complex_step module¶
Gradient approximation by complex step.
- 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.
- 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
- 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 (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
The approximated gradient.
- 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
- ALIAS = 'complex_step'¶
- f_pointer: Callable[[numpy.ndarray], numpy.ndarray]¶
The pointer to the function to derive.