论文标题
实际多项式根的新界限
New Bounds on the Real Polynomial Roots
论文作者
论文摘要
提出的分析确定了方程根的几个新界限，$ a_n x^n + a_ {n-1} x^{n-1} + \ cdots + a_0 = 0 $（使用$ a_n> 0 $）。所有提出的新边界都低于cauchy绑定最大$ \ {1，\ sum_ {j = 0}^{n-1} | a_j/a_n | \} $。首先，通过递归以新的方式呈现Cauchy Bound公式是通过以新的方式呈现的。结果表明，该递归可以在较早的阶段退出，并且递归终止越早，所得的根界限就越低。经过单独的分析，进一步证明，如果在cauchy绑定公式中的求和不超过每个系数$ a_0，a_1，\ ldots，a_ {n-1} $，而是仅在负值的情况下，则可以找到明显较低的根结合。在这种分析线中，最鲜明的根部结合被证明是1的较大，而方程式所有负系数的绝对值之和除以最大的正系数。本文还可以找到以下范围：最大$ \ {1，（\ sum_ {j = 1}^{q} b_j/a_l）^{1/（l-k）} $，其中$ b_1，b_1，b_1，b_2，\ ldots b_q $是所有等级的绝对值，是$ iS $ a $ a $ a $ a $ a $ a $ a $ k k $ k k a $ k k a $ k a $ k系数，$ a_l $是$ k <l \ le n $的术语$ a_l x^l $的正系数。
The presented analysis determines several new bounds on the roots of the equation $a_n x^n + a_{n-1} x^{n-1} + \cdots + a_0 = 0$ (with $a_n > 0$). All proposed new bounds are lower than the Cauchy bound max$\{1, \sum_{j=0}^{n-1} |a_j/a_n| \}$. 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 $a_0, a_1, \ldots, a_{n-1}$, 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$\{ 1, ( \sum_{j = 1}^{q} B_j/A_l )^{1/(l-k)}\}$, where $B_1, B_2, \ldots B_q$ 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, $A_l$ is the positive coefficient of the term $A_l x^l$ for which $k< l \le n$.
