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# Definitions
C = ComplexField(1000)
U = RealDistribution('uniform',[0,1])
R.<z> = C['z']
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# Using the code for Method 2
rndpoly1 = C(1)
for i in range(100):
rndpoly1 = rndpoly1 * (z - C(U.get_random_element()) - C(U.get_random_element())*C(I))
rootarray = []
for root in rndpoly1.roots():
j = root[1]
for i in range(j):
rootarray.append(root[0])
Gout = Graphics()
Gout += points(rootarray,color='white', markeredgecolor = 'red', pointsize=20,zorder = 1, marker = "o")
show(Gout,aspect_ratio=1)
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# Explore Problem (1)
# We want to see what happens if we construct a polynomial by taking the product of (z-x_i)*(z-conjugate(x_i)) n/2 times
# The polynomials are constructed almost the exact same way
rndpoly2 = C(1)
for i in range(50):
randRoot = C(U.get_random_element()) + C(U.get_random_element())*C(I)
rndpoly2 = rndpoly2 * (z - randRoot) * (z - conjugate(randRoot))
rootarray = []
for root in rndpoly2.roots():
j = root[1]
for i in range(j):
rootarray.append(root[0])
Gout = Graphics()
Gout += points(rootarray,color='white', markeredgecolor = 'red', pointsize=20,zorder = 1, marker = "o")
show(Gout,aspect_ratio=1)
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# Let's see what happens when we take a derivative
derpoly = rndpoly2.derivative(z)
cparray = []
for root in derpoly.roots():
j = root[1]
for i in range(j):
cparray.append(root[0])
Gout = Graphics()
Gout += points(rootarray,color='white', markeredgecolor = 'red', pointsize=20,zorder = 1, marker = "o")
Gout += points(cparray,color='white', markeredgecolor = 'blue', pointsize=20,zorder = 1, marker = "o")
show(Gout,aspect_ratio=1)
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# What about a lot of derivatives?
Gout = Graphics()
Gout += points(rootarray,color='white', markeredgecolor = 'red', pointsize=20,zorder = 1, marker = "o")
for i in range(25):
derpoly = rndpoly2.derivative(z)
cparray = []
for root in derpoly.roots():
j = root[1]
for i in range(j):
cparray.append(root[0])
Gout += points(cparray,color='white', markeredgecolor = 'blue', pointsize=20,zorder = 1, marker = "o")
rndpoly2 = derpoly
show(Gout,aspect_ratio=1)
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# How about some other distributions?
# Gaussian
sigma = 1
G = RealDistribution('gaussian', sigma)
rndpoly = C(1)
for i in range(50):
randRoot = C(G.get_random_element()) + C(G.get_random_element())*C(I)
rndpoly = rndpoly * (z - randRoot) * (z - conjugate(randRoot))
rootarray = []
for root in rndpoly.roots():
j = root[1]
for i in range(j):
rootarray.append(root[0])
Gout = Graphics()
Gout += points(rootarray,color='white', markeredgecolor = 'red', pointsize=20,zorder = 1, marker = "o")
for i in range(25):
derpoly = rndpoly.derivative(z)
cparray = []
for root in derpoly.roots():
j = root[1]
for i in range(j):
cparray.append(root[0])
Gout += points(cparray,color='white', markeredgecolor = 'blue', pointsize=20,zorder = 1, marker = "o")
rndpoly = derpoly
show(Gout,aspect_ratio=1)
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# Rayleigh
sigma = 1
RAY = RealDistribution('rayleigh', sigma)
rndpoly = C(1)
for i in range(50):
randRoot = C(RAY.get_random_element()) + C(RAY.get_random_element())*C(I)
rndpoly = rndpoly * (z - randRoot) * (z - conjugate(randRoot))
rootarray = []
for root in rndpoly.roots():
j = root[1]
for i in range(j):
rootarray.append(root[0])
Gout = Graphics()
Gout += points(rootarray,color='white', markeredgecolor = 'red', pointsize=20,zorder = 1, marker = "o")
for i in range(25):
derpoly = rndpoly.derivative(z)
cparray = []
for root in derpoly.roots():
j = root[1]
for i in range(j):
cparray.append(root[0])
Gout += points(cparray,color='white', markeredgecolor = 'blue', pointsize=20,zorder = 1, marker = "o")
rndpoly = derpoly
show(Gout,aspect_ratio=1)
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# Comparing Gaussians
# Create 3 Gaussian distributions with different variances and see behavior of polynomial and derivatives
sigma = 1
sigma1 = 2
sigma2 = 5
G = RealDistribution('gaussian', sigma)
G1 = RealDistribution('gaussian', sigma1)
G2 = RealDistribution('gaussian', sigma2)
# Generate 1st
rndpoly = C(1)
for i in range(25):
randRoot = C(G.get_random_element()) + C(G.get_random_element())*C(I)
rndpoly = rndpoly * (z - randRoot) * (z - conjugate(randRoot))
rootarray = []
for root in rndpoly.roots():
j = root[1]
for i in range(j):
rootarray.append(root[0])
# Generate 2nd
rndpoly1 = C(1)
for i in range(25):
randRoot = C(G1.get_random_element()) + C(G1.get_random_element())*C(I)
rndpoly1 = rndpoly1 * (z - randRoot) * (z - conjugate(randRoot))
rootarray1 = []
for root in rndpoly1.roots():
j = root[1]
for i in range(j):
rootarray1.append(root[0])
# Generate 3rd
rndpoly2 = C(1)
for i in range(25):
randRoot = C(G2.get_random_element()) + C(G2.get_random_element())*C(I)
rndpoly2 = rndpoly2 * (z - randRoot) * (z - conjugate(randRoot))
rootarray2 = []
for root in rndpoly2.roots():
j = root[1]
for i in range(j):
rootarray2.append(root[0])
Gout = Graphics()
Gout += points(rootarray,color='white', markeredgecolor = 'red', pointsize=20,zorder = 1, marker = "o")
Gout += points(rootarray1,color='white', markeredgecolor = 'blue', pointsize=20,zorder = 1, marker = "o")
Gout += points(rootarray2,color='white', markeredgecolor = 'green', pointsize=20,zorder = 1, marker = "o")
# Derivatives 1st
for i in range(10):
derpoly = rndpoly.derivative(z)
cparray = []
for root in derpoly.roots():
j = root[1]
for i in range(j):
cparray.append(root[0])
Gout += points(cparray,color='white', markeredgecolor = 'red', pointsize=20,zorder = 1, marker = "o")
rndpoly = derpoly
# Derivatives 2nd
for i in range(10):
derpoly1 = rndpoly1.derivative(z)
cparray = []
for root in derpoly1.roots():
j = root[1]
for i in range(j):
cparray.append(root[0])
Gout += points(cparray,color='white', markeredgecolor = 'blue', pointsize=20,zorder = 1, marker = "o")
rndpoly1 = derpoly1
# Derivatives 3rd
for i in range(10):
derpoly2 = rndpoly2.derivative(z)
cparray = []
for root in derpoly2.roots():
j = root[1]
for i in range(j):
cparray.append(root[0])
Gout += points(cparray,color='white', markeredgecolor = 'green', pointsize=20,zorder = 1, marker = "o")
rndpoly2 = derpoly2
show(Gout,aspect_ratio=1)
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# Kabluchko with Problem 1
# We want to see what happens if we construct a polynomial by taking the product of (z-x_i)*(z-conjugate(x_i)) n/2 times
# The polynomials are constructed almost the exact same way
rndpoly2 = C(1)
for i in range(50):
randRoot = C(U.get_random_element()) + C(U.get_random_element())*C(I)
rndpoly2 = rndpoly2 * (z - randRoot) * (z - conjugate(randRoot))
rootarray = []
for root in rndpoly2.roots():
j = root[1]
for i in range(j):
rootarray.append(root[0])
derpoly = rndpoly2.derivative(z)
cparray = []
for root in derpoly.roots():
j = root[1]
for i in range(j):
cparray.append(root[0])
kabluchko_sum = (1/100)*sum(rootarray) - (1/100)*sum(cparray)
Gout = Graphics()
Gout += points(kabluchko_sum,color='white', markeredgecolor = 'red', pointsize=20,zorder = 1, marker = "o")
show(Gout,aspect_ratio=1)
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# Repeat for n/2 = 5:50
# We want to see what happens if we construct a polynomial by taking the product of (z-x_i)*(z-conjugate(x_i)) n/2 times
# Do we still see convergence to 0 (does Kabluchko still hold?)
from sage.plot.colors import red, blue, lime
N = 5
Gout = Graphics()
plot_pts=[]
while N < 50:
q=50/N
rndpoly2 = C(1)
for i in range(N):
randRoot = C(U.get_random_element()) + C(U.get_random_element())*C(I)
rndpoly2 = rndpoly2 * (z - randRoot) * (z - conjugate(randRoot))
rootarray = []
for root in rndpoly2.roots():
j = root[1]
for i in range(j):
rootarray.append(root[0])
derpoly = rndpoly2.derivative(z)
cparray = []
for root in derpoly.roots():
j = root[1]
for i in range(j):
cparray.append(root[0])
kabluchko_sum = abs((1/(2*N))*sum(rootarray)) - abs((1/(2*N))*sum(cparray))
plot_pts.append((2*N,kabluchko_sum))
#Gout += list_plot(kabluchko_sum,color='white', markeredgecolor = Color(1,0,0).blend(Color(0,0,1),fraction=q), pointsize=20,zorder = 1, marker = "o")
N = N + 1
list_plot(plot_pts,color="green")
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# Repeat for n/2 = 5:150
# We want to see what happens if we construct a polynomial by taking the product of (z-x_i)*(z-conjugate(x_i)) n/2 times
# Do we still see convergence to 0 (does Kabluchko still hold?)
from sage.plot.colors import red, blue, lime
N = 5
Gout = Graphics()
plot_pts=[]
while N < 150:
q=250/N
rndpoly2 = C(1)
for i in range(N):
randRoot = C(U.get_random_element()) + C(U.get_random_element())*C(I)
rndpoly2 = rndpoly2 * (z - randRoot) * (z - conjugate(randRoot))
rootarray = []
for root in rndpoly2.roots():
j = root[1]
for i in range(j):
rootarray.append(root[0])
derpoly = rndpoly2.derivative(z)
cparray = []
for root in derpoly.roots():
j = root[1]
for i in range(j):
cparray.append(root[0])
kabluchko_sum = abs((1/(2*N))*sum(rootarray)) - abs((1/(2*N))*sum(cparray))
plot_pts.append((2*N,kabluchko_sum))
#Gout += list_plot(kabluchko_sum,color='white', markeredgecolor = Color(1,0,0).blend(Color(0,0,1),fraction=q), pointsize=20,zorder = 1, marker = "o")
N = N + 1
list_plot(plot_pts,color="green")
^C Traceback (click to the left of this block for traceback) ... __SAGE__ ^C
Traceback (most recent call last): N = 5
File "", line 1, in <module>
File "/tmp/tmpJSwL4z/___code___.py", line 29, in <module>
for root in derpoly.roots():
File "sage/rings/polynomial/polynomial_element.pyx", line 7691, in sage.rings.polynomial.polynomial_element.Polynomial.roots (build/cythonized/sage/rings/polynomial/polynomial_element.c:67636)
File "cypari2/gen.pyx", line 4357, in cypari2.gen.Gen.polroots
File "src/cysignals/signals.pyx", line 98, in cysignals.signals.sig_raise_exception
KeyboardInterrupt
__SAGE__
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