Error estimation for indirect measurements is exponentially hard

For many physical quantities, it is impossible or too costly to measure them directly. In such cases, we measure whatever quantities x1, ..., xn we can, and then we use the measurement results x̃1, ..., x̃n to estimate the value of the desired quantity y. Namely, as an estimate, we use ỹ = f(x̃1, ..., x̃n), where an algorithm f describes how the value y is related to xi. This procedure is called an indirect measurement. Measurements of xi cannot be absolutely precise. The resulting measurement errors ∆xi = x̃i−xi cause the estimate ỹ to differ from the actual value y. How to estimate this difference (i.e., the error of the indirect measurement)? Several error estimation algorithms are known for the case when f describes the exact relationship between xi and y, and the only sources of error are the errors of direct measurements (i.e., ∆xi). In general, the problem of calculating the largest possible value of ∆y = ỹ− y is proved to be intractable (NP-hard); however, in the realistic case when we can neglect squares of errors ∆xi, we can use a simple and fast (linear-time) algorithm for error estimation. In many real-life situations, we know f only approximately. We show that for such situations, the problem of estimating the exact bound for ∆y is intractable even when we can neglect the squares of errors ∆xi.