On accurate asymptotic approximations of roots for polynomial equations containing a small, but fixed parameter

In this paper, polynomial equations with real coefficients and in one variable were considered which contained a small, positive but specified and fixed parameter ${\epsilon }_0\ne 0$. By using the classical asymptotic method, roots of the polynomial equations have been constructed in the literature, which were proved to be valid for sufficiently small $\epsilon$-values (or equivalently for $\epsilon \to 0$). In this paper, it was assumed that for some or all roots of a polynomial equation, the first few terms in a Taylor or Laurent series in a small parameter depending on $\epsilon$ exist and can be constructed. We also assumed that at least two approximations $x_1(\epsilon )$ and $x_2(\epsilon )$ for the real roots exist and can be constructed. For a complex root, we assumed that at least two real approximations $a_1(\epsilon )$ and $a_2(\epsilon )$ for the real part of this root, and that at least two real approximations $b_1(\epsilon )$ and $b_2(\epsilon )$ for the imaginary part of this root, exist and can be constructed. Usually it was not clear whether for $\epsilon ={\epsilon }_0$ the approximations were valid or not. It was shown in this paper how the classical asymptotic method in combination with the bisection method could be used to prove how accurate the constructed approximations of the roots were for a given interval in $\epsilon$ (usually including the specified and fixed value ${\epsilon }_0\ne 0$). The method was illustrated by studying a polynomial equation of degree five with a small but fixed parameter ${\epsilon }_0=0.1$. It was shown how (absolute and relative) error estimates for the real and imaginary parts of the roots could be obtained for all values of the small parameter in the interval $(0,{\epsilon }_0]$.