Gaussian elimination with partial pivoting is * almost always*
numerically stable, so the error bound one expects from
solving **Ax=b** this way is

The notation means something close to
. In practice, one sees errors as large
as or so, where **n** is the dimension of **A**.
So for example, if and
we compute in IEEE standard
single precision, we can expect to get 3 correct decimal
places in the answer, since .

We can only say that Gaussian elimination with partial pivoting is
almost always numerically stable, because matrices **A** do exist where
is as large as
. Since grows very quickly with **n**, the error
bound rapidly becomes enormous (and the actual solution also becomes
poor). These examples are found in numerical analysis textbooks
[4,5,6], but
not in practice.

Rather than formally proving that Gaussian elimination with
partial pivoting is generally stable, let us instead illustrate
how omitting pivoting can destroy stability. Let us apply
Gaussian elimination without pivoting to compute the factorization **A=LU** of

To keep the example simple, we will use
3 decimal digit floating point arithmetic. Note that
,
so **A** is well conditioned and we should expect to be able to solve **Ax=b**
accurately. We will use the notation
to mean the
rounded, floating point result of , where **op** is one of the operations **+**, **-**,
and .