def _hex_orthob_at(order, p, q, r):
sk = [mp.sqrt(k + 0.5) for k in xrange(order)]
pa = [c*jp for c, jp in zip(sk, jacobi(order - 1, 0, 0, p))]
pb = [c*jp for c, jp in zip(sk, jacobi(order - 1, 0, 0, q))]
pc = [c*jp for c, jp in zip(sk, jacobi(order - 1, 0, 0, r))]
return [pi*pj*pk for pi in pa for pj in pb for pk in pc]
def jac_ortho_basis_at_mp(self, p, q, r):
a = 2 * (1 + p) / (1 - q) - 1 if q != 1 else -1
b = q
c = r
f = jacobi(self.order - 1, 0, 0, a)
df = jacobi_diff(self.order - 1, 0, 0, a)
pab = []
for i, (fi, dfi) in enumerate(zip(f, df)):
g = jacobi(self.order - i - 1, 2 * i + 1, 0, b)
dg = jacobi_diff(self.order - i - 1, 2 * i + 1, 0, b)
for j, (gj, dgj) in enumerate(zip(g, dg)):
cij = mp.sqrt((2 * i + 1) * (2 * i + 2 * j + 2)) / 2 ** (i + 1)
tmp = (1 - b) ** (i - 1) if i > 0 else 1
pij = 2 * tmp * dfi * gj
qij = tmp * (-i * fi + (1 + a) * dfi) * gj + (1 - b) ** i * fi * dgj
rij = (1 - b) ** i * fi * gj
pab.append([cij * pij, cij * qij, cij * rij])
sk = [mp.sqrt(k + 0.5) for k in range(self.order)]
hc = [s * jp for s, jp in zip(sk, jacobi(self.order - 1, 0, 0, c))]
dhc = [s * jp for s, jp in zip(sk, jacobi_diff(self.order - 1, 0, 0, c))]
return [[pij * hk, qij * hk, rij * dhk] for pij, qij, rij in pab for hk, dhk in zip(hc, dhc)]
def jac_ortho_basis_at_mp(self, p, q, r):
a = 2 * p / (1 - r) if r != 1 else 0
b = 2 * q / (1 - r) if r != 1 else 0
c = r
sk = [mp.mpf(2) ** (-k - 0.25) * mp.sqrt(k + 0.5) for k in range(self.order)]
fc = [s * jp for s, jp in zip(sk, jacobi(self.order - 1, 0, 0, a))]
gc = [s * jp for s, jp in zip(sk, jacobi(self.order - 1, 0, 0, b))]
dfc = [s * jp for s, jp in zip(sk, jacobi_diff(self.order - 1, 0, 0, a))]
dgc = [s * jp for s, jp in zip(sk, jacobi_diff(self.order - 1, 0, 0, b))]
ob = []
for i, (fi, dfi) in enumerate(zip(fc, dfc)):
for j, (gj, dgj) in enumerate(zip(gc, dgc)):
h = jacobi(self.order - max(i, j) - 1, 2 * (i + j + 1), 0, c)
dh = jacobi_diff(self.order - max(i, j) - 1, 2 * (i + j + 1), 0, c)
for k, (hk, dhk) in enumerate(zip(h, dh)):
ck = mp.sqrt(2 * (k + j + i) + 3)
tmp = (1 - c) ** (i + j - 1) if i + j > 0 else 1
pijk = 2 * tmp * dfi * gj * hk
qijk = 2 * tmp * fi * dgj * hk
rijk = (
tmp * (a * dfi * gj + b * fi * dgj - (i + j) * fi * gj) * hk
+ (1 - c) ** (i + j) * fi * gj * dhk
)
ob.append([ck * pijk, ck * qijk, ck * rijk])
return ob
def ortho_basis_at_mp(self, p, q, r):
r = r if r != 1 else r + mp.eps
a = 2*p/(1 - r)
b = 2*q/(1 - r)
c = r
sk = [mp.mpf(2)**(-k - 0.25)*mp.sqrt(k + 0.5)
for k in xrange(self.order)]
pa = [s*jp for s, jp in zip(sk, jacobi(self.order - 1, 0, 0, a))]
pb = [s*jp for s, jp in zip(sk, jacobi(self.order - 1, 0, 0, b))]
ob = []
for i, pi in enumerate(pa):
for j, pj in enumerate(pb):
cij = (1 - c)**(i + j)
pij = pi*pj
pc = jacobi(self.order - max(i, j) - 1, 2*(i + j + 1), 0, c)
for k, pk in enumerate(pc):
ck = mp.sqrt(2*(k + j + i) + 3)
ob.append(cij*ck*pij*pk)
return ob
def ortho_basis_at_mp(self, p, q, r):
sk = [mp.sqrt(k + 0.5) for k in range(self.order)]
pa = [c * jp for c, jp in zip(sk, jacobi(self.order - 1, 0, 0, p))]
pb = [c * jp for c, jp in zip(sk, jacobi(self.order - 1, 0, 0, q))]
pc = [c * jp for c, jp in zip(sk, jacobi(self.order - 1, 0, 0, r))]
return [pi * pj * pk for pi in pa for pj in pb for pk in pc]
def jac_ortho_basis_at_mp(self, p, q):
sk = [mp.sqrt(k + 0.5) for k in range(self.order)]
pa = [c * jp for c, jp in zip(sk, jacobi(self.order - 1, 0, 0, p))]
pb = [c * jp for c, jp in zip(sk, jacobi(self.order - 1, 0, 0, q))]
dpa = [c * jp for c, jp in zip(sk, jacobi_diff(self.order - 1, 0, 0, p))]
dpb = [c * jp for c, jp in zip(sk, jacobi_diff(self.order - 1, 0, 0, q))]
return [[dpi * pj, pi * dpj] for pi, dpi in zip(pa, dpa) for pj, dpj in zip(pb, dpb)]
def _tri_orthob_at(order, p, q):
a = 2*(1 + p)/(1 - q) - 1 if q != 1 else 0
b = q
ob = []
for i, pi in enumerate(jacobi(order - 1, 0, 0, a)):
pa = pi*(1 - b)**i
for j, pj in enumerate(jacobi(order - i - 1, 2*i + 1, 0, b)):
cij = mp.sqrt((2*i + 1)*(2*i + 2*j + 2)) / 2**(i + 1)
ob.append(cij*pa*pj)
return ob
def ortho_basis_at_mp(self, p, q):
a = 2 * (1 + p) / (1 - q) - 1 if q != 1 else -1
b = q
ob = []
for i, pi in enumerate(jacobi(self.order - 1, 0, 0, a)):
pa = pi * (1 - b) ** i
for j, pj in enumerate(jacobi(self.order - i - 1, 2 * i + 1, 0, b)):
cij = mp.sqrt((2 * i + 1) * (2 * i + 2 * j + 2)) / 2 ** (i + 1)
ob.append(cij * pa * pj)
return ob
def _pri_orthob_at(order, p, q, r):
a = 2*(1 + p)/(1 - q) - 1 if q != 1 else 0
b = q
c = r
pab = []
for i, pi in enumerate(jacobi(order - 1, 0, 0, a)):
ci = (1 - b)**i / 2**(i + 1)
for j, pj in enumerate(jacobi(order - i - 1, 2*i + 1, 0, b)):
cij = mp.sqrt((2*i + 1)*(2*i + 2*j + 2))*ci
pab.append(cij*pi*pj)
sk = [mp.sqrt(k + 0.5) for k in xrange(order)]
pc = [s*jp for s, jp in zip(sk, jacobi(order - 1, 0, 0, c))]
return [pij*pk for pij in pab for pk in pc]
def ortho_basis_at_mp(self, p, q, r):
q = q if q != 1 else q + mp.eps
a = 2*(1 + p)/(1 - q) - 1
b = q
c = r
pab = []
for i, pi in enumerate(jacobi(self.order - 1, 0, 0, a)):
ci = (1 - b)**i / 2**(i + 1)
for j, pj in enumerate(jacobi(self.order - i - 1, 2*i + 1, 0, b)):
cij = mp.sqrt((2*i + 1)*(2*i + 2*j + 2))*ci
pab.append(cij*pi*pj)
sk = [mp.sqrt(k + 0.5) for k in xrange(self.order)]
pc = [s*jp for s, jp in zip(sk, jacobi(self.order - 1, 0, 0, c))]
return [pij*pk for pij in pab for pk in pc]
def ortho_basis_at_mp(self, p, q, r):
a = -2 * (1 + p) / (q + r) - 1 if r != -q else -1
b = 2 * (1 + q) / (1 - r) - 1 if r != 1 else -1
c = r
ob = []
for i, pi in enumerate(jacobi(self.order - 1, 0, 0, a)):
ci = mp.mpf(2) ** (-2 * i - 1) * mp.sqrt(2 * i + 1) * (1 - b) ** i
for j, pj in enumerate(jacobi(self.order - i - 1, 2 * i + 1, 0, b)):
cj = mp.sqrt(i + j + 1) * 2 ** -j * (1 - c) ** (i + j)
cij = ci * cj
pij = pi * pj
jp = jacobi(self.order - i - j - 1, 2 * (i + j + 1), 0, c)
for k, pk in enumerate(jp):
ck = mp.sqrt(2 * (k + j + i) + 3)
ob.append(cij * ck * pij * pk)
return ob
def _tet_orthob_at(order, p, q, r):
a = -2*(1 + p)/(q + r) - 1 if q + r != 0 else 0
b = 2*(1 + q)/(1 - r) - 1 if r != 1 else 0
c = r
ob = []
for i, pi in enumerate(jacobi(order - 1, 0, 0, a)):
ci = mp.mpf(2)**(-2*i - 1.5)*mp.sqrt(4*i + 2)*(1 - b)**i
for j, pj in enumerate(jacobi(order - i - 1, 2*i + 1, 0, b)):
cj = mp.sqrt(i + j + 1)*2**-j*(1 - c)**(i + j)
cij = ci*cj
pij = pi*pj
jp = jacobi(order - i - j - 1, 2*(i + j + 1), 0, c)
for k, pk in enumerate(jp):
ck = mp.sqrt(2*(k + j + i) + 3)
ob.append(cij*ck*pij*pk)
return ob
def jac_ortho_basis_at_mp(self, p, q, r):
a = -2 * (1 + p) / (q + r) - 1 if r != -q else -1
b = 2 * (1 + q) / (1 - r) - 1 if r != 1 else -1
c = r
f = jacobi(self.order - 1, 0, 0, a)
df = jacobi_diff(self.order - 1, 0, 0, a)
ob = []
for i, (fi, dfi) in enumerate(zip(f, df)):
ci = mp.mpf(2) ** (-2 * i - 1) * mp.sqrt(2 * i + 1)
g = jacobi(self.order - i - 1, 2 * i + 1, 0, b)
dg = jacobi_diff(self.order - i - 1, 2 * i + 1, 0, b)
for j, (gj, dgj) in enumerate(zip(g, dg)):
cj = mp.sqrt(i + j + 1) * 2 ** -j
cij = ci * cj
h = jacobi(self.order - i - j - 1, 2 * (i + j + 1), 0, c)
dh = jacobi_diff(self.order - i - j - 1, 2 * (i + j + 1), 0, c)
for k, (hk, dhk) in enumerate(zip(h, dh)):
ck = mp.sqrt(2 * (k + j + i) + 3)
cijk = cij * ck
tmp1 = (1 - c) ** (i + j - 1) if i + j > 0 else 1
tmp2 = tmp1 * (1 - b) ** (i - 1) if i > 0 else 1
pijk = 4 * tmp2 * dfi * gj * hk
qijk = 2 * (tmp2 * (-i * fi + (1 + a) * dfi) * gj + tmp1 * (1 - b) ** i * fi * dgj) * hk
rijk = (
2 * (1 + a) * tmp2 * dfi * gj * hk
+ (1 + b) * tmp1 * (1 - b) ** i * fi * dgj * hk
+ (1 - c) ** (i + j) * (1 - b) ** i * fi * gj * dhk
- (i * (1 + b) * tmp2 + (i + j) * tmp1 * (1 - b) ** i) * fi * gj * hk
)
ob.append([cijk * pijk, cijk * qijk, cijk * rijk])
return ob
def jac_ortho_basis_at_mp(self, p, q):
a = 2 * (1 + p) / (1 - q) - 1 if q != 1 else -1
b = q
f = jacobi(self.order - 1, 0, 0, a)
df = jacobi_diff(self.order - 1, 0, 0, a)
ob = []
for i, (fi, dfi) in enumerate(zip(f, df)):
g = jacobi(self.order - i - 1, 2 * i + 1, 0, b)
dg = jacobi_diff(self.order - i - 1, 2 * i + 1, 0, b)
for j, (gj, dgj) in enumerate(zip(g, dg)):
cij = mp.sqrt((2 * i + 1) * (2 * i + 2 * j + 2)) / 2 ** (i + 1)
tmp = (1 - b) ** (i - 1) if i > 0 else 1
pij = 2 * tmp * dfi * gj
qij = tmp * (-i * fi + (1 + a) * dfi) * gj + (1 - b) ** i * fi * dgj
ob.append([cij * pij, cij * qij])
return ob
def ortho_basis_at_mp(self, p, q):
sk = [mp.sqrt(k + 0.5) for k in range(self.order)]
pa = [c * jp for c, jp in zip(sk, jacobi(self.order - 1, 0, 0, p))]
pb = [c * jp for c, jp in zip(sk, jacobi(self.order - 1, 0, 0, q))]
return [pi * pj for pi in pa for pj in pb]
def _quad_orthob_at(order, p, q):
sk = [mp.sqrt(k + 0.5) for k in xrange(order)]
pa = [c*jp for c, jp in zip(sk, jacobi(order - 1, 0, 0, p))]
pb = [c*jp for c, jp in zip(sk, jacobi(order - 1, 0, 0, q))]
return [pi*pj for pi in pa for pj in pb]
def _line_orthob_at(order, p):
jp = jacobi(order - 1, 0, 0, p)
return [mp.sqrt(i + 0.5)*p for i, p in enumerate(jp)]
def ortho_basis_at_mp(self, p):
jp = jacobi(self.order - 1, 0, 0, p)
return [mp.sqrt(i + 0.5) * p for i, p in enumerate(jp)]
def ortho_basis_at_mp(self, p):
jp = jacobi(self.order - 1, 0, 0, p)
return [mp.sqrt(i + 0.5) * p for i, p in enumerate(jp)]