迭代算法终止准则

Clarifying iterative condition

The user asks about a condition in iterative methods where $(x_{k+1} - x_k)/ x_k < 0.01$, likely referring to a relative error or convergence criterion.

Pinpointing a term

I’m thinking through various terms like "relative difference," "relative error," or "terminating condition" to describe the iterative algorithm's convergence criterion.

Thought about 迭代算法收敛条件，持续 6 秒

在数值迭代方法中，$\dfrac{x_{k+1} - x_k}{x_k} < 0.01$ 这一形式通常被称为“相对误差小于 1%”或“相对变化量（relative difference）小于 1%”的终止判据/收敛准则。

• 绝对误差判据一般指 $\bigl|x_{k+1} - x_k\bigr| < \varepsilon$。
• 相对误差判据指 $\dfrac{\bigl|x_{k+1} - x_k\bigr|}{\lvert x_k\rvert} < \varepsilon$。

这里的 0.01 就是设定的允许误差（公差），即相对误差小于 1%。当迭代过程中满足此条件时，我们通常就可以认为算法已经达到所需的精度而停止。