Scope of DualNumbers.jl

The DualNumbers package defines the Dual type to represent dual numbers and supports standard mathematical operations on them. Conversions and promotions are defined to allow performing operations on combinations of dual numbers with predefined Julia numeric types.

Dual numbers extend the real numbers, similar to complex numbers. They adjoin a new element ϵ such that ϵ*ϵ=0, in a similar way that complex numbers adjoin the imaginary unit i with the property i*i=-1. So the typical representation of a dual number takes the form x+y*ϵ, where x and y are real numbers.

Apart from their mathematical role in algebraic and differential geometry (they are mainly interpreted as angles between lines), they also find applications in physics (the real part of a dual represents the bosonic direction, while the epsilon part represents the fermionic direction), in screw theory, in motor and spatial vector algebra, and in computer science due to its relation with the forward mode of automatic differentiation.

The ForwardDiff package implements forward mode automatic differentiation in Julia using several approaches. One of these approaches employs dual numbers. For this reason, the ForwardDiff package relies on DualNumbers. The user is referred to ForwardDiff for some examples on how to perform forward mode automatic differentiation using dual numbers in Julia.

Supported functions

We aim for complete support for Dual types for numerical functions within Julia's Base. Currently, basic mathematical operations and trigonometric functions are supported.

The following functions are specific to dual numbers:

• dual,
• dual128,
• dual64,
• epsilon,
• isdual,
• dual_show,
• conjdual,
• absdual,
• abs2dual.

In some cases the mathematical definition of functions of Dual numbers is in conflict with their use as a drop-in replacement for calculating numerical derivatives, for example, conj, abs and abs2. In these cases, we choose to follow the rule f(x::Dual) = Dual(f(real(x)),epsilon(x)*f'(real(x))), where f' is the derivative of f. The mathematical definitions are available using the functions with the suffix dual. Similarly, comparison operators <, >, and == are overloaded to compare only real components.

A walk-through example

The example below demonstrates basic usage of dual numbers by employing them to perform automatic differentiation. The code for this example can be found in test/automatic_differentiation_test.jl.

First install the package by using the Julia package manager:

Pkg.update()
Pkg.add("DualNumbers")

Then make the package available via

using DualNumbers

Use the dual() function to define the dual number 2+1*du:

x = dual(2, 1)

Define a function that will be differentiated, say

f(x) = x^3

Perform automatic differentiation by passing the dual number x as argument to f:

y = f(x)

Use the functions real() and epsilon() to get the real and imaginary (dual) parts of x, respectively:

println("f(x) = x^3")
println("f(2) = ", real(y))
println("f'(2) = ", epsilon(y))