Barna’s inequality

December 30, 2014

(Disclaimer: This is planned to be part of a larger set of notes on the dynamics of Newton’s method.)

We present Barna’s proof of the following result: Suppose that f is a polynomial all of whose roots are real and N is is associated Newton’s function. If r is a root of f and its immediate basin of attraction I_r=(a,b) is bounded, then |N'(a)|>1 and |N'(b)|>1.

1. Introduction

If f is a differentiable function, we define N=N_f, Newton’s function for f, by

\displaystyle N_f(x)=x-\frac{f(x)}{f'(x)}

for all values of x such that f'(x)\ne 0. Under reasonable assumptions on f, N can be extended (by continuity) so that it is also defined at those points x such that f(x)=f'(x)=0. Note that if f'(x)\ne 0, then N'(x) exists, and we have

\displaystyle N'(x)=\frac{f(x)f''(x)}{(f'(x))^2}.

The function N is of course the function obtained through the application of the familiar Newton’s method for approximating roots of f. Recall that (under reasonable assumptions on f) the method starts with a guess x_0 for a root of f, and refines this guess successively, with each new guess x_{n+1} being obtained from the previous one x_n by replacing f with its linear approximation at x=x_n, that is, with the line y-x_n=f'(x_n)(x-x_n), and letting x_{n+1} be the value of x where this approximation equals 0, that is, x_{n+1}=N_f(x_n).

One can say much about what makes a function “reasonable”, but for the purposes of this note it suffices that polynomials certainly fall under this category. One easily verifies that if (f is a real valued polynomial and) f(x^*)=0 then there is an open neighborhood I of x^* such that if x_0\in I, then for all n we have that x_n is defined, x_n\in I, and \lim_{n\to \infty}x_n=x^*. This is perhaps easiest to see if we assume that x^* is a simple zero, that is, f(x^*)=0\ne f'(x^*). In this case N'(x^*)=0 and, by continuity, N(x) and N'(x) are defined, and |N'(x)|<1, for all x in a neighborhood (x^*-\epsilon,x^*+\epsilon) of x^*. But if x is in this neighborhood, by the mean value theorem we have that |N(x)-x^*|=|N(x)-N(x^*)|<|x-x^*|. It follows that N(x) is also in this neighborhood, and that successive applications of N result in a sequence that converges to x^*.

The largest interval I about x^* such that for any point x_0 in I the sequence x_0,x_1,\dots,x_{n+1}=N(x_n),\dots is well defined and converges to x^*, we call the immediate basin of attraction of x^*. One can verify that I is open, that N(I)=I, and that if I is bounded, say I=(a,b), then N(a)=b and N(b)=a, so that \{a,b\} is a cycle of period 2 for N. By the observations above, we know that if c is whichever of a and b is closest to x^*, then |N'(c)|\ge 1.

The study of the dynamics of Newton’s method is very interesting. Nowadays, most work on the subject is part of the more general topic of complex dynamics, but the study of the dynamical behavior on \mathbb R deserves to be better known. In this note I present a result of Barna, who in the late 1950s and early 1960s began such a study. The result is an inequality that improves the observation at the end of the previous paragraph. This inequality is useful in studying the chaotic behavior of the method for polynomials of degree at least 4, see for instance [SU84].

2. Barna inequality

Theorem (Barna [Bar61]). Suppose that f is a polynomial all of whose roots are real. If r is a root of f and its immediate basin of attraction I_r=(a,b) is bounded, then |N'(a)|>1 and |N'(b)|>1.

Barna’s argument is completely elementary, but it is not easy to locate it in modern literature.

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Analysis – HW 5 – Newton’s method

November 16, 2013

This set is due Friday, December 6, at the beginning of lecture.

Newton’s method was introduced by Newton on De analysi in 1669. It was originally restricted to polynomials; his example in Methodus fluxionum was the cubic equation

x^3-2x-5=0.

Raphson simplified its description in 1690. The modern presentation, in full generality, is due to Simpson in 1740. Here, we are mostly interested in the dynamics of Newton’s method on polynomials.

TeX source.

Pdf file.


170 – Extra credit

September 20, 2013

Choose a mathematician whose work is somehow related to calculus. The connection may be loose (of course, Newton or Leibniz qualify, but so does John Nash). To avoid repetitions, email me your choice, and I’ll let you know whether your choice has not yet been claimed.

Write (well, type) an essay on their life and mathematical work. It may be short. Make sure to follow reasonable standards of style when citing references, etc. Due December 2 (after Thanksgiving break).


170 – The birth of calculus, Quiz 1 and extra credit project

March 4, 2013

The short BBC documentary “The birth of calculus”, produced by The Open University, and narrated by Jeremy Gray, can be found here.

Quiz 1 was Friday, March 1. Here it is.

As an extra credit project: Write (type) a short essay on the life and mathematical contributions of one of the mathematicians responsible for the development of calculus. Please email me your choice before you get started, to avoid repetitions. Make sure to cite all your references appropriately. Turn this in by FridayMonday, April 1.