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*Image credit: "xkcd: Error Bars" (CC-BY-NC
2.5)*

Physical measures are typically reported with an error, a quantification of the uncertainty of the accuracy of the measurement. Whenever you perform mathematical operations involving these quantities you have also to propagate the uncertainty, so that the resulting number will also have an attached error to quantify the confidence about its accuracy. Measurements.jl relieves you from the hassle of propagating uncertainties coming from physical measurements, when performing mathematical operations involving them. The linear error propagation theory is employed to propagate the errors.

This library is written in Julia, a modern high-level, high-performance dynamic programming language designed for technical computing.

When used in the Julia interactive session, it can serve also as an easy-to-use calculator.

- Support for most mathematical operations available in Julia standard library
and special functions
from
`SpecialFunctions.jl`

package, involving real and complex numbers. All existing functions that accept`AbstractFloat`

(and`Complex{AbstractFloat}`

as well) arguments and internally use already supported functions can in turn perform calculations involving numbers with uncertainties without being redefined. This greatly enhances the power of`Measurements.jl`

without effort for the users - Functional correlation between variables is correctly handled, so
`x-x ≈ zero(x)`

,`x/x ≈ one(x)`

,`tan(x) ≈ sin(x)/cos(x)`

,`cis(x) ≈ exp(im*x)`

, etc... - Support for arbitrary precision (also called multiple precision) numbers with uncertainties. This is useful for measurements with very low relative error
- Define arrays of measurements and perform calculations with them. Some linear algebra functions work out-of-the-box
- Propagate uncertainty for any function of real arguments (including functions
based on
C/Fortran calls),
using
`@uncertain`

macro - Function to get the derivative and the gradient of an expression with respect to one or more independent measurements
- Functions to calculate standard score and weighted mean
- Parse strings to create measurement objects
- Easy way to attach the uncertainty to a number using the
`±`

sign as infix operator. This syntactic sugar makes the code more readable and visually appealing - Extensible in combination with external packages: you can propagate errors of
measurements with their physical units, perform numerical integration
with
`QuadGK.jl`

, numerical and automatic differentiation, and much more. - Integration with
`Plots.jl`

.

The method used to handle functional correlation is described in this paper:

- M. Giordano, 2016, "Uncertainty propagation with functionally correlated
quantities", arXiv:1610.08716
(Bibcode:
`2016arXiv161008716G`

)

A current limitation of the package is that it is not yet possible to define quantities related by a correlation matrix.

If you use use this package for your research, please cite it.

The complete manual of `Measurements.jl`

is available at
https://juliaphysics.github.io/Measurements.jl/stable/. There, people
interested in the details of the package, in order integrate the package in
their workflow, can find a technical appendix explaining how the package
internally works.

The latest version of `Measurements.jl`

is available for Julia v1.0 and later
releases, and can be installed with Julia built-in package
manager. In a Julia session, after
entering the package manager mode with `]`

, run the command

```
pkg> update
pkg> add Measurements
```

Older versions of this package are also available for Julia 0.4-0.7.

After installing the package, you can start using it with

`using Measurements`

The module defines a new `Measurement`

data type. `Measurement`

objects can be
created with the two following constructors:

```
measurement(value, uncertainty)
value ± uncertainty
```

where

`value`

is the nominal value of the measurement`uncertainty`

is its uncertainty, assumed to be a standard deviation.

Here is a quick taster of the functionalities of the package:

```
julia> using Measurements
julia> a = measurement(4.5, 0.1)
4.5 ± 0.1
julia> b = 3.8 ± 0.4
3.8 ± 0.4
julia> 2a + b
12.8 ± 0.4472135954999579
julia> x = 8.4 ± 0.7
julia> x - x
0.0 ± 0.0
julia> x/x
1.0 ± 0.0
julia> x*x*x - x^3
0.0 ± 0.0
julia> sin(x)/cos(x) - tan(x)
-2.220446049250313e-16 ± 0.0 # They are equal within numerical accuracy
```

For more details about the use of the package read the documentation, in particular the Usage and Examples sections.

The `Measurements.jl`

package is licensed under the MIT "Expat" License. The
original author is Mosè Giordano.

Please, cite the paper Giordano 2016 (http://arxiv.org/abs/1610.08716) if you
employ this package in your research work. For your convenience, a BibTeX entry
is provided in the `CITATION.bib`

file.