MATH.21:4.5 - Obtain a sufficient finite result

Translate the receiving request into a condition on a stage. If an established bound gives d(x_n,x)≤e_n, a stage with e_n within the requested tolerance supplies the approximation. For nested numerical intervals, the midpoint has error at most half their width. For compatible prefixes, a stage containing all requested entries supplies them.

When execution must find that stage, provide either an obtaining rule or a recognizable stopping condition with a reason it will be reached. An explicit rate of convergence is one option. An enclosure whose width can be checked during refinement may suffice without a rate fixed in advance.

A residual is useful only through its established relation to the requested error. A small equation residual can accompany a large error in the unknown; the comparison needed by the use must connect them.

Return the approximation with the mathematical bound or settled observation it supplies. If only existence has been established, retain that result and identify the missing obtaining procedure for a computational continuation. Numerical conditioning, finite arithmetic and execution cost belong to that continuation.