参数三次方程实根的孤立区间与改进的完全根分类
Isolation Intervals of the Real Roots of the Parametric Cubic Equation and Improved Complete Root Classification
摘要：
利用实符号一次三次多项式$p(x) = x^3 + a x^2 + b x + c\,\,$的系数的简单函数（如：$-a$，$-a/3$，$-c/b$，当$b$为负时，$\pm \sqrt{-b}$）和系数也是三次多项式系数的简单函数的辅助二次方程的根，求出了其实根的隔离区间.所有可能的情况都以清晰和非常详细的图表提出。对于任何一个给定的三次方程，很容易识别出这些图中哪一个是相关的，并从中读出方程的实根的孤立区间。一个大大改进的完整的根分类，解决的迹象（连同给予隔离间隔）的个别根，也提出了。没有使用数值近似或寻根技术。本文不考虑三次方程的判别式，而是将存在一个或三个实根的判别式作为由辅助二次方程的根得到的三次方程系数的条件。它还表明，如果一个三次方程有三个实根，那么这些谎言在一个区间$I$，使$\sqrt{3}\sqrt{a^2/3 - b} \le I \le 2 \sqrt{a^2/3 - b}$，独立于$c$。文中还给出了应用三次根隔离法的详细算法，并通过实例加以说明，包括对与Rayleigh弹性波相联系的三次方程进行全面的数学分析，并求出其实根的隔离区间。
英文摘要：
The isolation intervals of the real roots of the real symbolic monic cubic polynomial $p(x) = x^3 + a x^2 + b x + c\,\,$ are found in terms of simple functions of the coefficients of the polynomial (such as: $-a$, $-a/3$, $-c/b$, $\pm \sqrt{-b}$, when $b$ is negative), and the roots of some auxiliary quadratic equations whose coefficients are also simple functions of the coefficients of the cubic. All possible cases are presented with clear and very detailed diagrams. It is very easy to identify which of these diagrams is the relevant one for any given cubic equation and to read from it the isolation intervals of the real roots of the equation. A much-improved complete root classification, addressing the signs (together with giving the isolation intervals) of the individual roots, is also presented. No numerical approximations or root finding techniques are used. Instead of considering the discriminant of the cubic, criterion for the existence of a single real root or three real roots is found as conditions on the coefficients of the cubic, resulting from the roots of the auxiliary quadratic equations. It is also shown that, if a cubic equation has three real roots, then these lie in an interval $I$ such that $\sqrt{3}\sqrt{a^2/3 - b} \le I \le 2 \sqrt{a^2/3 - b}$, independent of $c$. A detailed algorithm for applying the method for isolation of the roots of the cubic is also given and it is illustrated through examples, including the full mathematical analysis of the cubic equation associated with the Rayleigh elastic waves and finding the isolation intervals of its real roots.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
