gemseo / utils / derivatives

complex_step module

Gradient approximation by complex step.

Classes:

ComplexStep(f_pointer[, step, parallel, ...])

Complex step approximator, performing a second-order gradient calculation.
class gemseo.utils.derivatives.complex_step.ComplexStep(f_pointer, step=1e-20, parallel=False, design_space=None, normalize=True, **parallel_args)[source]

Bases: gemseo.utils.derivatives.gradient_approximator.GradientApproximator

Complex step approximator, performing a second-order gradient calculation.

Enable a much lower step than real finite differences, typically 1e-30, since there is no cancellation error due to a difference calculation.

\[\begin{split}\frac{df(x)}{dx} \approx Im\left( \frac{f(x+j*\\delta x)}{\\delta x} \right)\end{split}\]

See Martins, Joaquim RRA, Peter Sturdza, and Juan J. Alonso. “The complex-step derivative approximation.” ACM Transactions on Mathematical Software (TOMS) 29.3 (2003): 245-262.

f_pointer

The pointer to the function to derive.

Type

Callable

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

Parameters

  • f_pointer (Callable[[ndarray],ndarray]) – The pointer to the function to derive.

  • step (complex) –

    The default differentiation step.

    By default it is set to 1e-20.

  • parallel (bool) –

    Whether to differentiate the function in parallel.

    By default it is set to False.

  • design_space (Optional[DesignSpace]) –

    The design space containing the upper bounds of the input variables. If None, consider that the input variables are unbounded.

    By default it is set to None.

  • normalize (bool) –

    If True, then the functions are normalized.

    By default it is set to True.

  • **parallel_args (Union[int,bool,float]) – The parallel execution options, see gemseo.core.parallel_execution.

Return type

None

Attributes:

ALIAS

step

The default approximation step.

Methods:

f_gradient(x_vect[, step, x_indices])

Approximate the gradient of the function for a given input vector.

generate_perturbations(n_dim, x_vect[, ...])

Generate the input perturbations from the differentiation step.
ALIAS = 'complex_step'
f_gradient(x_vect, step=None, x_indices=None, **kwargs)[source]

Approximate the gradient of the function for a given input vector.

Parameters

  • x_vect (numpy.ndarray) – The input vector.

  • step (Optional[complex]) –

    The differentiation step. If None, use the default differentiation step.

    By default it is set to None.

  • x_indices (Optional[Sequence[int]]) –

    The components of the input vector to be used for the differentiation. If None, use all the components.

    By default it is set to None.

  • **kwargs (Any) – The optional arguments for the function.

Returns

The approximated gradient.

Return type

numpy.ndarray

generate_perturbations(n_dim, x_vect, x_indices=None, step=None)

Generate the input perturbations from the differentiation step.

These perturbations will be used to compute the output ones.

Parameters

  • n_dim (int) – The input dimension.

  • x_vect (numpy.ndarray) – The input vector.

  • x_indices (Optional[Sequence[int]]) –

    The components of the input vector to be used for the differentiation. If None, use all the components.

    By default it is set to None.

  • step (Optional[float]) –

    The differentiation step. If None, use the default differentiation step.

    By default it is set to None.

Returns

  • The input perturbations.

  • The differentiation step, either one global step or one step by input component.

Return type

Tuple[numpy.ndarray, Union[float, numpy.ndarray]]

property step

The default approximation step.