The example below demonstrates basic usage of hyperdual numbers by employing them to perform automatic differentiation. The code for this example can be found in
test/runtests.jl.
First install the package by using the Julia package manager:
Pkg.add("HyperDualNumbers")
Then make the package available via
using HyperDualNumbers
Use the Hyper() function to define a hyperdual number, e.g.:
hd0 = Hyper()
hd1 = Hyper(1.0)
hd2 = Hyper(3.0, 1.0, 1.0, 0.0)
hd3 = Hyper(3//2, 1//1, 1//1,0//1)
Let's say we want to calculate the first and second derivative of
f(x) = ℯ^x / (sqrt(sin(x)^3 + cos(x)^3))
To calculate these derivatives at a location x, evaluate your function at hyper(x, 1.0, 1.0, 0.0). For example:
t0 = Hyper(1.5, 1.0, 1.0, 0.0)
y = f(t0)
For this example, you'll get the result
4.497780053946162 + 4.053427893898621ϵ1 + 4.053427893898621ϵ2 + 9.463073681596601ϵ1ϵ2
The first term is the function value, the coefficients of both ϵ1 and ϵ2 (which correspond to the second and third arguments of hyper) are equal to the first derivative, and the coefficient of ϵ1ϵ2 is the second derivative.
You can extract these coefficients from the hyperdual number using the functions realpart(), ε₁part() or ε₂part() and ε₁ε₂part():
println("f(1.5) = ", f(1.5))
println("f(t0) = ", realpart(f(t0)))
println("f'(t0) = ", ε₁part(f(t0)))
println("f'(t0) = ", ε₂part(f(t0)))
println("f''(t0) = ", ε₁ε₂part(f(t0)))