gemseo.utils.derivatives.complex_step module

Gradient approximation by complex step.

class ComplexStep(f_pointer, step=None, design_space=None, normalize=True, parallel=False, **parallel_args)

Bases: BaseGradientApproximator

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, Any, ...], ndarray]) -- The pointer to the function to derive.

• step (complex | ndarray | None) -- The default differentiation step.

• design_space (DesignSpace | None) -- The design space containing the upper bounds of the input variables. If None, consider that the input variables are unbounded.

• normalize (bool) --

  Whether to normalize the function.

  By default it is set to True.

• parallel (bool) --

  Whether to differentiate the function in parallel.

  By default it is set to False.

• **parallel_args (Any) -- The parallel execution options, see gemseo.core.parallel_execution.

f_gradient(x_vect, step=None, x_indices=(), **kwargs)

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.

• x_indices (Sequence[int]) --

  The components of the input vector to be used for the differentiation. If empty, use all the components.

  By default it is set to ().

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

Returns:

The approximated gradient.

Return type:

ndarray

property step: float

The default approximation step.