计算函数零点的积分
An integral that counts the zeros of a function
摘要：
给定区间$[a,b]$上满足弱正则性条件的实函数$f$，通过计算某个积分来确定$f$的零点个数.被积函数依赖于$f, f'$和$f''$。特别地，通过在足够精细的网格上用梯形法则近似积分，我们可以通过评估$f,f'$和$f''$的许多值来计算$f$的零点的数量。一个变量的积分甚至允许确定的数量的零分解其多重性。
英文摘要：
Given a real function $f$ on an interval $[a,b]$ satisfying mild regularity conditions, we determine the number of zeros of $f$ by evaluating a certain integral. The integrand depends on $f, f'$ and $f''$. In particular, by approximating the integral with the trapezoidal rule on a fine enough grid, we can compute the number of zeros of $f$ by evaluating finitely many values of $f,f'$ and $f''$. A variant of the integral even allows to determine the number of the zeros broken down by their multiplicity.
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。
