HyperJet — Algorithmic Differentiation with Hyper-Dual numbers for Python and C++
A header-only library for algorithmic differentiation with hyper-dual numbers. Written in C++17 with an extensive Python interface.
Installation
pip install hyperjet
Quickstart
Import the module:
import hyperjet as hjCreate a set of variables e.g. x=3 and y=6:
x, y = hj.variables([3, 6])x and y are hyper-dual numbers. This is indicated by the postfix hj:
x
>>> 3hjGet the value as a simple float:
x.f
>>> 3The hyper-dual number stores the derivatives as a numpy array.
Get the first order derivatives (Gradient) of a hyper-dual number:
x.g # = [dx/dx, dx/dy]
>>> array([1., 0.])Get the second order derivatives (Hessian matrix):
x.hm() # = [[d^2 x/ dx**2 , d^2 x/(dx*dy)],
# [d^2 x/(dx*dy), d^2 x/ dy**2 ]]
>>> array([[0., 0.],
[0., 0.]])For a simple variable these derivatives are trivial.
Now do some computations:
f = (x * y) / (x - y)
f
>>> -6hjThe result is again a hyper-dual number.
Get the first order derivatives of f with respect to x and y:
f.g # = [df/dx, df/dy]
>>> array([-4., 1.])Get the second order derivatives of f:
f.hm() # = [[d^2 f/ dx**2 , d^2 f/(dx*dy)],
# [d^2 f/(dx*dy), d^2 f/ dy**2 ]]
>>> array([[-2.66666667, 1.33333333],
[ 1.33333333, -0.66666667]])You can use numpy to perform vector and matrix operations.
Compute the nomalized cross product of two vectors u = [1, 2, 2] and v = [4, 1, -1] with hyper-dual numbers:
import numpy as np
variables = hj.DDScalar.variables([1, 2, 2,
4, 1, -1])
u = np.array(variables[:3]) # = [1hj, 2hj, 2hj]
v = np.array(variables[3:]) # = [4hj, 1hj, -1hj]
normal = np.cross(u, v)
normal /= np.linalg.norm(normal)
normal
>>> array([-0.331042hj, 0.744845hj, -0.579324hj], dtype=object)The result is a three-dimensional numpy array containing hyper-dual numbers.
Get the value and derivatives of the x-component:
normal[0].f
>>> -0.3310423554409472
normal[0].g
>>> array([ 0.00453483, -0.01020336, 0.00793595, 0.07255723, -0.16325376, 0.12697515])
normal[0].hm()
>>> array([[ 0.00434846, -0.01091775, 0.00647611, -0.0029818 , -0.01143025, -0.02335746],
[-0.01091775, 0.02711578, -0.01655522, 0.00444165, 0.03081974, 0.04858632],
[ 0.00647611, -0.01655522, 0.0093492 , -0.00295074, -0.02510461, -0.03690759],
[-0.0029818 , 0.00444165, -0.00295074, -0.02956956, 0.03025289, -0.01546811],
[-0.01143025, 0.03081974, -0.02510461, 0.03025289, 0.01355789, -0.02868433],
[-0.02335746, 0.04858632, -0.03690759, -0.01546811, -0.02868433, 0.03641839]])Reference
If you use HyperJet, please refer to the official GitHub repository:
@misc{HyperJet,
author = "Thomas Oberbichler",
title = "HyperJet",
howpublished = "\url{http://github.com/oberbichler/HyperJet}",
}
