def rootsAnnSec(m, rMin, rMax, aMin, aMax):
f0 = lambda k: ivp(m, η * k) * hankel1(m, k) / η + iv(m, η * k) * h1vp(
m, k)
f1 = (lambda k: ivp(m, η * k, 2) * hankel1(m, k) + c * ivp(m, η * k) *
h1vp(m, k) + iv(m, η * k) * h1vp(m, k, 2))
A = AnnulusSector(center=0.0, radii=(rMin, rMax), phiRange=(aMin, aMax))
z = A.roots(f0, df=f1)
return z.roots
def test_PolarRect_contains():
r0 = 8.938
r1 = 9.625
phi0 = 6.126
phi1 = 6.519
z = (9 - 1.04825594683e-18j)
C = AnnulusSector(0, [r0, r1], [phi0, phi1])
# import matplotlib.pyplot as plt
# plt.scatter((9),(0))
# C.show()
assert C.contains(z)
def setUp(self):
self.roots = roots = [0, -1.234, 1 + 1j, 1 - 1j, 2.345]
self.multiplicities = [1, 1, 1, 1, 1]
self.f = lambda z: (z - roots[0]) * (z - roots[1]) * (z - roots[2]) * (
z - roots[3]) * (z - roots[4])
self.df = lambda z: (z - roots[1]) * (z - roots[2]) * (z - roots[
3]) * (z - roots[4]) + (z - roots[0]) * (z - roots[2]) * (
z - roots[3]) * (z - roots[4]) + (z - roots[0]) * (z - roots[
1]) * (z - roots[3]) * (z - roots[4]) + (z - roots[0]) * (
z - roots[1]) * (z - roots[2]) * (z - roots[4]) + (
z - roots[0]) * (z - roots[1]) * (z - roots[2]) * (
z - roots[3])
self.Circle = Circle(0, 3)
self.Rectangle = Rectangle([-2, 2], [-2, 2])
self.halfAnnulus = AnnulusSector(0, [0.5, 3], [-pi / 2, pi / 2])
self.Annulus = Annulus(0, [1, 2])
def boundStateRegion(vRange, maxFreq=1):
from cxroots import AnnulusSector
# get the region in which to look for bound state eigenvalues
radiusBuffer = 0.05
# With lambda=mu in Eq.9 of "Breaking integrability at the boundary"
# As in Eq. II.2 in "Spectral theory for the periodic sine-Gordon equation: A concrete viewpoint" with lambda=Sqrt[E]
mu = lambda v: 0.25j * sqrt((1 - v) / (1 + v))
acosFeq = arccos(maxFreq)
phiRange = [acosFeq, pi - acosFeq]
radiusRange = np.sort(np.abs(mu(np.array(vRange))))
radiusRange = [
radiusRange[0] - radiusBuffer, radiusRange[1] + radiusBuffer
]
center = 0
return AnnulusSector(center, radiusRange, phiRange)
from numpy import pi from cxroots import AnnulusSector annulusSector = AnnulusSector(center=0.2, radii=(0.5, 1.25), phiRange=(-pi/4, pi/4)) annulusSector.show()
class TestRootfindingContours(unittest.TestCase):
def setUp(self):
self.roots = roots = [0, -1.234, 1 + 1j, 1 - 1j, 2.345]
self.multiplicities = [1, 1, 1, 1, 1]
self.f = lambda z: (z - roots[0]) * (z - roots[1]) * (z - roots[2]) * (
z - roots[3]) * (z - roots[4])
self.df = lambda z: (z - roots[1]) * (z - roots[2]) * (z - roots[
3]) * (z - roots[4]) + (z - roots[0]) * (z - roots[2]) * (
z - roots[3]) * (z - roots[4]) + (z - roots[0]) * (z - roots[
1]) * (z - roots[3]) * (z - roots[4]) + (z - roots[0]) * (
z - roots[1]) * (z - roots[2]) * (z - roots[4]) + (
z - roots[0]) * (z - roots[1]) * (z - roots[2]) * (
z - roots[3])
self.Circle = Circle(0, 3)
self.Rectangle = Rectangle([-2, 2], [-2, 2])
self.halfAnnulus = AnnulusSector(0, [0.5, 3], [-pi / 2, pi / 2])
self.Annulus = Annulus(0, [1, 2])
def test_rootfinding_circle_fdf(self):
roots_approx_equal(self.Circle.roots(self.f, self.df, verbose=True),
(self.roots, self.multiplicities),
decimal=7)
def test_rootfinding_circle_f(self):
roots_approx_equal(self.Circle.roots(self.f, self.df, verbose=True),
(self.roots, self.multiplicities),
decimal=7)
def test_rootfinding_rectangle_fdf(self):
roots_approx_equal(self.Rectangle.roots(self.f, self.df, verbose=True),
(self.roots[:-1], self.multiplicities[:-1]),
decimal=7)
def test_rootfinding_rectangle_f(self):
roots_approx_equal(self.Rectangle.roots(self.f, self.df, verbose=True),
(self.roots[:-1], self.multiplicities[:-1]),
decimal=7)
def test_rootfinding_halfAnnulus_fdf(self):
roots_approx_equal(self.halfAnnulus.roots(self.f,
self.df,
verbose=True),
(self.roots[2:], self.multiplicities[2:]),
decimal=7)
def test_rootfinding_halfAnnulus_f(self):
roots_approx_equal(self.halfAnnulus.roots(self.f,
self.df,
verbose=True),
(self.roots[2:], self.multiplicities[2:]),
decimal=7)
def test_rootfinding_Annulus_fdf(self):
roots_approx_equal(self.Annulus.roots(self.f, self.df, verbose=True),
(self.roots[1:-1], self.multiplicities[1:-1]),
decimal=7)
def test_rootfinding_Annulus_f(self):
roots_approx_equal(self.Annulus.roots(self.f, self.df, verbose=True),
(self.roots[1:-1], self.multiplicities[1:-1]),
decimal=7)
from numpy import pi
from cxroots import Circle, Annulus, AnnulusSector, Rectangle
# make the contours for the tutorial
Circle(0, 2).show('circle.png')
Rectangle([-2,2],[-1,1]).show('rectangle.png')
Annulus(0, [1,2]).show('annulus.png')
AnnulusSector(0, [1,2], [0,pi]).show('annulussefig.png')