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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.
- 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 acceptAbstractFloat
(andComplex{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 ofMeasurements.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 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 measurementuncertainty
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.
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
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.