332 lines
10 KiB
Python
Executable file
332 lines
10 KiB
Python
Executable file
# Automatic first-order derivatives
|
|
#
|
|
# Written by Konrad Hinsen <hinsen@cnrs-orleans.fr>
|
|
# last revision: 2006-6-12
|
|
#
|
|
|
|
"""
|
|
Automatic differentiation for functions with any number of variables
|
|
|
|
Instances of the class DerivVar represent the values of a function and
|
|
its partial X{derivatives} with respect to a list of variables. All
|
|
common mathematical operations and functions are available for these
|
|
numbers. There is no restriction on the type of the numbers fed into
|
|
the code; it works for real and complex numbers as well as for any
|
|
Python type that implements the necessary operations.
|
|
|
|
This module is as far as possible compatible with the n-th order
|
|
derivatives module Derivatives. If only first-order derivatives
|
|
are required, this module is faster than the general one.
|
|
|
|
Example::
|
|
|
|
print sin(DerivVar(2))
|
|
|
|
produces the output::
|
|
|
|
(0.909297426826, [-0.416146836547])
|
|
|
|
The first number is the value of sin(2); the number in the following
|
|
list is the value of the derivative of sin(x) at x=2, i.e. cos(2).
|
|
|
|
When there is more than one variable, DerivVar must be called with
|
|
an integer second argument that specifies the number of the variable.
|
|
|
|
Example::
|
|
|
|
>>>x = DerivVar(7., 0)
|
|
>>>y = DerivVar(42., 1)
|
|
>>>z = DerivVar(pi, 2)
|
|
>>>print (sqrt(pow(x,2)+pow(y,2)+pow(z,2)))
|
|
|
|
produces the output
|
|
|
|
>>>(42.6950770511, [0.163953328662, 0.98371997197, 0.0735820818365])
|
|
|
|
The numbers in the list are the partial derivatives with respect
|
|
to x, y, and z, respectively.
|
|
|
|
Note: It doesn't make sense to use DerivVar with different values
|
|
for the same variable index in one calculation, but there is
|
|
no check for this. I.e.::
|
|
|
|
>>>print DerivVar(3, 0)+DerivVar(5, 0)
|
|
|
|
produces
|
|
|
|
>>>(8, [2])
|
|
|
|
but this result is meaningless.
|
|
"""
|
|
|
|
|
|
from Scientific import N; Numeric = N
|
|
|
|
|
|
# The following class represents variables with derivatives:
|
|
|
|
class DerivVar:
|
|
|
|
"""
|
|
Numerical variable with automatic derivatives of first order
|
|
"""
|
|
|
|
"""Variable with derivatives
|
|
|
|
Constructor: DerivVar(|value|, |index| = 0)
|
|
|
|
Arguments:
|
|
|
|
|value| -- the numerical value of the variable
|
|
|
|
|index| -- the variable index (an integer), which serves to
|
|
distinguish between variables and as an index for
|
|
the derivative lists. Each explicitly created
|
|
instance of DerivVar must have a unique index.
|
|
|
|
Indexing with an integer yields the derivatives of the corresponding
|
|
order.
|
|
"""
|
|
|
|
def __init__(self, value, index=0, order=1):
|
|
"""
|
|
@param value: the numerical value of the variable
|
|
@type value: number
|
|
@param index: the variable index, which serves to
|
|
distinguish between variables and as an index for
|
|
the derivative lists. Each explicitly created
|
|
instance of DerivVar must have a unique index.
|
|
@type index: C{int}
|
|
@param order: the derivative order, must be zero or one
|
|
@type order: C{int}
|
|
@raise ValueError: if order < 0 or order > 1
|
|
"""
|
|
if order < 0 or order > 1:
|
|
raise ValueError('Only first-order derivatives')
|
|
self.value = value
|
|
if order == 0:
|
|
self.deriv = []
|
|
elif type(index) == type([]):
|
|
self.deriv = index
|
|
else:
|
|
self.deriv = index*[0] + [1]
|
|
|
|
def __getitem__(self, order):
|
|
"""
|
|
@param order: derivative order
|
|
@type order: C{int}
|
|
@return: a list of all derivatives of the given order
|
|
@rtype: C{list}
|
|
@raise ValueError: if order < 0 or order > 1
|
|
"""
|
|
if order < 0 or order > 1:
|
|
raise ValueError('Index out of range')
|
|
if order == 0:
|
|
return self.value
|
|
else:
|
|
return self.deriv
|
|
|
|
def __repr__(self):
|
|
return `(self.value, self.deriv)`
|
|
|
|
def __str__(self):
|
|
return str((self.value, self.deriv))
|
|
|
|
def __coerce__(self, other):
|
|
if isDerivVar(other):
|
|
return self, other
|
|
else:
|
|
return self, DerivVar(other, [])
|
|
|
|
def __cmp__(self, other):
|
|
return cmp(self.value, other.value)
|
|
|
|
def __neg__(self):
|
|
return DerivVar(-self.value, map(lambda a: -a, self.deriv))
|
|
|
|
def __pos__(self):
|
|
return self
|
|
|
|
def __abs__(self): # cf maple signum # derivate of abs
|
|
absvalue = abs(self.value)
|
|
return DerivVar(absvalue, map(lambda a, d=self.value/absvalue:
|
|
d*a, self.deriv))
|
|
def __nonzero__(self):
|
|
return self.value != 0
|
|
|
|
def __add__(self, other):
|
|
return DerivVar(self.value + other.value,
|
|
_mapderiv(lambda a,b: a+b, self.deriv, other.deriv))
|
|
__radd__ = __add__
|
|
|
|
def __sub__(self, other):
|
|
return DerivVar(self.value - other.value,
|
|
_mapderiv(lambda a,b: a-b, self.deriv, other.deriv))
|
|
|
|
def __rsub__(self, other):
|
|
return DerivVar(other.value - self.value,
|
|
_mapderiv(lambda a,b: a-b, other.deriv, self.deriv))
|
|
|
|
def __mul__(self, other):
|
|
return DerivVar(self.value*other.value,
|
|
_mapderiv(lambda a,b: a+b,
|
|
map(lambda x,f=other.value:f*x, self.deriv),
|
|
map(lambda x,f=self.value:f*x, other.deriv)))
|
|
__rmul__ = __mul__
|
|
|
|
def __div__(self, other):
|
|
if not other.value:
|
|
raise ZeroDivisionError('DerivVar division')
|
|
inv = 1./other.value
|
|
return DerivVar(self.value*inv,
|
|
_mapderiv(lambda a,b: a-b,
|
|
map(lambda x,f=inv: f*x, self.deriv),
|
|
map(lambda x,f=self.value*inv*inv: f*x,
|
|
other.deriv)))
|
|
def __rdiv__(self, other):
|
|
return other/self
|
|
|
|
def __pow__(self, other, z=None):
|
|
if z is not None:
|
|
raise TypeError('DerivVar does not support ternary pow()')
|
|
val1 = pow(self.value, other.value-1)
|
|
val = val1*self.value
|
|
deriv1 = map(lambda x, f=val1*other.value: f*x, self.deriv)
|
|
if isDerivVar(other) and len(other.deriv) > 0:
|
|
deriv2 = map(lambda x, f=val*Numeric.log(self.value): f*x,
|
|
other.deriv)
|
|
return DerivVar(val, _mapderiv(lambda a,b: a+b, deriv1, deriv2))
|
|
else:
|
|
return DerivVar(val, deriv1)
|
|
|
|
def __rpow__(self, other):
|
|
return pow(other, self)
|
|
|
|
def exp(self):
|
|
v = Numeric.exp(self.value)
|
|
return DerivVar(v, map(lambda x, f=v: f*x, self.deriv))
|
|
|
|
def log(self):
|
|
v = Numeric.log(self.value)
|
|
d = 1./self.value
|
|
return DerivVar(v, map(lambda x, f=d: f*x, self.deriv))
|
|
|
|
def log10(self):
|
|
v = Numeric.log10(self.value)
|
|
d = 1./(self.value * Numeric.log(10))
|
|
return DerivVar(v, map(lambda x, f=d: f*x, self.deriv))
|
|
|
|
def sqrt(self):
|
|
v = Numeric.sqrt(self.value)
|
|
d = 0.5/v
|
|
return DerivVar(v, map(lambda x, f=d: f*x, self.deriv))
|
|
|
|
def sign(self):
|
|
if self.value == 0:
|
|
raise ValueError("can't differentiate sign() at zero")
|
|
return DerivVar(Numeric.sign(self.value), 0)
|
|
|
|
def sin(self):
|
|
v = Numeric.sin(self.value)
|
|
d = Numeric.cos(self.value)
|
|
return DerivVar(v, map(lambda x, f=d: f*x, self.deriv))
|
|
|
|
def cos(self):
|
|
v = Numeric.cos(self.value)
|
|
d = -Numeric.sin(self.value)
|
|
return DerivVar(v, map(lambda x, f=d: f*x, self.deriv))
|
|
|
|
def tan(self):
|
|
v = Numeric.tan(self.value)
|
|
d = 1.+pow(v,2)
|
|
return DerivVar(v, map(lambda x, f=d: f*x, self.deriv))
|
|
|
|
def sinh(self):
|
|
v = Numeric.sinh(self.value)
|
|
d = Numeric.cosh(self.value)
|
|
return DerivVar(v, map(lambda x, f=d: f*x, self.deriv))
|
|
|
|
def cosh(self):
|
|
v = Numeric.cosh(self.value)
|
|
d = Numeric.sinh(self.value)
|
|
return DerivVar(v, map(lambda x, f=d: f*x, self.deriv))
|
|
|
|
def tanh(self):
|
|
v = Numeric.tanh(self.value)
|
|
d = 1./pow(Numeric.cosh(self.value),2)
|
|
return DerivVar(v, map(lambda x, f=d: f*x, self.deriv))
|
|
|
|
def arcsin(self):
|
|
v = Numeric.arcsin(self.value)
|
|
d = 1./Numeric.sqrt(1.-pow(self.value,2))
|
|
return DerivVar(v, map(lambda x, f=d: f*x, self.deriv))
|
|
|
|
def arccos(self):
|
|
v = Numeric.arccos(self.value)
|
|
d = -1./Numeric.sqrt(1.-pow(self.value,2))
|
|
return DerivVar(v, map(lambda x, f=d: f*x, self.deriv))
|
|
|
|
def arctan(self):
|
|
v = Numeric.arctan(self.value)
|
|
d = 1./(1.+pow(self.value,2))
|
|
return DerivVar(v, map(lambda x, f=d: f*x, self.deriv))
|
|
|
|
def arctan2(self, other):
|
|
den = self.value*self.value+other.value*other.value
|
|
s = self.value/den
|
|
o = other.value/den
|
|
return DerivVar(Numeric.arctan2(self.value, other.value),
|
|
_mapderiv(lambda a, b: a-b,
|
|
map(lambda x, f=o: f*x, self.deriv),
|
|
map(lambda x, f=s: f*x, other.deriv)))
|
|
|
|
def gamma(self):
|
|
from transcendental import gamma, psi
|
|
v = gamma(self.value)
|
|
d = v*psi(self.value)
|
|
return DerivVar(v, map(lambda x, f=d: f*x, self.deriv))
|
|
|
|
# Type check
|
|
|
|
def isDerivVar(x):
|
|
"""
|
|
@param x: an arbitrary object
|
|
@return: True if x is a DerivVar object, False otherwise
|
|
@rtype: bool
|
|
"""
|
|
return hasattr(x,'value') and hasattr(x,'deriv')
|
|
|
|
# Map a binary function on two first derivative lists
|
|
|
|
def _mapderiv(func, a, b):
|
|
nvars = max(len(a), len(b))
|
|
a = a + (nvars-len(a))*[0]
|
|
b = b + (nvars-len(b))*[0]
|
|
return map(func, a, b)
|
|
|
|
|
|
# Define vector of DerivVars
|
|
|
|
def DerivVector(x, y, z, index=0):
|
|
|
|
"""
|
|
@param x: x component of the vector
|
|
@type x: number
|
|
@param y: y component of the vector
|
|
@type y: number
|
|
@param z: z component of the vector
|
|
@type z: number
|
|
@param index: the DerivVar index for the x component. The y and z
|
|
components receive consecutive indices.
|
|
@type index: C{int}
|
|
@return: a vector whose components are DerivVar objects
|
|
@rtype: L{Scientific.Geometry.Vector}
|
|
"""
|
|
|
|
from Scientific.Geometry.VectorModule import Vector
|
|
if isDerivVar(x) and isDerivVar(y) and isDerivVar(z):
|
|
return Vector(x, y, z)
|
|
else:
|
|
return Vector(DerivVar(x, index),
|
|
DerivVar(y, index+1),
|
|
DerivVar(z, index+2))
|