/usr/share/pyshared/brian/tools/autodiff.py is in python-brian 1.4.1-2.
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Automatic differentiation.
(C) R. Brette 2008
This file is released under the terms of the BSD licence
--------------------------------------------------------
Method: forward accumulation by operator overloading.
Use::
df=differentiate(f) # where f is a single-variable function
y=differentiate(f,x0) # differentiate at x=x0
y=differentiate(f,x0,order=2) # second-order derivative at x=x0
y=differentiate(f,(a,b),order=(1,2)) # d2f/dxdy(a,b)
y=differentiate(f,order=(1,2)) # d2f/dxdy
a0,a1,a2=taylor_series(f,x0,order=2) # 2nd order Taylor series of f at x0
J=gradient(f,(a,b)) # gradient of f at (a,b) (returns a list)
H=hessian(f,(a,b)) # hessian of f at (a,b) (returns a numpy array)
TODO:
* replace x by x0
* merge HigherDifferentiable in Differentiable?
* simplify code
* gradient, taylor_series, hessian: x0 should be optional
* iadd, etc. (more operators)
* NonDifferentiableException and maybe use this on conditions (when cmp=0)
* Differential operators: Laplacian...
* gradient, taylor_series, hessian: x0 should be optional
* test with units, functions Rp->something, alternative differential operators,
differential (f:Rp->Rn), complex numbers, arithmetic derivative
* remove most dependency with numpy (except Hessian)
* implicitly defined functions
If several partial derivatives of the same function need to be
calculated, it is faster to use the lower level object Differentiable.
Example::
y=f(Differentiable('x',3),Differentiable('y',2))
will return a Differentiable object with y.val=f(3,2)
and y.diff={'x': df/dx(3,2), 'y': df/dy(3,2)}
'''
from numpy import exp, sin, cos, log, zeros # todo: look in frame instead?
import types
__all__ = ['differentiate', 'Differentiable', 'taylor_series', 'gradient', 'hessian']
def differentiate(f, x=None, order=1):
'''
Calculate the derivative of f at point x.
Higher-order derivatives are possible.
Ex.: differentiate(lambda x:3*x*x+2,x=2,order=2)
(Returns: 6.0)
Partial derivatives:
1) Give a tuple with the order.
Ex.: differentiate(lambda x,y:x*y+2*y+1,x=(1,2),order=(0,1))
(Returns: 3.0 (partial derivative d/dy) )
2) Give a dictionnary with the order:
Ex.: differentiate(lambda x,y:x*y+2*y+1,x=(1,2),order={'y':1})
'''
if x is None:
return lambda x:differentiate(f, x, order)
if (type(order) == types.ListType) or (type(order) == types.TupleType): # several variables
# Build the list of arguments
n = sum(order) # total order
args = [HigherDifferentiable(i, x[i], n) for i in range(len(x))]
y = f(*args)
for i in xrange(len(x)):
for _ in xrange(order[i]):
if isinstance(y, Differentiable) and (i in y.diff):
y = y.diff[i]
else: # constant
return 0.
return y
elif (type(order) == types.DictType): # named variables
# Build the list of arguments
n = sum(order.itervalues())
vars = list(f.func_code.co_varnames)
args = []
args.extend(x)
for name in order.iterkeys():
i = vars.index(name)
args[i] = HigherDifferentiable(name, x[i], n)
y = f(*args)
for name, n in order.iteritems():
for j in range(n):
if isinstance(y, Differentiable) and (name in y.diff):
y = y.diff[name]
else: # constant
return 0.
return y
elif order == 0:
return f(x)
elif order == 1:
y = f(Differentiable('x', x))
if isinstance(y, Differentiable) and ('x' in y.diff):
return y.diff['x']
else: # constant
return 0.
else:
return differentiate(lambda y:differentiate(f, y, order=order - 1), x)
def taylor_series(f, x, order=1):
'''
Returns the list of coefficients of the Taylor Series of f
at x (scalar).
'''
y = f(HigherDifferentiable('x', x, order))
series = []
n = 1.
for i in range(order + 1):
# Value
val = y
while isinstance(val, Differentiable):
val = val.val
series.append(val / n)
n *= (i + 1)
# Next order
if isinstance(y, Differentiable) and ('x' in y.diff):
y = y.diff['x']
else:
y = 0.
return series
def gradient(f, x):
'''
Gradient of f at x (= tuple).
Result = list.
'''
args = [Differentiable(i, val=x[i]) for i in range(len(x))]
y = f(*args)
if isinstance(y, Differentiable):
result = []
for i in range(len(x)):
if i in y.diff:
result.append(y.diff[i])
else:
result.append(0.)
return result
else: # constant
return [0.] * len(x)
def hessian(f, x):
'''
Hessian of f at x (= tuple).
Result = array.
N.B.: not working with vectors.
'''
args = [HigherDifferentiable(i, x[i], 2) for i in range(len(x))]
y = f(*args)
result = zeros((len(x), len(x)))
if isinstance(y, Differentiable): # non-zero Hessian
for i in range(len(x)):
if i in y.diff and isinstance(y.diff[i], Differentiable):
for j in range(len(x)):
if j in y.diff[i].diff:
result[i, j] = y.diff[i].diff[j]
return result
class Differentiable(object):
'''
A differentiable variable.
Implemented operations: +,-,*,[],len,exp,sin,cos,/,abs,sqrt,comparisons,**
'''
def __init__(self, name=None, val=0.):
'''
Initializes a variable and its derivative.
If the name is None, then it is a constant.
'''
self.val = val # value
if name == None: # constant
self.diff = {} # derivative
else:
self.diff = {name:1.} # or jacobian? (eye)
def sqrt(self):
return self ** .5
def __cmp__(self, x):
if isinstance(x, Differentiable):
return cmp(self.val, x.val)
else:
return cmp(self.val, x)
def __abs__(self):
# NOT WORKING WITH VECTORS
if self.val == 0:
# TODO: specific exception
raise ZeroDivisionError, "x:abs(x) is not differentiable at x=0."
elif self.val > 0:
return self
else:
return - self
def __add__(self, y):
# VECTOR-READY
if isinstance(y, Differentiable):
zdict = {}
zdict.update(self.diff)
zdict.update(y.diff)
for key in zdict.iterkeys():
if (key in self.diff) and (key in y.diff):
zdict[key] = self.diff[key] + y.diff[key]
z = Differentiable()
z.val = self.val + y.val
z.diff = zdict
return z
else: # Adding a constant
z = Differentiable(val=self.val + y)
z.diff = self.diff
return z
def __radd__(self, x):
# VECTOR-READY
if isinstance(x, Differentiable):
zdict = {}
zdict.update(self.diff)
zdict.update(x.diff)
for key in zdict.iterkeys():
if (key in self.diff) and (key in x.diff):
zdict[key] = x.diff[key] + self.diff[key]
z = Differentiable()
z.val = x.val + self.val
z.diff = zdict
return z
else: # Adding a constant
z = Differentiable(val=x + self.val)
z.diff = self.diff
return z
def __mul__(self, y):
# !Not working with vectors!
if isinstance(y, Differentiable):
#TODO: with vectors
zdict = {}
zdict.update(self.diff)
zdict.update(y.diff)
for key in zdict.iterkeys():
if (key in self.diff) and (key in y.diff):
zdict[key] = self.diff[key] * y.val + self.val * y.diff[key]
elif (key in self.diff):
zdict[key] = self.diff[key] * y.val
else:
zdict[key] = self.val * y.diff[key]
z = Differentiable()
z.val = self.val * y.val
z.diff = zdict
return z
else: # Multiplying by a constant
z = Differentiable(val=self.val * y)
for key in self.diff:
z.diff[key] = self.diff[key] * y
return z
def __rmul__(self, x):
if isinstance(x, Differentiable):
#TODO: with vectors
zdict = {}
zdict.update(self.diff)
zdict.update(x.diff)
for key in zdict.iterkeys():
if (key in self.diff) and (key in x.diff):
zdict[key] = x.val * self.diff[key] + x.diff[key] * self.val
elif (key in self.diff):
zdict[key] = x.val * self.diff[key]
else:
zdict[key] = x.diff[key] * self.val
z = Differentiable()
z.val = x.val * self.val
z.diff = zdict
return z
else: # Multiplying by a constant
z = Differentiable(val=x * self.val)
for key in self.diff:
z.diff[key] = x * self.diff[key]
return z
def __div__(self, x):
return self * (x ** -1)
def __rdiv__(self, x):
return x * (self ** -1)
def __sub__(self, y):
# VECTOR-READY
if isinstance(y, Differentiable):
zdict = {}
zdict.update(self.diff)
zdict.update(y.diff)
for key in zdict.iterkeys():
if (key in self.diff) and (key in y.diff):
zdict[key] = self.diff[key] - y.diff[key]
z = Differentiable()
z.val = self.val - y.val
z.diff = zdict
return z
else: # Subtracting a constant
z = Differentiable(val=self.val - y)
z.diff = self.diff
return z
def __rsub__(self, x):
# VECTOR-READY
if isinstance(x, Differentiable):
zdict = {}
zdict.update(self.diff)
zdict.update(x.diff)
for key in zdict.iterkeys():
if (key in self.diff) and (key in x.diff):
zdict[key] = x.diff[key] - self.diff[key]
z = Differentiable()
z.val = x.val - self.val
z.diff = zdict
return z
else: # Subtracting a constant
z = Differentiable(val=x - self.val)
for key in self.diff:
z.diff[key] = -self.diff[key]
return z
def __neg__(self):
# VECTOR-READY
z = Differentiable(val= -self.val)
for key in self.diff:
z.diff[key] = -self.diff[key]
return z
def __pow__(self, x):
# NOT WORKING WITH VECTORS
if isinstance(x, Differentiable):
return exp(x * log(self))
elif x == 0:
z = Differentiable(val=self.val ** x)
for key in self.diff:
z.diff[key] = 0 * self.diff[key]
return z
else:
z = Differentiable(val=self.val ** x)
for key in self.diff:
z.diff[key] = x * self.val ** (x - 1) * self.diff[key]
return z
def __rpow__(self, x):
# NOT WORKING WITH VECTORS
return exp(self * log(x))
def __getitem__(self, i):
z = Differentiable(val=self.val[i])
z.diff = {}
for key in self.diff.iterkeys():
z.diff[key] = self.diff[key][i, :]
return z
def __len__(self):
return len(self.val)
def __str__(self):
s = str(self.val)
for key, value in self.diff.iteritems():
s += ' ; d/d' + key + '=' + str(value)
return s
def exp(self):
# !! Not working for vectors !!
zdict = {}
zdict.update(self.diff)
for key in zdict.iterkeys():
zdict[key] = exp(self.val) * self.diff[key]
z = Differentiable(val=exp(self.val))
z.diff = zdict
return z
def cos(self):
# !! Not working for vectors !!
zdict = {}
zdict.update(self.diff)
for key in zdict.iterkeys():
zdict[key] = -sin(self.val) * self.diff[key]
z = Differentiable(val=cos(self.val))
z.diff = zdict
return z
def sin(self):
# !! Not working for vectors !!
zdict = {}
zdict.update(self.diff)
for key in zdict.iterkeys():
zdict[key] = cos(self.val) * self.diff[key]
z = Differentiable(val=sin(self.val))
z.diff = zdict
return z
def log(self):
# !! Not working for vectors !!
zdict = {}
zdict.update(self.diff)
for key in zdict.iterkeys():
zdict[key] = (1. / (self.val)) * self.diff[key]
z = Differentiable(val=log(self.val))
z.diff = zdict
return z
class HigherDifferentiable(Differentiable):
'''
A differentiable variable with order n.
'''
def __init__(self, name=None, val=0., order=1):
Differentiable.__init__(self, name, val)
if order > 1:
self.val = HigherDifferentiable(name, val, order - 1)
if __name__ == '__main__':
print "Derivative:"
print "> differentiate(lambda x:3*x*x+2,x=2,order=2)"
print differentiate(lambda x:3 * x * x + 2, x=2, order=2)
print "> differentiate(lambda x:3*x*x+2,order=2)(2)"
print differentiate(lambda x:3 * x * x + 2, order=2)(2)
print "Partial derivative:"
print "> differentiate(lambda x,y:x*y+2*y+1,x=(1,2),order=(0,1))"
print differentiate(lambda x, y:x * y + 2 * y + 1, x=(1, 2), order=(0, 1))
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