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.
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.
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))