Catastrophic cancellation

In numerical analysis, catastrophic cancellation is the phenomenon that subtracting good approximations to two nearby numbers may yield a very bad approximation to the difference of the original numbers.

For example, if there are two studs, one ${\displaystyle L_{1}=253.5\,{\text{cm}}}$ long and the other ${\displaystyle L_{2}=252.5\,{\text{cm}}}$ long, and they are measured with a ruler that is good only to the centimeter, then the approximations could come out to be ${\displaystyle {\tilde {L}}_{1}=254\,{\text{cm}}}$ and ${\displaystyle {\tilde {L}}_{2}=252\,{\text{cm}}}$. These may be good approximations, in relative error, to the true lengths: the approximations are in error by less than 2% of the true lengths, ${\displaystyle |L_{1}-{\tilde {L}}_{1}|/|L_{1}|<2\%}$.

However, if the approximate lengths are subtracted, the difference will be ${\displaystyle {\tilde {L}}_{1}-{\tilde {L}}_{2}=254\,{\text{cm}}-252\,{\text{cm}}=2\,{\text{cm}}}$, even though the true difference between the lengths is ${\displaystyle L_{1}-L_{2}=253.5\,{\text{cm}}-252.5\,{\text{cm}}=1\,{\text{cm}}}$. The difference of the approximations, ${\displaystyle 2\,{\text{cm}}}$, is in error by 100% of the magnitude of the difference of the true values, ${\displaystyle 1\,{\text{cm}}}$.

Catastrophic cancellation isn't affected by how large the inputs are—it applies just as much to large and small inputs. It depends only on how large the difference is, and on the error of the inputs. Exactly the same error would arise by subtracting ${\displaystyle 52\,{\text{cm}}}$ from ${\displaystyle 54\,{\text{cm}}}$ as approximations to ${\displaystyle 52.5\,{\text{cm}}}$ and ${\displaystyle 53.5\,{\text{cm}}}$, or by subtracting ${\displaystyle 2.00052\,{\text{km}}}$ from ${\displaystyle 2.00054\,{\text{km}}}$ as approximations to ${\displaystyle 2.000525\,{\text{km}}}$ and ${\displaystyle 2.000535\,{\text{km}}}$.

Catastrophic cancellation may happen even if the difference is computed exactly, as in the example above—it is not a property of any particular kind of arithmetic like floating-point arithmetic; rather, it is inherent to subtraction, when the inputs are approximations themselves. Indeed, in floating-point arithmetic, when the inputs are close enough, the floating-point difference is computed exactly, by the Sterbenz lemma—there is no rounding error introduced by the floating-point subtraction operation.