LagrangePoles.py
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	from matplotlib import pyplot as plt
	import numpy as np
	from rrompy.hfengines.linear_problem import \
	HelmholtzSquareBubbleProblemEngine as HSBPE
	from rrompy.reduction_methods.lagrange import ApproximantLagrangePade as Pade
	from rrompy.utilities.parameter_sampling import QuadratureSampler as QS
	from rrompy.utilities.base import squareResonances

	verb = 0

	ks = [1, 46 ** .5]
	solver = HSBPE(kappa = 12 ** .5, theta = np.pi / 3, n = 20,
	verbosity = verb)
	k0 = np.mean(np.power(ks, 2.)) ** .5
	k0 = 3.46104724
	solver.omega = np.real(k0)

	rescaling = lambda x: np.power(x, 2.)
	rescalingInv = lambda x: np.power(x, .5)
	sampler = QS(ks, "UNIFORM", rescaling, rescalingInv)

	nsets = 15
	paramsPade = {'S':2, 'POD':True, 'basis':"LEGENDRE", 'sampler':sampler}
	approx = Pade(solver, mu0 = k0, approxParameters = paramsPade,
	verbosity = verb)

	poles = [None] * (nsets - 1)
	polesexact = np.unique(np.power(squareResonances(ks[0]2., ks[1]2., False),
	.5))
	for i in range(1, nsets):
	print("N = {}".format(4 * i))
	approx.approxParameters = {'N': 4 * i, 'M': 4 * i, 'S': 4 * i + 1}
	approx.setupApprox()
	poles[i - 1] = approx.getPoles()


	for i in range(1, nsets):
	plt.figure()
	plt.plot(np.real(poles[i - 1]), np.imag(poles[i - 1]), 'kx')
	plt.plot(polesexact, np.zeros_like(polesexact), 'm.')
	plt.plot(k0, 0, 'r*')
	plt.xlim(ks)
	plt.ylim((ks[0] - ks[1]) / 2., (ks[1] - ks[0]) / 2.)
	plt.title("N = {}, Neff = {}".format(4 * i, len(poles[i - 1])))
	plt.grid()
	plt.show()
	plt.close()