NumericalDerivative

class odl.functionals.derivatives.NumericalDerivative(*args, **kwargs)[source]

Bases: Operator

The derivative of an operator by finite differences.

See Also

NumericalGradient : Compute gradient of a functional

__init__(operator, point, method='forward', step=None)[source]

Initialize a new instance.

Parameters

operatorOperator: The operator whose derivative should be computed numerically. Its domain and range must be TensorSpace spaces.
pointoperator.domain element-like: The point to compute the derivative in.
method{‘backward’, ‘forward’, ‘central’}, optional: The method to use to compute the derivative.
stepfloat, optional: The step length used in the derivative computation. Default: selects the step according to the dtype of the space.

Examples

Compute a numerical estimate of the derivative (Hessian) of the squared L2 norm:

>>> space = odl.rn(3)
>>> func = odl.functionals.L2NormSquared(space)
>>> hess = NumericalDerivative(func.gradient, [1, 1, 1])
>>> hess([0, 0, 1])
rn(3).element([ 0.,  0.,  2.])

Find the Hessian matrix:

>>> hess_matrix = odl.matrix_representation(hess)
>>> np.allclose(hess_matrix, 2 * np.eye(3))
True

Notes

If the operator is $A$ and step size $h$ is used, the derivative in the point $x$ and direction $dx$ is computed as follows.

method='backward':

$\partial A(x)(dx) = (A(x) - A(x - dx \cdot h / \| dx \|)) \cdot \frac{\| dx \|}{h}$

method='forward':

$\partial A(x)(dx) = (A(x + dx \cdot h / \| dx \|) - A(x)) \cdot \frac{\| dx \|}{h}$

method='central':

$\partial A(x)(dx) = (A(x + dx \cdot h / (2 \| dx \|)) - A(x - dx \cdot h / (2 \| dx \|)) \cdot \frac{\| dx \|}{h}$

The number of operator evaluations is 2, regardless of parameters.