def test_guess_symmetry_1(symmetry): C = Circle(0, 3) f = lambda z: z**4 + z**3 + z**2 + z roots = [0,-1,1j,-1j] multiplicities = [1,1,1,1] roots_approx_equal(C.roots(f, verbose=True, guessRootSymmetry=symmetry), (roots, multiplicities))
def test_reevaluation_of_N(): from cxroots import Circle C = Circle(0, 2) f = lambda z: (z - 1) * (z - 0.2)**2 roots = [1, 0.2] multiplicities = [1, 2] roots_approx_equal( C.roots(f, NIntAbsTol=10, intMethod='romb', verbose=True), (roots, multiplicities))
def test_guess_root(guesses): C = Circle(0, 3) f = lambda z: (z-2.5)**2 * (exp(-z)*sin(z/2.) - 1.2*cos(z)) roots = [2.5, 1.44025113016670301345110737, -0.974651035111059787741822566 - 1.381047768247156339633038236j, -0.974651035111059787741822566 + 1.381047768247156339633038236j] multiplicities = [2,1,1,1] roots_approx_equal(C.roots(f, guessRoots=[2.5], verbose=True), (roots, multiplicities))
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 setUp(self): # Ex 1.4.1 from [KB] self.C = Circle(0, 3) e = 1e-2 self.f = lambda z: (z - e) * (1 + (z - sqrt(3))**2) self.df = lambda z: (1 + (z - sqrt(3))**2) + (z - e) * 2 * (z - sqrt(3)) self.roots = [e, sqrt(3) + 1j, sqrt(3) - 1j] self.multiplicities = [1, 1, 1]
def calcInt(): plaTrue = ε > -1.0 if plaTrue: Int = open(f"eps_{ε}_int", "w") Pla = open(f"eps_{ε}_pla", "w") else: Int = open(f"eps_{ε}_int", "w") for m in range(65): print(f"m = {m}") 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)) t = np.linspace(0.2, 65.0, num=1024) k = 1j * t rf = np.real(f0(k)) ind = np.where(rf[1:] * rf[:-1] < 0.0)[0] roots = np.zeros(np.shape(ind), dtype=complex) for a, i in enumerate(ind): C = Circle(center=1j * (t[i] + t[i + 1]) / 2.0, radius=(t[i + 1] - t[i])) z = C.roots(f0, df=f1) roots[a] = z.roots[0] if plaTrue: if m: writeFile(Int, m, roots[1:]) writeFile(Pla, m, roots[[0]]) else: writeFile(Int, m, roots) else: writeFile(Int, m, roots) if plaTrue: Int.close() Pla.close() else: Int.close()
def setUp(self): # Ex 1.4.2 from [KB] self.C = Circle(0, 2) self.f = lambda z: exp(3 * z) + 2 * z * cos(z) - 1 self.df = lambda z: 3 * exp(3 * z) + 2 * cos(z) - 2 * z * sin(z) self.roots = [ 0, -1.844233953262213, 0.5308949302929305 + 1.33179187675112098j, 0.5308949302929305 - 1.33179187675112098j ] self.multiplicities = [1, 1, 1, 1]
def setUp(self): # Ex 1.4.5 from [KB] self.C = Circle(0, 11) self.f = lambda z: np.prod([z - k for k in range(1, 11)], axis=0) self.df = lambda z: np.sum([ np.prod([z - k for k in range(1, 11) if k != m], axis=0) for m in range(1, 11) ], axis=0) self.roots = [1, 2, 3, 4, 5, 6, 7, 8, 9, 10] self.multiplicities = np.ones(10)
def setUp(self): # Ex 1.4.3 from [KB] self.C = Circle(0, 5) self.f = lambda z: z**2 * (z - 1) * (z - 2) * (z - 3) * (z - 4 ) + z * sin(z) self.df = lambda z: 2 * z * (3 * z**4 - 25 * z**3 + 70 * z**2 - 75 * z + 24) + sin(z) + z * cos(z) self.roots = [ 0, 1.18906588973011365517521756, 1.72843498616506284043592924, 3.01990732809571222812005354, 4.03038191606046844562845941 ] self.multiplicities = [2, 1, 1, 1, 1]
def test_guess_symmetry_2(usedf): C = Circle(0, 1.5) f = lambda z: z**27-2*z**11+0.5*z**6-1 df = (lambda z: 27*z**26-22*z**10+3*z**5) if usedf else None symmetry = lambda z: [z.conjugate()] roots = [-1.03509521179240, -0.920332541459108, 1.05026721944263, -0.983563736801535 - 0.382365167035741j, -0.983563736801535 + 0.382365167035741j, -0.792214346729517 - 0.520708613101932j, -0.792214346729517 + 0.520708613101932j, -0.732229626596468 - 0.757345327222341j, -0.732229626596468 + 0.757345327222341j, -0.40289002582335 - 0.825650446354661j, -0.40289002582335 + 0.825650446354661j, -0.383382611408318 - 0.967939747947639j, -0.383382611408318 + 0.967939747947639j, -0.02594227096144 - 1.05524415820652j, -0.02594227096144 + 1.05524415820652j, 0.160356899544475 - 0.927983420797727j, 0.160356899544475 + 0.927983420797727j, 0.41133738621461 - 0.967444751898913j, 0.41133738621461 + 0.967444751898913j, 0.576737152896681 - 0.719511178392941j, 0.576737152896681 + 0.719511178392941j, 0.758074415348703 - 0.724716122470435j, 0.758074415348703 + 0.724716122470435j, 0.903278407433416 - 0.22751872334709j, 0.903278407433416 + 0.22751872334709j, 0.963018623787179 - 0.427294816877434j, 0.963018623787179 + 0.427294816877434j] multiplicities = np.ones_like(roots) roots_approx_equal(C.roots(f, df, verbose=True, guessRootSymmetry=symmetry), (roots, multiplicities))
def setUp(self): self.C = Circle(0, 3) self.f = lambda z: (z * (z + 2))**2 * (exp(2 * z) * cos(z) - 1 - sin(z) + z**5) self.df = lambda z: 2 * (exp(2 * z) * cos(z) - 1 - sin(z) + z**5) * ( z**2 * (z + 2) + (z + 2)**2 * z) + (z * (z + 2))**2 * (2 * exp( 2 * z) * cos(z) - exp(2 * z) * sin(z) - cos(z) + 5 * z**4) self.roots = [ 0, -2, 2.23755778246706002284084684, -0.6511140702635986824274097994 - 0.3904257190882864369857773146j, -0.6511140702635986824274097994 + 0.3904257190882864369857773146j, 0.64857808095387581293067569277 - 1.35662268398824203963215495605j, 0.64857808095387581293067569277 + 1.35662268398824203963215495605j ] self.multiplicities = [3, 2, 1, 1, 1, 1, 1]
def setUp(self): # Ex 1.4.4 from [KB] self.C = Circle(0, 3) self.f = lambda z: (z * (z - 2))**2 * (exp(2 * z) * cos(z) + z**3 - 1 - sin(z)) self.df = lambda z: 2 * z * (z - 2)**2 * (exp(2 * z) * cos( z) + z**3 - 1 - sin(z)) + 2 * (z - 2) * z**2 * (exp(2 * z) * cos( z) + z**3 - 1 - sin(z)) + (z * (z - 2))**2 * (2 * exp( 2 * z) * cos(z) - exp(2 * z) * sin(z) + 3 * z**2 - cos(z)) self.roots = [ -0.4607141197289707542294459477 - 0.6254277693477682516688207854j, -0.4607141197289707542294459477 + 0.6254277693477682516688207854j, 0, 2, 1.66468286974551654134568653 ] self.multiplicities = [1, 1, 3, 2, 1]
F = (Z**3 - 2 * Z**2 + 0.5 * Z + 1) / (1) # Taking the derivative FP = sym.diff(F, Z) def fp(z): return complex(FP.evalf(subs={Z: z, C: c})) print(FP) #print(complex(FP.evalf(subs={Z: 4+1j, C: 1}))) # Finding the roots roots = [] cont = Circle(0, 3) zeroes = cont.roots(f) zeroes.show() print(zeroes) numroots = len(zeroes[0]) for j in range(numroots): roots.append(complex(zeroes[0][j].real, zeroes[0][j].imag)) print(roots) # Assign hues for each root colours = [(90, 190, 155), (0, 190, 155), (150, 255, 255), (90, 190, 155), (90, 190, 155), (90, 190, 155)] # Converting to latex # Write the latex expression to file exp = sym.Eq(FZ, F)
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')
from numpy import exp, cos, sin f = lambda z: (exp(2 * z) * cos(z) - 1 - sin(z) + z**5) * (z * (z + 2))**2 from matplotlib import pyplot as plt plt.figure(figsize=(4.8, 4.8), dpi=80) from cxroots import Circle C = Circle(0, 3) roots = C.roots(f) roots.show('readmeEx.png') print(roots)
from numpy import exp, sin, cos from cxroots import Circle C = Circle(0, 3) f = lambda z: (exp(-z) * sin(z / 2) - 1.2 * cos(z)) * (z + 1.2) * (z - 2.5)**2 C.demo_roots(f, guessRoots=[2.5, -1.2], saveFile='guessRoots.gif', writer='imagemagick')
from cxroots import Circle circle = Circle(center=1, radius=0.5) circle.show()
from cxroots import Circle C = Circle(0, 2) f = lambda z: z**6 + z**3 df = lambda z: 6 * z**5 + 3 * z**2 r = C.roots(f, df) r.show()
import pytest import numpy as np from scipy import cos, sin from cxroots import Circle, Rectangle from cxroots import CxDerivative @pytest.mark.parametrize('C', [ pytest.param(Circle(0, 2), id='circle'), pytest.param(Rectangle([-1.5, 1.5], [-2, 2]), id='rect'), pytest.param(None, id='default') ]) def test_CxDerivative(C): f = lambda z: z**10 - 2 * z**5 + sin(z) * cos(z / 2) df = lambda z: 10 * (z**9 - z**4) + cos(z) * cos(z / 2) - 0.5 * sin( z) * sin(z / 2) z = np.array([-1.234, 0.3 + 1j, 0.1j, -0.9 - 0.5j]) assert CxDerivative(f, z, n=1, contour=C) == pytest.approx(df(z))
pytest.param([3, 3.0001, 3.0002, 8, 8.0002, 8 + 0.0001j], [1, 1, 1, 1, 1, 1], id='cluster_10^-4', marks=pytest.mark.slow), pytest.param([3, 3.00001, 3.00002, 8, 8.00002, 8 + 0.00001j], [1, 1, 1, 1, 1, 1], id='cluster_10^-5', marks=[ pytest.mark.slow, pytest.mark.xfail(reason='Cluster of roots too tight') ]) ] contours = [ pytest.param(Rectangle([2, 9], [-1, 1]), id='rect'), pytest.param(Circle(0, 8.5), id='circle') ] @pytest.mark.parametrize('C', contours) @pytest.mark.parametrize('roots,multiplicities', funcs) def test_rootfinding_df(C, roots, multiplicities): f = lambda z: np.prod([z - r for r in roots], axis=0) df = lambda z: np.sum([ np.prod([z - r for r in np.delete(roots, i)], axis=0) for i in range(len(roots)) ], axis=0) roots_approx_equal(C.roots(f, df, verbose=True), (roots, multiplicities))
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 sin, cos f = lambda z: 1j*z**5 + z*sin(z) # Define f(z) df = lambda z: 5j*z**4 + z*cos(z) + sin(z) # Define f'(z) from cxroots import Circle C = Circle(0, 2) # Define a circle, centered at 0 and with radius 2 r = C.roots(f, df) # Find the roots of f(z) within the circle r.show('tutorial_roots.png')
from cxroots import Circle from numpy import exp, cos, sin f = lambda z: (z * (z + 2))**2 * (exp(2 * z) * cos(z) - 1 - sin(z) + z**5) C = Circle(0, 3) C.demo_roots(f, saveFile='rootsDemo.gif', writer='imagemagick')