04/20/2013

19 days ago

79 commits

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 `ɛ`

such that `ɛ*ɛ=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*ɛ`

, where `x`

and `y`

are
real numbers. Duality can further extend complex numbers by adjoining one new
element to each of the real and imaginary parts.

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`

,`realpart`

,`dualpart`

,`epsilon`

,`isdual`

,`dual_show`

,`conjdual`

,`absdual`

,`abs2dual`

.

The dual number `f(a+bɛ)`

is defined by the limit:

```
f(a+bɛ) := f(a) + lim_{h→0} (f(a + bɛh) - f(a))/h .
```

For complex differentiable functions, this is equivalent to differentiation:

```
f(a+bɛ) := f(a) + b f'(a) ɛ.
```

For functions that are not complex differentiable, the dual part returns the limit
and can be identified with a directional derivative in `R²`

.

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`

. The mathematical
definitions are available using the functions with the suffix `dual`

.
Similarly, comparison operators `<`

, `>`

, and `==`

are overloaded to compare only value
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()`

constructor to define the dual number `2+1*ɛ`

:

```
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 `realpart`

and `dualpart`

to get the concrete and dual
parts of `x`

, respectively:

```
println("f(x) = x^3")
println("f(2) = ", realpart(y))
println("f'(2) = ", dualpart(y))
```