如果在各个步骤中它不会过度放大数据上的错误，您可以定义一个稳定的方法/算法。

所以你引入了一些有界估计。请注意，这种由稳定性检查监控的错误与由问题的病态发生的错误不同，即在稳定性中，您检查算法本身传播的错误。

另一个需要注意的是稳定性与问题有关，例如，相同的方法对于一个问题是稳定的，对于不同的问题是不稳定的，参见 [1] 的第 1.16 章。

作为参考可以是[1]。Kendal 的书 [2] 涵盖了不同的论点，您可以找到相对稳定性证明（我推荐它是因为我喜欢它）。一般来说，关于特定论点的书籍已经为所涵盖的方法提供了稳定性证明。

[1] Nicholas J. Higham，MR 1927606数值算法的准确性和稳定性，ISBN：0-89871-521-0。

[2] 阿特金森、肯德尔和韩伟民。理论数值分析。卷。39. 柏林：斯普林格，2005 年。

如果在数值方法中，解$\phi$在迭代级别$n+1$是（谁）给的，

$B_1 \phi^{n+1}=B_2 \phi^n +f$

然后对于初始假设扰动$\delta^0$

$B_1 \delta^{n+1}=B_2 \delta^n$

Then the numerical method is stable if for all $\delta^0$, $\delta^n$ remains bounded as $n\rightarrow \infty$

You got two questions. I can answer the first one about real coding and design.

The stability is for your algorithm. If the algorithm is not sensitive to the approximations made at every step, then the stability is implied.

Consider an example:

Suppose you have a lot of steps (iterations) to deal your data, and you make an approximate to the result of every step. At the end of your algorithm, you will get a huge deviation through enough steps. Especially in the iteration process.

This kind of error produced by your algorithm has nothing to do with your data. The deviation is acceptable if it is under your control, thus the algorithm is stable. Here is some advice to avoid such problems in your experiment:

1. Deduce calculation times.

2. Avoid addition of a small number to a big number.

3. Avoid subtraction of two numbers of similar magnitude

4. Avoid division by a small number and multiplication by a large number.