def steiltjes_from_gamma(s_vector, gamma_function, itr=200):
return (1 / s_vector) * (
1 + fixed_point(gamma_function, 1 / s_vector, itr, 1e-2))
def half_circle_stieltjes_gamma(s_vector, iterations):
return -(1 / s_vector) * (
fixed_point(half_circle_gamma_s_recip, s_vector, iterations) + 1)
def steiltjes_from_r_transform(s_vector, R_Transform, itr=200):
return fixed_point(R_Transform, s_vector, itr, 1e-3)
def quarter_gamma_s_fixed(z):
return fixed_point(quarter_gamma_s, z, 1000)
def quarter_gamma_fixed(z, alpha):
return fixed_point(quarter_gamma4, z, 100, func_param=alpha)
def gamma(z):
check = fixed_point(gamma_fixed, z, 100)
return check
def inverse_gamma(z_vector):
check = fixed_point(inv_gamma_fixed, z_vector, 100, unique_half_plane=True)
return check
import seaborn
size = 100
rows = 100
cols = 100
def gamma(z, gamma_z):
return np.power(gamma_z / z, 1 / 4) - 1
y = 1e-6
x, step = np.linspace(0.01, 10, 100, retstep=True)
s_vector = x + 1j * y
steiltjes_values = (1 / s_vector) * (
1 + fixed_point(gamma, 1 / s_vector, 100, 1e-2))
aed = estimated_pdf(steiltjes_values)
theoretic_capacity = aed_capacity(x, aed, 1 / cols, rows, step=step)
bins = int(size / 2)
main_terms = []
no_irs = []
irs = []
numeric_cacacity = []
average = 10
for i in range(average):
H = c_rand(rows, cols) @ c_rand(rows, cols) @ c_rand(rows, cols)
HH = H @ np.conj(H.T)
total_irs = HH
e_val_total_irs, e_vec_total_irs = np.linalg.eig(total_irs)
irs.append(np.real(e_val_total_irs))