The maximum error

Determining the transformed maximum error

In this section a method will be postulated and derived under certain assumptions to determine the maximum error, after a transformation with a map \(f\).

Definition 1: let \(f: \mathbb{R}^n \to \mathbb{R} :(x_1 \pm \Delta_{x_1}, \dots, x_n \pm \Delta_{x_n}) \mapsto f(x_1 \pm \Delta_{x_1}, \dots, x_n \pm \Delta_{x_n}) \overset{.} = y \pm \Delta_y\) be a function that maps independent measurements with a corresponding maximum error to a new quantity \(y\) with maximum error \(\Delta_y\) for \(n \in \mathbb{N}\).

In assumption that the maximum errors of the independent measurements are small the following may be posed.

Postulate 1: let \(f:(x_1 \pm \Delta_{x_1}, \dots, x_n \pm \Delta_{x_n}) \mapsto f(x_1 \pm \Delta_{x_1}, \dots, x_n \pm \Delta_{x_n}) \overset{.} = y \pm \Delta_y\), the maximum error \(\Delta_y\) may be given by

\[ \Delta_y = \sum_{i=1}^n | \partial_i f(x_1, \dots, x_n) | \Delta_{x_i}, \]

and \(y = f(x_1, \dots, x_n)\) correspondingly for all \((x_1 \pm \Delta_{x_1}, \dots, x_n \pm \Delta_{x_n}) \in \mathbb{R}^n\).

Derivation:

Will be added later.

With this general expression the following properties may be derived.

Properties

The sum of the independently measured quantities is posed in the following corollary.

Corollary 1: let \(f:(x_1 \pm \Delta_{x_1}, \dots, x_n \pm \Delta_{x_n}) \mapsto f(x_1 \pm \Delta_{x_1}, \dots, x_n \pm \Delta_{x_n}) \overset{.} = y \pm \Delta_y\) with \(y\) given by

\[ y = f(x_1, \dots, x_n) = x_1 + \dots x_n, \]

then the maximum error \(\Delta_y\) may be given by

\[ \Delta_y = \Delta_{x_1} + \dots + \Delta_{x_n}, \]

for all \((x_1 \pm \Delta_{x_1}, \dots, x_n \pm \Delta_{x_n}) \in \mathbb{R}^n\).

Proof:

Will be added later.

The multiplication of a constant with the independently measured quantities is posed in the following corollary.

Corollary 2: let \(f:(x_1 \pm \Delta_{x_1}, \dots, x_n \pm \Delta_{x_n}) \mapsto f(x_1 \pm \Delta_{x_1}, \dots, x_n \pm \Delta_{x_n}) \overset{.} = y \pm \Delta_y\) with \(y\) given by

\[ y = f(x_1, \dots, x_n) = \lambda(x_1 + \dots x_n), \]

for \(\lambda \in \mathbb{R}\) then the maximum error \(\Delta_y\) may be given by

\[ \Delta_y = |\lambda| (\Delta_{x_1} + \dots + \Delta_{x_n}), \]

for all \((x_1 \pm \Delta_{x_1}, \dots, x_n \pm \Delta_{x_n}) \in \mathbb{R}^n\).

Proof:

Will be added later.

The product of two independently measured quantities is posed in the following corollary.

Corollary 3: let \(f: (x_1 \pm \Delta_{x_1}, x_2 \pm \Delta_{x_2}) \mapsto f(x_1 \pm \Delta_{x_1}, x_2 \pm \Delta_{x_2}) \overset{.} = y \pm \Delta_y\) with \(y\) given by

\[ y = f(x_1, x_2) = x_1 x_2, \]

then the maximum error \(\Delta_y\) may be given by

\[ \Delta_y = \frac{\Delta_{x_1}}{|x_1|} + \frac{\Delta_{x_2}}{|x_2|}, \]

for all \((x_1 \pm \Delta_{x_1}, x_2 \pm \Delta_{x_2}) \in \mathbb{R}^2\).

Proof:

Will be added later.

Combining measurements

If by a measurement series \(m \in \mathbb{N}\) values \(\{y_1 \pm \Delta_{y_1}, \dots, y_m \pm \Delta_{y_m}\}\) have been found for the same quantity then

\[ [y \pm \Delta_y] = \bigcap_{i \in \mathbb{N}[i \leq m]} [y_i \pm \Delta_{y_i}], \]

the overlap of all the intervals with \([y \pm \Delta_y]\) denoting the interval in which the real value exists.