Stability of a Numerical Method

Topics: Numerical Analysis - Numerical Method

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Any given a numerical method can be said to be stable or non-stable.

Let $E_n$ denote the growth of the error after $n$ subsequent operations, and $\xi$ denote the error in any given step of the calculations.

Then:

• If $|E(n)|\approx C_n\xi$, we say that the error growth is linear ($C$ being a constant that’s independent from $n$)
  • We call a method with a linear error growth a stable method
• If $|E(n)|\approx k^n\xi$ for a given $k>1$, the growth of the error is exponential
  • We call a method with an exponential error growth an unstable method

Things to Consider

Most of the time, a linear error growth is inevitable. When $C$ and $\xi$ are small, the results of the method are generally acceptable.

An exponential error growth should be avoided, since $k^n$ will be big even for small values of $n$. This results in unacceptable inaccuracies.