5.1.6.3. numdifftools.nd_algopy.Jacobian

class Jacobian(fun, n=1, method='forward', full_output=False)[source]

Calculate Jacobian with Algorithmic Differentiation method

Parameters:
fun : function

function of one array fun(x, *args, **kwds)

method : string, optional {‘forward’, ‘reverse’}

defines method used in the approximation

Returns:
jacob : array

Jacobian

Notes

Algorithmic differentiation is a set of techniques to numerically evaluate the derivative of a function specified by a computer program. AD exploits the fact that every computer program, no matter how complicated, executes a sequence of elementary arithmetic operations (addition, subtraction, multiplication, division, etc.) and elementary functions (exp, log, sin, cos, etc.). By applying the chain rule repeatedly to these operations, derivatives of arbitrary order can be computed automatically, accurately to working precision, and using at most a small constant factor more arithmetic operations than the original program.

References

Sebastian F. Walter and Lutz Lehmann 2013, “Algorithmic differentiation in Python with AlgoPy”, in Journal of Computational Science, vol 4, no 5, pp 334 - 344, http://www.sciencedirect.com/science/article/pii/S1877750311001013

https://en.wikipedia.org/wiki/Automatic_differentiation

Examples

>>> import numdifftools.nd_algopy as nda

#(nonlinear least squares)

>>> xdata = np.arange(0,1,0.1)
>>> ydata = 1+2*np.exp(0.75*xdata)
>>> fun = lambda c: (c[0]+c[1]*np.exp(c[2]*xdata) - ydata)**2

Jfun = nda.Jacobian(fun) # Todo: This does not work Jfun([1,2,0.75]).T # should be numerically zero array([[ 0., 0., 0., 0., 0., 0., 0., 0., 0., 0.],

[ 0., 0., 0., 0., 0., 0., 0., 0., 0., 0.], [ 0., 0., 0., 0., 0., 0., 0., 0., 0., 0.]])
>>> Jfun2 = nda.Jacobian(fun, method='reverse')
>>> res = Jfun2([1,2,0.75]).T # should be numerically zero
>>> np.allclose(res,
...                [[ 0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.],
...                 [ 0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.],
...                 [ 0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.]])
True
>>> f2 = lambda x : x[0]*x[1]*x[2]**2
>>> Jfun2 = nda.Jacobian(f2)
>>> np.allclose(Jfun2([1., 2., 3.]), [[ 18., 9., 12.]])
True
>>> Jfun21 = nda.Jacobian(f2, method='reverse')
>>> np.allclose(Jfun21([1., 2., 3.]), [[ 18., 9., 12.]])
True
>>> def fun3(x):
...     n = int(np.prod(np.shape(x[0])))
...     out = nda.algopy.zeros((2, n), dtype=x)
...     out[0] = x[0]*x[1]*x[2]**2
...     out[1] = x[0]*x[1]*x[2]
...     return out
>>> Jfun3 = nda.Jacobian(fun3)
>>> np.allclose(Jfun3([1., 2., 3.]), [[[18., 9., 12.]], [[6., 3., 2.]]])
True
>>> np.allclose(Jfun3([4., 5., 6.]), [[[180., 144., 240.]],
...                                   [[30., 24., 20.]]])
True
>>> np.allclose(Jfun3(np.array([[1.,2.,3.], [4., 5., 6.]]).T),
...             [[[18.,    0.,    9.,    0.,   12.,    0.],
...               [0.,  180.,    0.,  144.,    0.,  240.]],
...              [[6.,    0.,    3.,    0.,    2.,    0.],
...               [0.,   30.,    0.,   24.,    0.,   20.]]])
True

Methods

__call__(x, *args, **kwds) Call self as a function.
__init__(fun, n=1, method='forward', full_output=False)

Initialize self. See help(type(self)) for accurate signature.

Methods

__init__(fun[, n, method, full_output]) Initialize self.
computational_graph(x, *args, **kwds)

Attributes

fun