Graeffe's method

Abstract: Let an entire functionF(z) of finite genus have infinitely many zeros which are all positive, and take real values for realz. Then it is shown how to give two-sided bounds for all the zeros ofF in terms of the coefficients of the power series ofF, in fact in terms of the coefficients obtained byGraeffe's algorithm applied toF. A simple numerical illustration is given for a Bessel function.

Abstract: The only really useful practical method for solving numerical algebraic equations of higher orders, possessing complex roots, is that devised by C. H. Graeffe early in the nineteenth century. When an equation with real coefficients has only one or two pairs of complex roots, the Graeffe process leads to the evaluation of these roots without great labour. If, however, the equation has a number of pairs of complex roots there is considerable difficulty in completing the solution: the moduli of the roots are found easily, but the evaluation of the arguments often leads to long and wearisome calculations. The best method that has yet been suggested for overcoming this difficulty is that by C. Runge (Praxis der Gleichungen, Sammlung Schubert). It consists in making a change in the origin of the Argand diagram by shifting it to some other point on the real axis of the original Argand plane. The new moduli and the old moduli of the complex roots can then be used as bipolar coordinates for deducing the complex roots completely: this also checks the real roots.

Abstract: — The problem considérée her e is to find a good upper bound for the largest modulus of the roots of a given complex polynomial. We propose to first use a few itérations of Graeffe's method and then an upper bound given by Knuth. Resumé. — Le problème considéré ici est la recherche d'un bon majorant pour le plus grand module des racines d'un polynôme donné. Nous proposons d'abord d'appliquer quelques itérations de la méthode de Graeffe puis une borne due à Knuth. We consider a polynomial with complex coefficients The question we wish to solve is « fïnd an R such that ail roots of ƒ have absolute value at most R ». This quantity appears in many bounds in computer algebra, and figures to a very high power in the bounds for factoring polynomials over algebraic number fields (see [6]). 1. CAUCHY'S METHOD AND ITS INHERENT WEAKNESS Since Cauchy [1] (p. 122), it is known that R can be chosen as the unique positive real root C(f) of the polynomial (*) Received in April 1989. () University of Bath, School of Mathematics, Claverton Down, Bath BA2 7AY, England. () Université Louis Pasteur, Mathématique, 7 rue René Descartes, 67084 Strasbourg, France. MAN Modélisation mathématique et Analyse numérique 0764-583X/90/06/693/04/$ 2.40 Mathematical Modelling and Numerical Analysis © AFCET Gauthier-Villars 694 J H. DA VENPORT, M. MIGNOTTE Let p be the absolute value of the largest root of the polynomial ƒ lt is easy to see that, for any positive real x, f*(x)^2x(x + p)*. Hence C(f) satisfies where both inequalities are sharp. The left inequality is an equality when ƒ = / *, whereas the right inequality is an equality when ƒ (x) = (x + p). This shows in particular that C(f) may be too big by a factor / / L o g 2. One would prefer a bound based on the | at | which did not require the explicit computation of the root C(f) of/*, There are many such bounds Cauchy [1], Knuth [5] (ex. 4.6.2 : 20), Dieudonné [3] (p. 66) : all based on an analysis of/*. Knuth's is (**) * * = * ( ƒ ) = 2 m a x {\\ak_,\\9 \\ak^\\ \\ K _ 3 | 1 / 3 , . . . , and can also be found in Henrici [4] (cor. 6.4k, p. 457). Knuth shows that K(f ) === 2 kp. In our notation this follows from lV 1 f 1 7 k for 1 ̂ i ^ k . 2. THE POWER OF GRAEFFE'S METHOD In this note we use Graeffe's method to bound the roots of/as closely as we require. We remark that this method was also used in [2] to compute a good upper bound of the measure of a polynomial. If we apply Graeffe's method t o / we obtain a polynomial fx whose roots are the squares of the roots of ƒ This process can be repeated, to obtain fn, whose roots are the 2\"-th powers of the roots of the polynomial/. The computation is very easy : suppose that then fl + ](X)=g (X)-Xh(X). If we apply (*) to the polynomial ƒ„ we get MAN Modélisation mathématique et Analyse numérique Mathematical Modelling and Numerical Analysis ON FINDING THE LARGEST ROOT OF A POLYNOMIAL 695 and the last term tends rapidly to p as n increases (n = max {3, [Log k]} gives a very accurate upper bound). As previously remarked, one would prefer an explicit bound. Using (**) instead of (*), we get the same behaviour : rapid convergence for small n. More precisely, we have Conclusion : A few itérations of Graeffe's root-squaring method followed by an application of Knuth's inequality will give a very tight bound for the absolute value of the roots of a polynomial, with comparatively little effort.