def compute_preconditioned_basis_matrix_condition_number(
basis_matrix_func, samples):
basis_matrix = basis_matrix_func(samples)
basis_matrix = basis_matrix * np.sqrt(
christoffel_weights(basis_matrix))[:, np.newaxis]
cond = np.linalg.cond(basis_matrix)
return cond
def candidate_based_christoffel_leja_rule_1d(
recursion_coeffs,
generate_candidate_samples,
num_candidate_samples,
level,
initial_points=None,
growth_rule=leja_growth_rule,
samples_filename=None,
return_weights_for_all_levels=True):
num_vars = 1
num_leja_samples = growth_rule(level)
leja_sequence = get_candidate_based_christoffel_leja_sequence_1d(
num_leja_samples, recursion_coeffs, generate_candidate_samples,
num_candidate_samples, initial_points, samples_filename)
from pyapprox.polynomial_sampling import christoffel_weights
generate_basis_matrix = lambda x: evaluate_orthonormal_polynomial_deriv_1d(
x[0, :], num_leja_samples, recursion_coeffs, deriv_order=0)
weight_function = lambda x: christoffel_weights(generate_basis_matrix(x))
ordered_weights_1d = get_leja_sequence_quadrature_weights(
leja_sequence, growth_rule, generate_basis_matrix, weight_function,
level, return_weights_for_all_levels)
return leja_sequence[0, :], ordered_weights_1d
def candidate_based_leja_rule(recursion_coeffs,
generate_candidate_samples,
num_candidate_samples,
level,
initial_points=None,
growth_rule=leja_growth_rule,
samples_filename=None,
return_weights_for_all_levels=True):
from pyapprox.orthonormal_polynomials_1d import \
evaluate_orthonormal_polynomial_1d
from pyapprox.polynomial_sampling import get_lu_leja_samples,\
christoffel_preconditioner, christoffel_weights
num_leja_samples = growth_rule(level)
generate_basis_matrix = lambda x: evaluate_orthonormal_polynomial_1d(
x[0, :], num_leja_samples, recursion_coeffs)
if samples_filename is None or not os.path.exists(samples_filename):
leja_sequence, __ = get_lu_leja_samples(
generate_basis_matrix,
generate_candidate_samples,
num_candidate_samples,
num_leja_samples,
preconditioning_function=christoffel_preconditioner,
initial_samples=initial_points)
if samples_filename is not None:
np.savez(samples_filename, samples=leja_sequence)
else:
leja_sequence = np.load(samples_filename)['samples']
assert leja_sequence.shape[1] >= growth_rule(level)
leja_sequence = leja_sequence[:, :growth_rule(level)]
weight_function = lambda x: christoffel_weights(generate_basis_matrix(x))
ordered_weights_1d = get_leja_sequence_quadrature_weights(
leja_sequence, growth_rule, generate_basis_matrix, weight_function,
level, return_weights_for_all_levels)
return leja_sequence[0, :], ordered_weights_1d
def weight_function(x): return christoffel_weights(
generate_basis_matrix(x))
ordered_weights_1d = get_leja_sequence_quadrature_weights(
def precond_func(matrix, samples): return christoffel_weights(matrix) canonical_samples, data_structures = get_fekete_samples(
def precond_func(matrix, samples): return christoffel_weights(matrix) initial_samples, data_structures = get_lu_leja_samples(