Newton-method obeys Benford’s law. (English) Zbl 1136.65048

Berger, Arno; Hill, Theodore P.

Am. Math. Mon. 114, No. 7, 588-601 (2007).

From the introduction: One of the most popular methods in all of applied mathematics is Newton’s method, used for computing successive approximations of zeros of functions. The main purpose of this article is to show that Newton’s method exhibits exactly the type of nonuniformity of significant digits not only do the first few significant digits of the distances from the successive approximations to any root, and of the distances between approximations, tend to be small, but – much more specifically – they typically follow a well-known logarithmic (and thus highly nonuniform) distribution called Benford’s law [cf. F. Benford, Proc. Am. Philos. Soc. 78, 551–572 (1938; Zbl 0018.26502)].

MSC:

65H05	Numerical computation of solutions to single equations
65G50	Roundoff error
11K16	Normal numbers, radix expansions, Pisot numbers, Salem numbers, good lattice points, etc.

Keywords:

roundoff error; numerical examples; Benford’s law; Newton’s method; zeros of functions; nonuniformity of significant digits

Citations:

Zbl 0018.26502