# Tolerances

### From Glossary

Small positive values to control elements of a computer implementation of an algorithm. When
determining whether a value, is non-negative, the actual test is where is an *absolute tolerance*. When comparing two values to determine if the actual test is

where is the absolute tolerance (as above), and is the *relative tolerance* (some make the relative deviation depend on as well as on such as the sum of magnitudes,
). Most MPSs have a tolerance for every action it takes during its progression; in particular, one zero tolerance is not enough - one to test feasibility is usually less than one that is used to determine an acceptable pivot element. In fact, the use of tolerances is a crucial part of an MPS, including any presolve that would fix a variable when its upper and lower bounds are "sufficiently close" (i.e., within some tolerance). A tolerance is *dynamic* if it can change during the algorithm. An example is that a high tolerance might be used for line search early in an algorithm, reducing it as the sequence gets close to an optimal solution. The Nelder-Mead simplex method illustrates how tolerances might change up and down during the algorithm.

Another typical tolerance test applies to residuals to determine if is a solution to In this case, the residal is and the test has the form:

where is some norm. Further details are in the note on Tolerances.