# 施图姆定理

## 标准施图姆序列

${\displaystyle X=a_{n}x^{n}+\ldots +a_{1}x+a_{0}.}$

${\displaystyle {\begin{matrix}X_{2}&=&-{\rm {rem}}(X,X_{1})\\X_{3}&=&-{\rm {rem}}(X_{1},X_{2})\\&\vdots &\\0&=&-{\rm {rem}}(X_{r-1},X_{r}),\end{matrix}}}$

${\displaystyle X,X_{1},X_{2},\ldots ,X_{r}.\,}$

## 表述

${\displaystyle V_{\xi }}$为以下序列中符号变化的次数（零不计算在内）：

${\displaystyle X(\xi ),X_{1}(\xi ),X_{2}(\xi ),\ldots ,X_{r}(\xi ),\,\!}$

## 应用

${\displaystyle M=1+{\frac {\max _{i=0}^{n-1}|a_{i}|}{|a_{n}|}}.\,\!}$

${\displaystyle P(x)=a_{n}x^{n}+\cdots \,\!}$

${\displaystyle \operatorname {sgn}(a_{n})}$，而${\displaystyle \operatorname {sgn}(P(-x))}$则是${\displaystyle \operatorname {sgn}((-1)^{n}a_{n})}$

## 一般的施图姆序列

${\displaystyle [a,b]}$ 上的施图姆序列，是实系数多项式 ${\displaystyle X}$ 的一个有限序列${\displaystyle X_{0},X_{1},\ldots ,X_{r}}$，使得：

1. ${\displaystyle X_{r}}$${\displaystyle [a,b]}$ 上没有根
2. ${\displaystyle X_{0}(a)X_{0}(b)\neq 0}$
3. 如果对于${\displaystyle \xi \in [a,b],1\leq i\leq r-1,X_{i}(\xi )=0}$，那么${\displaystyle X_{i-1}(\xi )X_{i+1}(\xi )<0}$
4. 若对于${\displaystyle \xi \in [a,b],X(\xi )=0}$ ,则存在${\displaystyle \delta >0}$,使得 ${\displaystyle c\in (\xi -\delta ,\xi )}$时，${\displaystyle X_{0}(c)X_{1}(c)<0}$${\displaystyle c\in (\xi ,\xi +\delta )}$${\displaystyle X_{0}(c)X_{1}(c)>0}$

## 參考資料

• D.G. Hook and P.R. McAree, "Using Sturm Sequences To Bracket Real Roots of Polynomial Equations" in Graphic Gems I (A. Glassner ed.), Academic Press, p. 416-422, 1990.