Document Type

Conference Paper

Rights

Available under a Creative Commons Attribution Non-Commercial Share Alike 4.0 International Licence

Disciplines

1.1 MATHEMATICS

Abstract

The presented analysis determines several new bounds on the real roots of the equation anxn+an−1xn−1+⋯+a0=0 (with an>0). All proposed new bounds are lower than the Cauchy bound max{1,∑j=0n−1|aj/an|}. Firstly, the Cauchy bound formula is derived by presenting it in a new light – through a recursion. It is shown that this recursion could be exited at earlier stages and, the earlier the recursion is terminated, the lower the resulting root bound will be. Following a separate analysis, it is further demonstrated that a significantly lower root bound can be found if the summation in the Cauchy bound formula is made not over each one of the coefficients a0,a1,…,an−1, but only over the negative ones. The sharpest root bound in this line of analysis is shown to be the larger of 1 and the sum of the absolute values of all negative coefficients of the equation divided by the largest positive coefficient. The following bounds are also found in this paper: max{1,(∑j=1qBj/Al)1/(l−k)}, where B1,B2,…,Bq are the absolute values of all of the negative coefficients in the equation, k is the highest degree of a monomial with a negative coefficient, Al is the positive coefficient of the term Alxl for which k

DOI

https://doi.org/10.7546/CRABS.2022.02.02

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