def f(x,y):
# We can call other user defined functions
z = pow(x,5)
# and return more than one parameter
return z + x*math.sqrt(y), x+y
x,y = 3,6
print "value: ", f(x,y)
der = adolpy.derivate(f)
derivative = der(x,y)
print
print "Automatic derivatives"
for i in derivative:
print i.dot
print "Manually computed jacobian"
print [5 * x**4 + math.sqrt(y), x/(2*math.sqrt(y))]
print [1,1]
step = 1e-5
print "Numerically computed by central differences for step=", step
print [(f(x+step,y)[0] - f(x-step,y)[0])/(2*step), (f(x,y+step)[0] - f(x, y-step)[0])/(2*step)]
print [(f(x+step,y)[1] - f(x-step,y)[1])/(2*step), (f(x,y+step)[1] - f(x, y-step)[1])/(2*step)]
print
print "Now, a function defined by parts (huber loss), delta = 3"
print "For a < delta, huber(a) = .5 * a**2"
print "For a = delta, huber(a) = .5 * a**2 == delta * (abs(a) - delta/2)"
print "For a > delta, huber(a) = delta * (abs(a) - delta/2)"