capytaine.tools.prony_decomposition module

Prony decomposition: tool to approximate a function as a sum of exponentials. Used in particular in the finite depth Green function.

exception capytaine.tools.prony_decomposition.PronyDecompositionFailure

Bases: Exception

capytaine.tools.prony_decomposition.error_exponential_decomposition(X, F, a, lamda)

Mean square error of the exponential decomposition defined by the coefficients a and lamda with respect to the reference values in F.

Parameters:

• X (1D array) – sampling points

• F (1D array (same size as X)) – reference values

• a (1D array) – coefficients of the exponentials

• lamda (1D array (same size as a)) – growth rate of the exponentials

Returns:

error – mean square error of the decomposition

Return type:

float

capytaine.tools.prony_decomposition.exponential_decomposition(X, F, m)

Use Prony’s method to approximate the sampled real function F=f(X) as a sum of m exponential functions x → Σ a_i exp(lamda_i x).

Parameters:

• X (1D array) – sampling points.

• F (1D array (same size as X)) – values of the function to approximate at the points of x.

• m (integer) – number of exponential functions

Returns:

• a (1D array (size m)) – coefficients of the exponentials

• lamda (1D array (size m)) – growth rate of the exponentials

capytaine.tools.prony_decomposition.find_best_exponential_decomposition(f, x_min, x_max, n_exp_range, *, tol=0.0001, noise_on_domain_points_std=0.01)

Tries to construct an exponential decompositoin of the function f on the domain [x_min, x_max] by testing the number of exponentials in n_exp_range.

Parameters:

• f (callable) – The function ℝ→ℝ to be approximated. Should support vectorized calls (that is passing a vector of inputs and get the vector of corresponding outputs)

• x_min, x_max (floats) – The bounds of the domain of input in which f should be approximated

• n_exp_range (iterable of ints) – The decomposition sizes that will be tested

• tol (float, optional) – The target mean square error.

• noise_on_domain_points_std (float, optional) – Introduces some random variability on the points where the function is evaluated. Set this parameter to zero to disable randomness.