Python Automatic Differentiation and (Co)variance Calculation

pyderiv keeps the first and second derivatives with respect to all related variables in a Variable object. Variable objects are created with deriv.varying or noise.noise. They can then be manipulated like normal numbers, except for that when using mathematical functions, the replacements in deriv must be used (deriv.sin). A 'derivative' can be taken of a Variable object with deriv.d and the numerical value of any Variable object can be obtained with deriv.v. For noise, noise.E, noise.cov, noise.var, and noise.cov_matrix are easy to use.

Example (section of README.py)

r = (x * x + y * y) ** .5

print "r", deriv.v(r)
print "dr/dx", deriv.v(deriv.d(r, x))
print "dr/dy", deriv.v(deriv.d(r, y))
print "d2r/dx2", deriv.v(deriv.d(r, x, x))
print "d2r/dy2", deriv.v(deriv.d(r, y, y))
print "d2r/dx/dy", deriv.v(deriv.d(r, x, y))
print "d2r/dy/dx", deriv.v(deriv.d(r, y, x))

print
print "---"
print

x = -1 + noise.noise(3)
y = 2 + noise.noise(7)

r = x, y, x + y, x - y, x / y, x * x, x * y, y * y

print ' '.join("%7.03f" % noise.E(x) for x in r), "E"
print ' '.join("%7.03f" % noise.var(x) for x in r), "variance"
print
print "covariance matrix:"

for row in noise.cov_matrix(r):
    for col in row:
        print "%7.03f" % col,
    print

Result

r 8.60232526704
dr/dx 0.581238193719
dr/dy -0.813733471207
d2r/dx2 0.0769747878169
d2r/dy2 0.039272850927
d2r/dx/dy 0.0549819912978
d2r/dy/dx 0.0549819912978

---

 -1.000   2.000   1.000  -3.000  -1.375   4.000  -2.000  11.000 E
  3.000   7.000  10.000  10.000   4.031  30.000  40.000 210.000 variance

covariance matrix:
  3.000  -0.000   3.000   3.000   1.500  -6.000   6.000   0.000
 -0.000   7.000   7.000  -7.000   1.750  -0.000  -7.000  28.000
  3.000   7.000  10.000  -4.000   3.250  -6.000  -1.000  28.000
  3.000  -7.000  -4.000  10.000  -0.250  -6.000  13.000 -28.000
  1.500   1.750   3.250  -0.250   4.031  -3.000  -4.000  -5.250
 -6.000  -0.000  -6.000  -6.000  -3.000  30.000 -12.000   0.000
  6.000  -7.000  -1.000  13.000  -4.000 -12.000  40.000 -28.000
  0.000  28.000  28.000 -28.000  -5.250   0.000 -28.000 210.000