numericalderivative.SteplemanWinarskyInitialize

class numericalderivative.SteplemanWinarskyInitialize(algorithm, relative_precision=1e-15, verbose=False)

Compute an initial step for a search algorithm

Parameters:

algorithmSteplemanWinarsky

The algorithm to find a step

relative_precisionfloat, > 0, optional

The relative precision of evaluation of f.

Methods

find_initial_step(h_min, h_max[, ...])	Compute the initial step using bisection.
get_number_of_function_evaluations()	Returns the number of function evaluations.
get_step_history()	Return the history of steps during the search.
number_of_lost_digits(h)	Compute the number of (base 10) lost digits.

Returns:

None.

References

• Adaptive numerical differentiation. R. S. Stepleman and N. D. Winarsky. Journal: Math. Comp. 33 (1979), 1257-1264

Examples

The next example computes the step of a badly scaled function. We first compute an appropriate initial step using find_initial_step() and set it as the input of find_step().

>>> import numericalderivative as nd
>>>
>>> def scaled_exp(x):
>>>     alpha = 1.e6
>>>     return np.exp(-x / alpha)
>>>
>>> x = 1.0e-2
>>> initial_step = 1.0e8
>>> algorithm = nd.SteplemanWinarsky(scaled_exp, x)
>>> initialize = nd.SteplemanWinarskyInitialize(algorithm)
>>> initial_step, number_of_iterations = initialize.find_initial_step(
>>>     1.0e-5,
>>>     1.0e7,
>>> )
>>> h_optimal, number_of_iterations = algorithm.find_step(initial_step)
>>> f_prime_approx = algorithm.compute_first_derivative(h_optimal)

__init__(algorithm, relative_precision=1e-15, verbose=False)

Methods

__init__(algorithm[, relative_precision, ...])
find_initial_step(h_min, h_max[, ...])	Compute the initial step using bisection.
get_number_of_function_evaluations()	Returns the number of function evaluations.
get_step_history()	Return the history of steps during the search.
number_of_lost_digits(h)	Compute the number of (base 10) lost digits.

find_initial_step(h_min, h_max, maximum_bisection=53, log_scale=True)

Compute the initial step using bisection.

Search for an initial step \(h_0\) such that:

\[0 < N(h_0) < T := \log_{10}\left(\frac{\epsilon_r^{-1 / 3}}{\beta}\right)\]

where \(N\) is the number of lost digits (as computed by number_of_lost_digits()), \(h_0\) is the initial step and \(\epsilon_r\) is the relative precision of the function evaluation. This heuristic is based on the hypothesis that the absolute value of the third derivative is close to 1.

The value returned by find_initial_step() can be used as input of find_step().

This algorithm can fail if the required finite difference step is so large that the points \(x \pm h\) fall beyond the mathematical input domain of the function.

Parameters:

h_minfloat

The lower bound to bracket the initial differentiation step.

h_maxfloat, > h_min

The upper bound to bracket the initial differentiation step. We must have N(h_min) > N(h_max) where N is the number of lost digits.

maximum_bisectionint, optional

The maximum number of bisection iterations.

log_scalebool, optional

Set to True to bisect in log scale.

Returns:

initial_stepfloat

The initial step.

number_of_iterationsint

The number of required iterations.

get_number_of_function_evaluations()

Returns the number of function evaluations.

Returns:

number_of_function_evaluationsint

The number of function evaluations.

get_step_history()

Return the history of steps during the search.

Let n be the number of iterations. The best step h is not the last one (with index n-1), since this is the step which breaks the monotony. The best step has index n - 2.

Returns:

step_historylist(float)

The list of steps h during intermediate iterations of the search. This is updated by find_initial_step().

number_of_lost_digits(h)

Compute the number of (base 10) lost digits.

The loss of figures produced by the substraction in the finite difference formula is (see (Stepleman & Winarsky, 1979), eq. 3.10 page 1261):

\[\delta(h) = \left|\frac{2hd(h)}{f(x)}\right|\]

where \(h > 0\) is the step and \(d(h)\) is the central finite difference formula. The number of decimal digits losts in the substraction is (see (Stepleman & Winarsky, 1979), eq. 3.11 page 1261):

\[N(h) = -\log_{10}(\delta(h))\]

where \(\log_{10}\) is the base-10 decimal digit.

Parameters:

hfloat

Differentiation step.

Returns:

number_of_digitsfloat

The number of digits lost by cancellation.