\) is the tolerance. However in general, criterion (1) can fail: there exist sequences (e.g., the partial sums of the harmonic series) for which (1) is true but which nonetheless diverge.
We derive two functions, which exhibit this ``false convergence'' phenomenon. The first of these has no real root, but nevertheless generates a sequence under Newton's method for which (1) is satisfied for any \(\varepsilon\), namely, \(\{\sqrt {u_n}\}\) where \(u_n \in \mathbb{R}\), \(u_{n + 1} = u_n + 1\), and \(n = 0,1, \dots\) Although this sequence satisfies (1), it obviously does not converge. The second function, like the first one, appears to converge where there are no roots, but it has a real root, to which Newton's method will never converge.