__author__ = 'hwb'

# This file will calculate the trace of the higgs field squared for a point (x1, x2, x3) of space and a parameter k (between 0 and 1)

#

# Given a point x we associate to this four points P=(zeta, eta) on an elliptic curve. The four zeta values for these are determined by the roots of the quartic describing the curve

# quartic_roots(k, x1, x2, x3) but they are ordered by properties coming from the real structure order_roots(roots); then

# calc_zeta(k, x1, x2, x3)= order_roots(quartic_roots(k, x1, x2, x3))

# The second coordinate eta is given by calc_eta

#

# We need to calculate the Abel image of P. This will be done in calc_abel. To correctly identify this point with the correct sheet we will calculate the eta value of the Abel image by

# calc_eta_by_theta and if this agrees accept it, and if not shift by the half period to the correct sheet. This is done by abel_select and calc_abel returns the correct Abel images

# for each of the points.

#

# There is one transcendental function mu for each of the four points P given by calc_mu(k, x1, x2, x3, zeta, abel).

#

# All functions are then functions of (k, x1, x2, x3) and zeta_i, eta_i, mu_i (i=1..4).

#

from numpy import roots, complex, complex64, complex128, mat, dot, trace, pi, sqrt, sum, trace, linalg, matmul, array, matrix, conj, sort_complex

from math import *

from cmath import exp

from mpmath import ellipk, ellipe, j, taufrom, jtheta, qfrom, ellipf, asin, mfrom

from numpy import roots, complex64, conj, pi, sqrt, sum, trace, linalg, matmul, array

import os

from python_expressions.grams import grams

from python_expressions.phis import phis

def quartic_roots(k, x1, x2, x3):

# return roots([e4, e3, e2, e1, e0])

def order_roots(roots):

raise ValueError

def calc_zeta(k, x1, x2, x3):

return order_roots(quartic_roots(k, x1, x2, x3))

def calc_eta(k, x1, x2, x3):

zeta = calc_zeta(k, x1, x2, x3)

return map(lambda zetai : -(x2 + j*x1) * zetai**2 - 2*x3 * zetai + x2 - j*x1, zeta)

def calc_abel(k, zeta, eta):

return abel

def abel_select(k, abeli, etai):

return abeli

def calc_eta_by_theta(k, z):

/ ( ((jtheta(1, z*pi, qfrom(k=k), 0)) ** 2) * ((jtheta(3, z*pi, qfrom(k=k), 0)) ** 2) )

def calc_mu(k, x1, x2, x3, zeta, abel):

return mu

def is_awc_multiple_root(k, x1, x2, x3): # This will test if there are multiple roots; the analytic derivation assumes they are distinct

return False

def is_awc_branch_point(k, x1, x2, x3): # This will test if we get a branch point as a roots; these are numerically unstable

return False

def higgs_squared(k, x1, x2, x3):

return -(trace(matmul( matmul(higgs, inv_gram), matmul(higgs, inv_gram) )).real)/2

def higgs_squared_on_xy_plane(k_value, x0, x1, y0, y1, partition_size):

return points

def higgs_squared_on_xz_plane(k_value, x0, x1, z0, z1, partition_size):

return points

def write_point_to_file(points, filename):

fo.close()

def higgs_squared_on_x_axis(k_value, x0, x1, partition_size):