Residual (mathematics)

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In mathematics, a residual set is the complement of a meager set. A meager set is one which is the countable union of nowhere dense sets.

Also, loosely speaking, a residual is the error in a result. To be precise, suppose we want to find x such that

f(x)=b.\,

Given an approximation of x0 of x, the residual is

b-f(x_0)\,

whereas the error is

x_0 - x.\,

If we do not know x, we cannot compute the error but we can compute the residual.

Residuals appear in many areas in mathematics, from iterative solvers such as the generalized minimal residual method, which seeks solutions to equations by systematically minimizing the residual, to statistics.

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