Numerical differentiation¶
The functions I have defined use the numdifftools package,see https://pypi.python.org/pypi/Numdifftools. The curl, divergence, and laplacian operators, are not explicitly a part of the package (they probably should be), but the package does return the Jacobian matrix of all first-order partial derivatives of a vector-valued function (https://en.wikipedia.org/wiki/Jacobian_matrix_and_determinant), and the Hessian matrix of second-order partial derivatives of a scalar-valued function (https://en.wikipedia.org/wiki/Hessian_matrix), from which it is easy to compute the desired operators.
Marty Ligare, September 2019
import numpy as np
import numdifftools as nd # See https://pypi.python.org/pypi/Numdifftools
import matplotlib as mpl
import matplotlib.pyplot as plt
# Following is an Ipython magic command that puts figures in the notebook.
%matplotlib notebook
plt.style.use('classic')
# M.L. modification of matplotlib defaults
# Changes can also be put in matplotlibrc file,
# or effected using mpl.rcParams[]
plt.rc('figure', figsize = (6, 4.5)) # Reduces overall size of figures
plt.rc('axes', labelsize=14, titlesize=14)
plt.rc('xtick', labelsize=12)
plt.rc('ytick', labelsize=12)
plt.rc('figure', autolayout = True) # Adjusts supblot parameters for new size
# Divergence of vector-valued function f evaluated at x
def div(f,x):
jac = nd.Jacobian(f)(x)
return jac[0,0] + jac[1,1] + jac[2,2]
# Gradient of scalar-valued function f evaluated at x
def grad(f,x):
return nd.Gradient(f)(x)
# Curl of vector field f evaluated at x
def curl(f,x):
jac = nd.Jacobian(f)(x)
return np.array([jac[2,1]-jac[1,2],jac[0,2]-jac[2,0],jac[1,0]-jac[0,1]])
# Laplacian of scalar field f evaluated at x
def laplacian(f,x):
hes = nd.Hessdiag(f)(x)
return sum(hes)
Example from documentation¶
plt.figure(1)
x = np.linspace(-2, 2, 100)
for i in range(10):
df = nd.Derivative(np.tanh, n=i)
y = df(x)
h = plt.plot(x, y/np.absolute(y).max())
Scalar function single variable: $\quad f(x) = 4x^3, \quad\left.\frac{df}{dx}\right|_{x_0},\quad\left.\frac{d^2f}{dx^2}\right|_{x_0}$¶
def f(x):
return 4*x**3
x0 = 3
d1 = nd.Derivative(f,n=1) # OR nd.Derivative(f)
d2 = nd.Derivative(f,n=2)
print(d1(x0),d2(x0))
Scalar field (3D): $\quad g({\bf r}) = x^2 + y^3 + 1\left.\quad\nabla g\right|_{{\bf r}_0}\quad \left.\nabla^2g\right|_{{\bf r}_0}$¶
def g(x):
return x[0]**2+x[1]**3+1
r0 = np.array([1,2,3])
grad(g,r0)
laplacian(g,r0)
Vector field (3D): $\quad {\bf v}({\bf r})\quad\nabla\cdot {\bf v}\quad \nabla\times{\bf v}$¶
def v1(x):
return np.array([x[1]**3,x[0]**2,x[0]])
def v2(x):
return np.array([x[0]**3*x[2],x[1],x[2]*x[1]])
r0 = np.array([1,2,3])
curl(v1,r0)
div(v1,r0), div(v2,r0)
Version Information¶
version_information is from J.R. Johansson (jrjohansson at gmail.com)
See Introduction to scientific computing with Python:
http://nbviewer.jupyter.org/github/jrjohansson/scientific-python-lectures/blob/master/Lecture-0-Scientific-Computing-with-Python.ipynb
for more information and instructions for package installation.
If version_information has been installed system wide (as it has been on Bucknell linux computers with shared file systems), continue with next cell as written. If not, comment out top line in next cell and uncomment the second line.
%load_ext version_information
version_information numpy, matplotlib, numdifftools