3.4 KiB
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 quantityy
with maximum error\Delta_y
forn \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
withy
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
withy
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
withy
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.