The Graeffe Process as Applied to Power Series
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چکیده
Of the many methods which have been proposed for solving algebraic equations the most practical one, where complex roots are concerned, is the well known "root-squaring" method usually referred to as the Graeffe1 process. Nearly all accounts of the method show by examples how all the roots of an equation may be found by first finding all the absolute values of the roots, and later by various devices their arguments. The problem of finding the absolute values of the roots is comparatively simple and can be extended (with certain practical difficulties) to the case of the zeros of entire functions. This step was made early in the history of the problem by Euler2 in generalizing Bernoulli's method to determine the first few zeros of the Bessel function J0(z). More recent accounts of the problem of finding the absolute values of the zeros of entire functions have been given by Ostrowski1 and Polya.3 The problem of finding by the Graeffe process the arguments of the zeros of entire functions (when these zeros are complex) apparently has not been attempted. At first this may seem a little strange since the Graeffe process owes its popularity to the fact that it is so successful in finding the complex roots of algebraic equations. The explanation is that the various devices, referred to above, for finding the arguments of the complex roots depend on an equation which has a finite number of roots. There is however one modification of the Graeffe process for polynomials, given more than a score of years ago by Brodetsky & Smeal4 and apparently overlooked by recent writers, in which a pair of complex roots may be found without having to find any other roots. The purpose of this note is to call attention to the fact that this process may be extended to entire functions and their complex zeros. As mentioned above, accounts of the Graeffe process are nearly always given in terms of examples. To be sure, the "general ideas" of the process are also set forth, but I have yet to see a writer admit that the Graeffe process fails utterly in such simple cases as the equation
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The Graeffe Process as Applied to Power Series
Of the many methods which have been proposed for solving algebraic equations the most practical one, where complex roots are concerned, is the well known "root-squaring" method usually referred to as the Graeffe1 process. Nearly all accounts of the method show by examples how all the roots of an equation may be found by first finding all the absolute values of the roots, and later by various de...
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