C.16.IR:4.3 - Obtain values or bounds without losing alternatives

Choose the lightest adequate computation. Solve a small equation, eliminate an influential variable, enumerate a finite domain, or obtain a bound. A numerical formulation can use C.29.2 when its construction is nontrivial.

During elimination, preserve the conditions of each operation. Dividing by an unknown expression can discard its zero case. Squaring an equation can admit additional roots. Test the retained cases in the original relation and domains. Finding one root establishes a possible case; uniqueness requires an argument covering the admitted domain or a sufficient justified restriction.

For several unknowns, obtain the sought value from each compatible joint case. The other unknowns can remain unresolved when every compatible case gives the same sought answer. Section :5.1 gives a source voltage determined while its internal resistance remains unknown.

When a complete solution is costly, distinguish two useful computational results:

• A feasible example is one case that satisfies the original relations and conditions. Two such cases can establish a consequential ambiguity.
• An outer bound contains every compatible sought value but can also contain values that no case realizes. If the whole bound satisfies an inequality, that inequality holds for every compatible value. A bound spanning both sides of a threshold alone does not establish that both outcomes are possible.

For example, let an ideal indication obey r = x² with x ≥ 0, and let r = 9. A computation that has retained only the outer bound 0 ≤ x ≤ 4 leaves values on both sides of the threshold x = 2. The original relation, however, admits only x = 3. Thus x > 2 is resolved, and no feasible witness with x ≤ 2 exists. The coarse bound left the calculation unfinished; it did not establish ambiguity in the indication.

Use a validated enclosure method when the conclusion depends on retaining every solution through numerical calculation. Ordinary sampling can discover a counterexample but can miss another branch. When computation has not established existence, completeness or a needed bound, retain that limitation in the answer instead of interpreting solver termination as the missing result.