def inner0_j1k2(j, k):
'''
Calculates the L2 inner product <P_j1, P_k2>
Args:
j, k: indices for the monomials P_j1, P_k2
Returns:
L2 inner product <P_j1, P_k2>
'''
s1 = 0
for l in range(j + 1):
s1 += alpha(j - l) * alpha(k + l + 1) + beta(k + l + 1) * eta(j - l)
return -2 * s1
def norm_p_jk(addr, j, k):
'''
This function calculates the value of the normal derivative of the
monomial basis partial_{n}p_jk at a point on SG addressed by
addr.
Args:
j, k: indices mentioned in preceding paragraph.
addr: address of evaluation point F_w(q_n) given as a string of
digits whose first digit is i and whose following digits are
those in w.
Returns:
value of partial_{n}p_jk(F_w(q_n))
'''
# Based on (2.14), (2.20) of the Calculus paper
if k == 1:
res = norm_f_jk(addr, j, 0)
for l in range(j + 1):
res += alpha(j - l) * (norm_f_jk(addr, l, 1) +
norm_f_jk(addr, l, 2))
if k == 2:
res = 0
for l in range(j + 1):
res += beta(j - l) * (norm_f_jk(addr, l, 1) +
norm_f_jk(addr, l, 2))
if k == 3:
res = 0
for l in range(j + 1):
res += gamma(j - l) * (norm_f_jk(addr, l, 1) -
norm_f_jk(addr, l, 2))
return res
def p_jk(addr, j, k):
'''
This function calculates the value of the monomial basis p_jk at a
point on SG addressed by addr. This code is based on the Splines
on Fractals paper and Calculus on SG I.
Args:
j, k: indices mentioned in preceding paragraph.
addr: address of evaluation point F_w(q_i) given as a string of
digits whose first digit is i and whose following digits are
those in w.
Returns:
value of p_jk(F_w(q_i))
'''
# Based on (2.14), (2.20) of the Calculus paper
if k == 1:
res = f_jk(addr, j, 0)
for l in range(j + 1):
res += alpha(j - l) * (f_jk(addr, l, 1) + f_jk(addr, l, 2))
if k == 2:
res = 0
for l in range(j + 1):
res += beta(j - l) * (f_jk(addr, l, 1) + f_jk(addr, l, 2))
if k == 3:
res = 0
for l in range(j + 1):
res += gamma(j - l) * (f_jk(addr, l, 1) - f_jk(addr, l, 2))
return res
def inner0_j3k3(j, k):
'''
Calculates the L2 inner product <P_j3, P_k3>
Args:
j, k: indices for the monomials P_j3, P_k3
Returns:
L2 inner product <P_j3, P_k3>
'''
ms = min(j, k)
s1 = 0
for l in range(j - ms, j + 1):
s1 += alpha(j - l + 1) * eta(k + l + 2) - alpha(k + l + 2) * eta(j -
l + 1)
return 18 * s1
def inner0_j1k1(j, k):
'''
Calculates the L2 inner product <P_j1, P_k1>
Args:
j, k: indices for the monomials P_j1, P_k1
Returns:
L2 inner product <P_j1, P_k1>
'''
ms = min(j, k)
s1 = 0
for l in range(j - ms, j + 1):
s1 += alpha(j - l) * eta(k + l + 1) - alpha(k + l + 1) * eta(j - l)
return 2 * s1
def pnj1(nn, j, i):
'''
This function calculates the value of P_{j,1}^(n) at q_i
Args:
nn: corresponds to the value of (n) in P_{j,1}^(n)
j: corresponds to the value of j in P_{j,1}^(n)
i: corresponds to the boundary index q_i
Returns:
Value of P_{j,1}^(n) at q_i
'''
return int(j == 0) if i == nn else alpha(j)
def inner0_j2k2(j, k):
'''
Calculates the L2 inner product <P_j2, P_k2>
Args:
j, k: indices for the monomials P_j2, P_k
Returns:
L2 inner product <P_j2, P_k2>
'''
ms = min(j, k)
s1 = 0
for l in range(j - ms, j + 1):
s1 += beta(j - l) * alpha(k + l + 1) - beta(k + l + 1) * ap(j - l)
return -2 * s1
def dnpj2(j):
return -alpha(j)