gemseo / utils / derivatives

# complex_step module¶

class gemseo.utils.derivatives.complex_step.ComplexStep(f_pointer, step=1e-20, parallel=False, design_space=None, normalize=True, **parallel_args)[source]

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.

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 (DesignSpace | None) –

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 (int | bool | float) – The parallel execution options, see gemseo.core.parallel_execution.

Return type

None

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

Parameters
• x_vect (ndarray) – The input vector.

• step (complex | None) –

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

By default it is set to None.

• x_indices (Sequence[int] | None) –

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

Return type

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 (ndarray) – The input vector.

• x_indices (Sequence[int] | None) –

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 (float | None) –

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[ndarray, float | ndarray]

ALIAS = 'complex_step'
f_pointer: Callable[[numpy.ndarray], numpy.ndarray]

The pointer to the function to derive.

property step: float

The default approximation step.