Nearcancellation

Nearcancellation is a term used in numerical analysis and scientific computing to describe a situation in which the sum or difference of two large, nearly equal quantities produces a much smaller result. This effect can lead to a disproportionate loss of precision when calculations are performed with finite-precision arithmetic, such as floating-point representations. It is closely related to catastrophic cancellation, but nearcancellation emphasizes the sensitivity that arises when significant digits cancel out rather than a complete breakdown of arithmetic.

The underlying mechanism is straightforward: when two numbers have almost the same magnitude and opposite signs (or nearly the same magnitude and the same sign but nearly equal), subtracting or adding them can leave a residual that is small compared to the inputs. In floating-point systems, the finite number of representable digits means many leading digits cancel, leaving a result whose relative error is large even if the inputs were stored exactly. The severity depends on the relative sizes of the quantities, the precision of the arithmetic, and the conditioning of the problem.

Examples arise in many contexts. A classic illustration is subtracting numbers like 1e14 and 1e14+1 in double precision, where the spacing between representable numbers near 1e14 is larger than 1, so the result may be indistinguishable from zero. Another common scenario is evaluating expressions with alternating large terms, such as sums or differences of large positive and negative contributions, where cancellation yields a small net value.

Mitigation strategies include reformulating the computation to avoid subtracting nearly equal terms, using compensated or higher-precision arithmetic, or applying numerically stable algorithms that minimize cancellation and preserve significant digits. See also: floating-point arithmetic, catastrophic cancellation, numerical stability.