mathematics-physics-wiki/docs/en/physics/mathematical-physics/error-analysis/maximum-error.md

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.

??? note "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.

??? note "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.

??? note "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.

??? note "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.