def B(k, ay, by):
term1 = (beta(k) * alpha(k - 2) - alpha(k)) / (alpha(k) * alpha(k - 2))
term2 = -(2 * k - 1) / (-1.0) * \
(((beta(k) * alpha(k - 2) + beta(k - 1) * alpha(k)) /\
(alpha(k - 2) * alpha(k - 1) * alpha(k))) + \
(ay ** 2 + by ** 2))
return term1 + term2
def B(k, ay, by):
term1 = (beta(k) * alpha(k - 2) - alpha(k)) / (alpha(k) * alpha(k - 2))
term2 = -(2 * k - 1) / (-1.0) * \
(((beta(k) * alpha(k - 2) + beta(k - 1) * alpha(k)) /\
(alpha(k - 2) * alpha(k - 1) * alpha(k))) + \
(ay ** 2 + by ** 2))
return term1 + term2
def modify_divide_r2(k_max, moments, a, b, first_term, second_term):
"""
Create the modified moments for k(x,y) * Pn / ((x - a)^2 + b^2) from
the modified moments for k(x, y) * Pn
The first two terms must be computed explicitly from the
expression for the kernel, k(x, y).
See (iii) on page 35 of Diligenti, Monegato 1997
I assume use of the Legendre Polynomials
"""
new_list = []
# The first two terms are computed explicitly
new_list.append(first_term)
new_list.append(second_term)
# Compute m_2 explicitly
term1 = alpha(1) * (moments[0] + 2 * a * new_list[1])
term2 = -(alpha(1) * (a ** 2 + b ** 2) + beta(1)) * new_list[0]
m_2 = term1 + term2
new_list.append(m_2)
# Compute q_r and q_i (the real and imaginary parts of the helper
# integral)
q_r = initial_q_r(a, new_list)
q_i = initial_q_i(b, new_list)
for k in range(1, k_max):
q_r.append(next_q_r(k, a, b, moments, q_i, q_r))
q_i.append(next_q_i(k, a, b, q_i, q_r))
# Compute m_j through the relation to q_j^I
for k in range(3, k_max + 1):
new_list.append(q_i[k] / b)
return new_list
def next_q_r(k, a, b, moments, q_i, q_r):
"""Helper method for modify_divide_r"""
term1 = alpha(k) * a * q_r[k]
term2 = -alpha(k) * b * q_i[k]
term3 = -beta(k) * q_r[k - 1]
term4 = alpha(k) * moments[k]
return term1 + term2 + term3 + term4
def modify_divide_r2(k_max, moments, a, b, first_term, second_term):
"""
Create the modified moments for k(x,y) * Pn / ((x - a)^2 + b^2) from
the modified moments for k(x, y) * Pn
The first two terms must be computed explicitly from the
expression for the kernel, k(x, y).
See (iii) on page 35 of Diligenti, Monegato 1997
I assume use of the Legendre Polynomials
"""
new_list = []
# The first two terms are computed explicitly
new_list.append(first_term)
new_list.append(second_term)
# Compute m_2 explicitly
term1 = alpha(1) * (moments[0] + 2 * a * new_list[1])
term2 = -(alpha(1) * (a**2 + b**2) + beta(1)) * new_list[0]
m_2 = term1 + term2
new_list.append(m_2)
# Compute q_r and q_i (the real and imaginary parts of the helper
# integral)
q_r = initial_q_r(a, new_list)
q_i = initial_q_i(b, new_list)
for k in range(1, k_max):
q_r.append(next_q_r(k, a, b, moments, q_i, q_r))
q_i.append(next_q_i(k, a, b, q_i, q_r))
# Compute m_j through the relation to q_j^I
for k in range(3, k_max + 1):
new_list.append(q_i[k] / b)
return new_list
def next_q_r(k, a, b, moments, q_i, q_r):
"""Helper method for modify_divide_r"""
term1 = alpha(k) * a * q_r[k]
term2 = -alpha(k) * b * q_i[k]
term3 = -beta(k) * q_r[k - 1]
term4 = alpha(k) * moments[k]
return term1 + term2 + term3 + term4
def initial_q_r(a, new_list):
"""Helper method for modify_divide_r"""
# Compute q_0^R and q_1^R explicitly
q_r = []
q_r.append(1 / alpha(0) * new_list[1] - a * new_list[0])
q_r.append(1 / alpha(1) * new_list[2] - a * new_list[1] +
beta(1) / alpha(1) * new_list[0])
return q_r
def initial_q_r(a, new_list):
"""Helper method for modify_divide_r"""
# Compute q_0^R and q_1^R explicitly
q_r = []
q_r.append(1 / alpha(0) * new_list[1] - a * new_list[0])
q_r.append(1 / alpha(1) * new_list[2] -
a * new_list[1] +
beta(1) / alpha(1) * new_list[0])
return q_r
def modify_times_x_minus_a(k_max, moments, a):
"""
Create the modified moments for k(x,y) * (x - a) * Pn from the
modified moments for k(x, y) * Pn
See (i) on page 34 of Diligenti, Monegato 1997
I assume use of the Legendre Polynomials
"""
new_list = []
new_list.append(moments[1] - a * moments[0])
for k in range(1, k_max + 1):
term1 = (1.0 / alpha(k)) * moments[k + 1]
term2 = -a * moments[k]
term3 = beta(k) / alpha(k) * moments[k - 1]
new_moment = term1 + term2 + term3
new_list.append(new_moment)
return new_list
def modify_times_x_minus_a(k_max, moments, a):
"""
Create the modified moments for k(x,y) * (x - a) * Pn from the
modified moments for k(x, y) * Pn
See (i) on page 34 of Diligenti, Monegato 1997
I assume use of the Legendre Polynomials
"""
new_list = []
new_list.append(moments[1] - a * moments[0])
for k in range(1, k_max + 1):
term1 = (1.0 / alpha(k)) * moments[k + 1]
term2 = -a * moments[k]
term3 = beta(k) / alpha(k) * moments[k - 1]
new_moment = term1 + term2 + term3
new_list.append(new_moment)
return new_list
def modify_divide_x_minus_a(k_max, moments, a, first_term):
"""
Create the modified moments for k(x,y) * Pn / (x - a) from the
modified moments for k(x,y) * Pn
The first term depends on k(x,y) and thus, the correct value
must be provided
See (ii) on pages 34 and 35 of Diligenti, Monegato 1997
I assume use of the Legendre polynomials
"""
new_list = []
new_list.append(first_term)
new_list.append(a * new_list[0] + moments[0])
for k in range(1, k_max):
term1 = alpha(k) * a * new_list[k]
term2 = -beta(k) * new_list[k - 1]
term3 = alpha(k) * moments[k]
new_moment = term1 + term2 + term3
new_list.append(new_moment)
return new_list
def modify_divide_x_minus_a(k_max, moments, a, first_term):
"""
Create the modified moments for k(x,y) * Pn / (x - a) from the
modified moments for k(x,y) * Pn
The first term depends on k(x,y) and thus, the correct value
must be provided
See (ii) on pages 34 and 35 of Diligenti, Monegato 1997
I assume use of the Legendre polynomials
"""
new_list = []
new_list.append(first_term)
new_list.append(a * new_list[0] + moments[0])
for k in range(1, k_max):
term1 = alpha(k) * a * new_list[k]
term2 = -beta(k) * new_list[k - 1]
term3 = alpha(k) * moments[k]
new_moment = term1 + term2 + term3
new_list.append(new_moment)
return new_list
def D(k, ay, by):
return - ((2 * k - 1) / (-1.0) *\
((beta(k - 1) * beta(k - 2)) / (alpha(k - 2) * alpha(k - 1)))) - \
(beta(k - 2) / alpha(k - 2))
def C(k, ay, by):
return ay + (2 * k - 1) / (-1.0) * (2 * ay / alpha(k - 1) * beta(k - 1))
def next_q_i(k, a, b, q_i, q_r):
"""Helper method for modify_divide_r"""
term1 = alpha(k) * a * q_i[k]
term2 = alpha(k) * b * q_r[k]
term3 = -beta(k) * q_i[k - 1]
return term1 + term2 + term3
def D(k, ay, by):
return - ((2 * k - 1) / (-1.0) *\
((beta(k - 1) * beta(k - 2)) / (alpha(k - 2) * alpha(k - 1)))) - \
(beta(k - 2) / alpha(k - 2))
def C(k, ay, by):
return ay + (2 * k - 1) / (-1.0) * (2 * ay / alpha(k - 1) * beta(k - 1))
def next_q_i(k, a, b, q_i, q_r):
"""Helper method for modify_divide_r"""
term1 = alpha(k) * a * q_i[k]
term2 = alpha(k) * b * q_r[k]
term3 = -beta(k) * q_i[k - 1]
return term1 + term2 + term3