# Measurements.jl

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## Introduction

### What Is This Package Useful For?

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

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.

### Features List

- 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.

### Documentation

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 can find a technical appendix explaining how the package
internally works.

## Installation

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.

## Usage

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.

They are both subtype of `AbstractFloat`

. Some keyboard layouts provide an easy
way to type the `±`

sign, if your does not, remember you can insert it in Julia
REPL with `\pm`

followed by `TAB`

key. You can provide `value`

and
`uncertainty`

of any subtype of `Real`

that can be converted to `AbstractFloat`

.
Thus, `measurement(42, 33//12)`

and `pi ± 0.1`

are valid.

`measurement(value)`

creates a `Measurement`

object with zero uncertainty, like
mathematical constants. See below for further examples.

Every time you use one of the constructors above, you define a *new independent*
measurement. Instead, when you perform mathematical operations involving
`Measurement`

objects you create a quantity that is not independent, but rather
depends on really independent measurements.

Most mathematical operations are instructed, by
operator overloading, to
accept `Measurement`

type, and uncertainty is calculated exactly using analityc
expressions of functions’ derivatives.

In addition, it is possible to create a `Complex`

measurement with
`complex(measurement(a, b), measurement(c, d))`

.

`measurement(string)`

`measurement`

function has also a method that enables you to create a
`Measurement`

object from a string.

This module extends many methods defined in Julia’s mathematical standard
library, and some methods from widespread third-party packages as well. This is
the case for most special functions
in `SpecialFunctions.jl`

package, and the `quadgk`

integration routine
from `QuadGK.jl`

package. See the
full manual for details.

## Examples

```
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
```

## License

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 convience, a BibTeX entry
is provided in the `CITATION.bib`

file.