def XG_plusBA(n, RA, RB, rh, l, pm1, Om, lam, tau0, width, deltaphi, isP):
bA = mp.sqrt(RA**2 - rh**2) / l
bB = mp.sqrt(RB**2 - rh**2) / l
f = lambda y: 2/Om/(bB-bA) * fp.exp(-fp.j*Om/2*(bB+bA)*y)\
* h_n2(y,n,RA,RB,rh,l,pm1,deltaphi,isP)\
* fp.sin( Om/2*(bB-bA)*(2*(tau0+width)-y))
return fp.quad(f, [2 * tau0 + width, 2 * (tau0 + width)])
def lourenco_dist_mp(
mgamma,
z,
n,
m,
sigma,
Ne,
Ne_scal,
scal_fac=1): # not sure if I do it right with the scale factor...
s = mgamma / 2. / Ne_scal * scal_fac
if s == 0:
return numpy.nan
#s = mgamma
# prob = mp.power(2, (1-m)/2.)*mp.power(Ne, 0.5)*mp.power(mp.fabs(s), (m-1)/2.)*(1+1/(Ne*mp.power(sigma, 2.)))*mp.exp(-Ne*s) / (mp.power(mp.pi,0.5)*mp.power(sigma, m)*mp.gamma(m/2.)) * mp.besselk((m-1)/2., Ne*mp.fabs(s)*mp.power(1+1/(Ne*mp.power(sigma, 2.)),0.5))
prob = mp.power(2, -(m + 1) / 2) * mp.exp(
-n * s / 4 / mp.power(z, 2.)) * mp.sqrt(n) * mp.power(
(1 + 4. * mp.power(z, 2.) / n / mp.power(sigma, 2.)) /
mp.power(s, 2.), (1 - m) / 4) * mp.power(sigma, -m) / (
mp.sqrt(mp.pi) * z * mp.gamma(m / 2.)) * mp.besselk(
(m - 1) / 2., 1 / (4. * mp.sqrt(
mp.power(z, 2.) /
(n * mp.power(s, 2.) *
(n / mp.power(z, 2.) + 4. / mp.power(sigma, 2.))))))
return float(prob / 2 / Ne_scal * scal_fac)
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 XG_minusAB(n, RA, RB, rh, l, pm1, Om, lam, tau0, width, deltaphi, isP):
bA = mp.sqrt(RA**2 - rh**2) / l
bB = mp.sqrt(RB**2 - rh**2) / l
f = lambda y: 2/Om/(bB-bA) * fp.exp(-fp.j*Om/2*(bB+bA)*y)\
* h_n1(y,n,RA,RB,rh,l,pm1,deltaphi,isP)\
* fp.sin( Om/2*(bB-bA)*(y-2*tau0))
return fp.quad(f, [2 * tau0, 2 * tau0 + width])
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 integrandof_Xn_plus(y, n, RA, RB, rh, l, pm1, Om, lam, tau0, width,
deltaphi, isP):
bA = mp.sqrt(RA**2 - rh**2) / l
bB = mp.sqrt(RB**2 - rh**2) / l
return fp.j*4 * fp.exp(-fp.j*Om*(bB+bA)*(tau0+width/2))/(Om*(bB+bA))\
* fp.cos( Om/2*(bB+bA)*(y-width) ) * fp.cos(Om/2*(bB-bA)*y)\
* g_n2(y,n,RA,RB,rh,l,pm1,deltaphi,isP)
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 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)
]
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 test_svd_test_case():
# a test case from Golub and Reinsch
# (see wilkinson/reinsch: handbook for auto. comp., vol ii-linear algebra, 134-151(1971).)
eps = mp.exp(0.8 * mp.log(mp.eps))
a = [[22, 10, 2, 3, 7],
[14, 7, 10, 0, 8],
[-1, 13, -1, -11, 3],
[-3, -2, 13, -2, 4],
[ 9, 8, 1, -2, 4],
[ 9, 1, -7, 5, -1],
[ 2, -6, 6, 5, 1],
[ 4, 5, 0, -2, 2]]
a = mp.matrix(a)
b = mp.matrix([mp.sqrt(1248), 20, mp.sqrt(384), 0, 0])
S = mp.svd_r(a, compute_uv = False)
S -= b
assert mp.mnorm(S) < eps
S = mp.svd_c(a, compute_uv = False)
S -= b
assert mp.mnorm(S) < eps
def test_gauss_quadrature_dynamic(verbose = False):
n = 5
A = mp.randmatrix(2 * n, 1)
def F(x):
r = 0
for i in xrange(len(A) - 1, -1, -1):
r = r * x + A[i]
return r
def run(qtype, FW, R, alpha = 0, beta = 0):
X, W = mp.gauss_quadrature(n, qtype, alpha = alpha, beta = beta)
a = 0
for i in xrange(len(X)):
a += W[i] * F(X[i])
b = mp.quad(lambda x: FW(x) * F(x), R)
c = mp.fabs(a - b)
if verbose:
print(qtype, c, a, b)
assert c < 1e-5
run("legendre", lambda x: 1, [-1, 1])
run("legendre01", lambda x: 1, [0, 1])
run("hermite", lambda x: mp.exp(-x*x), [-mp.inf, mp.inf])
run("laguerre", lambda x: mp.exp(-x), [0, mp.inf])
run("glaguerre", lambda x: mp.sqrt(x)*mp.exp(-x), [0, mp.inf], alpha = 1 / mp.mpf(2))
run("chebyshev1", lambda x: 1/mp.sqrt(1-x*x), [-1, 1])
run("chebyshev2", lambda x: mp.sqrt(1-x*x), [-1, 1])
run("jacobi", lambda x: (1-x)**(1/mp.mpf(3)) * (1+x)**(1/mp.mpf(5)), [-1, 1], alpha = 1 / mp.mpf(3), beta = 1 / mp.mpf(5) )
def w(sigma, t, T0dash):
wterm1 = 1 + (sigma**2) / (T0dash**2)
wterm2 = 1 + ((1 - sigma)**2) / (T0dash**2)
wterm3 = (sigma - 1) * mp.log(wterm1) / 4.0 + nonnegative(
(T0dash / 2.0) * mp.atan(sigma / T0dash) -
sigma / 2.0) + 1 / (12.0 * (T0dash - 0.33))
return mp.sqrt(wterm1) * mp.sqrt(wterm2) * mp.exp(wterm3)
def createP_sigma(lx, ly, band):
C = table.C_Sigma(ly / (2 / mp.sqrt(3)), lx / (mp.sqrt(3) / 2), band)
#Values tabulated on a rectangular grid, flipped and compensated for.
C = [
C[0], C[1], C[2], C[3] / (3**(1 / 2)), C[4] / (3**(1 / 2)),
C[5] / (3**(1 / 2))
]
#For sqrt(3) normalisation, see paper for reference.
f = lambda kx, ky, kz: C[0] * fi_As(lx, ly, kx, ky, kz) + C[1] * fi_Ax(
lx, ly, kx, ky, kz) + C[2] * fi_Ay(lx, ly, kx, ky, kz) + C[3] * fi_Bs(
lx, ly, kx, ky, kz) + C[4] * fi_Bx(lx, ly, kx, ky, kz) + C[
5] * fi_By(lx, ly, kx, ky, kz)
nrm = 1 / mp.sqrt(
mp.conj(C[0]) * C[0] + mp.conj(C[1]) * C[1] + mp.conj(C[2]) * C[2] +
3 *
(mp.conj(C[3]) * C[3] + mp.conj(C[4]) * C[4] + mp.conj(C[5]) * C[5]) +
2 * mp.re((mp.conj(C[0]) * C[3] * pss + mp.conj(C[0]) * C[4] * psx +
mp.conj(C[0]) * C[5] * psy + mp.conj(C[1]) * C[3] * psx +
mp.conj(C[1]) * C[4] * pxx + mp.conj(C[1]) * C[5] * pxy +
mp.conj(C[2]) * C[3] * psy + mp.conj(C[2]) * C[4] * pxy +
mp.conj(C[2]) * C[5] * pyy) *
(mp.exp(1j * mp.fdot([lx, ly, 0], R_1)) +
mp.exp(1j * mp.fdot([lx, ly, 0], R_2)) +
mp.exp(1j * mp.fdot([lx, ly, 0], R_3)))))
return lambda kx, ky, kz: f(kx, ky, kz) * nrm
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 xdipole_wigner3j_recurrence(l, wig_1llp1_000, wig_1llm1_000,
wig_1llp1_1m10, wig_1llm1_1m10):
#routine for computing successive Wigner3j symbols for x-polarized dipole based on recurrence
#wig_1llp1_000 stores wigner symbols of the form (1 l l+1 ; 0 0 0)
#wig_1llm1_000 stores wigner symbols of the form (1 l l-1 ; 0 0 0)
mp.dps = mp.dps * 2
if len(wig_1llp1_000) < 2:
#the m1=m2=m3=0 wigner3j symbols
wig_1llp1_000.clear()
wig_1llm1_000.clear()
wig_1llp1_000.extend([
mp.mpf(wigner_3j(1, 1, 2, 0, 0, 0).evalf(mp.dps)),
mp.mpf(wigner_3j(1, 2, 3, 0, 0, 0).evalf(mp.dps))
])
wig_1llm1_000.extend([
mp.mpf(wigner_3j(1, 1, 0, 0, 0, 0).evalf(mp.dps)),
mp.mpf(wigner_3j(1, 2, 1, 0, 0, 0).evalf(mp.dps))
])
#the m1=1, m2=-1, m3=0 wigner3j symbols
wig_1llp1_1m10.clear()
wig_1llm1_1m10.clear()
wig_1llp1_1m10.extend([
mp.mpf(wigner_3j(1, 1, 2, 1, -1, 0).evalf(mp.dps)),
mp.mpf(wigner_3j(1, 2, 3, 1, -1, 0).evalf(mp.dps))
])
wig_1llm1_1m10.extend([
mp.mpf(wigner_3j(1, 1, 0, 1, -1, 0).evalf(mp.dps)),
mp.mpf(wigner_3j(1, 2, 1, 1, -1, 0).evalf(mp.dps))
])
i = len(wig_1llp1_000)
while i < l:
i = i + 1
mp_i = mp.mpf(i)
wig_1llm1_000.append(-mp.sqrt(mp_i * (2 * mp_i - 3) /
((mp_i - 1) * (2 * mp_i + 1))) *
wig_1llp1_000[i - 3])
wig_1llp1_000.append(-mp.sqrt(
(2 * mp_i - 1) * (mp_i + 1) / (mp_i * (2 * mp_i + 3))) *
wig_1llm1_000[i - 1])
wig_1llm1_1m10.append(
(wig3j_j2rec_j2m1_factor(1, i, i - 1, 1, -1, 0) *
wig3j_j2rec_j2m1_factor(1, i - 1, i - 1, 1, -1, 0) +
wig3j_j2rec_j2m2_factor(1, i, i - 1, 1, -1, 0)) *
wig_1llp1_1m10[i - 1 - 2])
wig_1llp1_1m10.append(
(wig3j_j3rec_j3m1_factor(1, i, i + 1, 1, -1, 0) *
wig3j_j3rec_j3m1_factor(1, i, i, 1, -1, 0) +
wig3j_j3rec_j3m2_factor(1, i, i + 1, 1, -1, 0)) *
wig_1llm1_1m10[i - 1])
mp.dps = mp.dps // 2
def XBTZ_n(n, RA, RB, rh, l, pm1, Om, lam, tau0, width, deltaphi=0, isP=False):
bA = mp.sqrt(RA**2 - rh**2) / l
bB = mp.sqrt(RB**2 - rh**2) / l
tau0 *= bA
width *= bA
return -lam**2*bA*bB * (Xn_minus(n,RA,RB,rh,l,pm1,Om,lam,tau0,width,deltaphi,isP)\
- pm1*Xn_plus(n,RA,RB,rh,l,pm1,Om,lam,tau0,width,deltaphi,isP))
def check_xdipole_spherical_expansion(k, R, xp, yp, zp, dist):
print("original xdipole field function")
print(xdipole_field(k, 0, 0, -R - dist, xp, yp, zp))
print("spherical wave expansion dipole field")
print(xdipole_field_from_spherical_wave(k, 0, 0, -R - dist, xp, yp, zp))
field = lambda x, y, z: xdipole_field(k, 0, 0, -R - dist, x, y, z)
fnormsqr = get_field_normsqr(R, field)
print(fnormsqr)
wig_1llp1_000 = []
wig_1llm1_000 = []
wig_1llp1_1m10 = []
wig_1llm1_1m10 = [] #setup wigner3j lists
xdipole_wigner3j_recurrence(2, wig_1llp1_000, wig_1llm1_000,
wig_1llp1_1m10, wig_1llm1_1m10)
cs = []
cnormsqr = 0.0
l = 0
while (fnormsqr - cnormsqr) / fnormsqr > 1e-4:
l += 1
#cl = get_real_RgNM_l_coeffs_for_xdipole_field(l,k,R+dist, wig_1llp1_000,wig_1llm1_000,wig_1llp1_1m10,wig_1llm1_1m10)
cl = get_normalized_real_RgNM_l_coeffs_for_xdipole_field(
l, k, R, dist, wig_1llp1_000, wig_1llm1_000, wig_1llp1_1m10,
wig_1llm1_1m10)
cs.extend(cl)
# rhoM = mp_rho_M(l,k*R)
# rhoN = mp_rho_N(l,k*R)
# rhol = [rhoM, rhoM, rhoN, rhoN]
#cnormsqr += mp.re( np.sum(np.conjugate(cl)*cl * rhol / (2*k**3)) )#factor of 2 since here m!=0
cnormsqr += mp.re(np.sum(np.conjugate(cl) * cl))
# print(cnormsqr)
expfield1 = np.array([mp.zero, mp.zero, mp.zero])
for i in range(1, l + 1):
RgMe_field = get_rgM(k, xp, yp, zp, i, 1, 0)
RgMo_field = get_rgM(k, xp, yp, zp, i, 1, 1)
RgNe_field = get_rgN(k, xp, yp, zp, i, 1, 0)
RgNo_field = get_rgN(k, xp, yp, zp, i, 1, 1)
# expfield1 += (RgMe_field*cs[4*i-4] + RgMo_field*cs[4*i-3] +
# RgNe_field*cs[4*i-2] + RgNo_field*cs[4*i-1])
rhoM = mp_rho_M(i, k * R)
rhoN = mp_rho_N(i, k * R)
normM = mp.sqrt(rhoM / (2 * k**3))
normN = mp.sqrt(rhoN / (2 * k**3))
expfield1 += (RgMe_field * cs[4 * i - 4] / normM +
RgMo_field * cs[4 * i - 3] / normM +
RgNe_field * cs[4 * i - 2] / normN +
RgNo_field * cs[4 * i - 1] / normN)
print("expansion field via real spherical waves is")
print(expfield1)
print(cs)
def _CalculateLeastUpperBoundInoperativeInterval(x0, x1, v0, v1, vm, am):
# All input must already be of mp.mpf type
d = x1 - x0
temp1 = Prod([
number('2'),
Neg(Sqr(am)),
Sub(Prod([number('2'), am, d]), Add(Sqr(v0), Sqr(v1)))
])
if temp1 < zero:
T0 = number('-1')
T1 = number('-1')
else:
term1 = mp.fdiv(Add(v0, v1), am)
term2 = mp.fdiv(mp.sqrt(temp1), Sqr(am))
T0 = Add(term1, term2)
T1 = Sub(term1, term2)
temp2 = Prod([
number('2'),
Sqr(am),
Add(Prod([number('2'), am, d]), Add(Sqr(v0), Sqr(v1)))
])
if temp2 < zero:
T2 = number('-1')
T3 = number('-1')
else:
term1 = Neg(mp.fdiv(Add(v0, v1), am))
term2 = mp.fdiv(mp.sqrt(temp2), Sqr(am))
T2 = Add(term1, term2)
T3 = Sub(term1, term2)
newDuration = max(max(T0, T1), max(T2, T3))
if newDuration > zero:
dStraight = Prod([pointfive, Add(v0, v1), newDuration])
if Sub(d, dStraight) > 0:
amNew = am
vmNew = vm
else:
amNew = -am
vmNew = -vm
# import IPython; IPython.embed()
vp = Mul(pointfive, Sum([Mul(newDuration, amNew), v0,
v1])) # the peak velocity
if (Abs(vp) > vm):
dExcess = mp.fdiv(Sqr(Sub(vp, vmNew)), am)
assert (dExcess > 0)
deltaTime = mp.fdiv(dExcess, vm)
newDuration = Add(newDuration, deltaTime)
newDuration = Mul(newDuration, number('1.01')) # add 1% safety bound
return newDuration
else:
log.debug('Unable to calculate the least upper bound: T0 = {0}; T1 = {1}; T2 = {2}; T3 = {3}'.\
format(mp.nstr(T0, n=_prec), mp.nstr(T1, n=_prec), mp.nstr(T2, n=_prec), mp.nstr(T3, n=_prec)))
return number('-1')
def L_transform(j1,j2):
d = mp.mpf(10**(-mp.dps+1))
z0,z1 = j1*j2+1/(j1*j2), j1/j2+j2/j1
jp = d + mp.sqrt(z1)/mp.sqrt(z0)
dF = mp.log(mp.sqrt(z1)*mp.sqrt(z0))
return jp,dF
def RSZ_upto_c1(x):
x = mp.mpf(x.real)
tau = mp.sqrt(x/(2*PI))
N = int(tau.real)
p = tau-N
running_sum=0
for n in range(1,N+1):
running_sum += mp.cos(RStheta(x) - x*mp.log(n))/mp.sqrt(n)
return (2*running_sum + mp.power(-1,N-1)*mp.power(tau,-0.5)*(c0(p) - (1/tau)*c1(p))).real
def RSZ_plain(x):
x = mp.mpf(x.real)
tau = mp.sqrt(x / (2 * mp.pi()))
N = int(tau.real)
z = 2 * (x - N) - 1
running_sum = 0
for n in range(1, N + 1):
running_sum += mp.cos(RStheta(x) - (x * mp.log(n))) / mp.sqrt(n)
return (2 * running_sum).real
def wig3j_j2rec_j2m2_factor(j1, j2, j3, m1, m2, m3):
num = -j2 * mp.sqrt(
(j2 - m2 - 1) * (j2 + m2 - 1) * (j1 - j2 + j3 + 2) *
(-j1 + j2 + j3 - 1) * (j1 + j2 - j3 - 1) * (j1 + j2 + j3))
denom = (j2 - 1) * mp.sqrt(
(j2 - m2) * (j2 + m2) * (j1 - j2 + j3 + 1) * (-j1 + j2 + j3) *
(j1 + j2 - j3) * (j1 + j2 + j3 + 1))
return num / denom
def wig3j_j3rec_j3m2_factor(j1, j2, j3, m1, m2, m3):
num = -j3 * mp.sqrt(
(j3 + m3 - 1) * (j3 - m3 - 1) * (-j1 + j2 + j3 - 1) *
(j1 - j2 + j3 - 1) * (j1 + j2 - j3 + 2) * (j1 + j2 + j3))
denom = (j3 - 1) * mp.sqrt(
(j3 - m3) * (j3 + m3) * (-j1 + j2 + j3) * (j1 - j2 + j3) *
(j1 + j2 - j3 + 1) * (j1 + j2 + j3 + 1))
return num / denom
def gauss_hermite(n):
d = mp.matrix([mp.mpf('0.0') for _ in range(n)])
e = [mp.sqrt(k / 2) for k in mp.arange(1, n)]
e.append(mp.mpf('0.0'))
e = mp.matrix(e)
z = mp.eye(n)[0, :]
tridiag_eigen(mp, d, e, z)
z = mp.sqrt(mp.pi) * z.apply(lambda x: x**2)
return d, z.T
def integrand_deltaPdot_n(s, tau, n, R, rh, l, pm1, Om, lam, sig, deltaphi):
Zm = mp.mpf(rh**2/(R**2-rh**2)\
*(R**2/rh**2 * fp.cosh(rh/l * (deltaphi - 2*fp.pi*(n+1/2))) - 1))
Zp = mp.mpf(rh**2/(R**2-rh**2)\
*(R**2/rh**2 * fp.cosh(rh/l * (deltaphi - 2*fp.pi*(n+1/2))) + 1))
K = lam**2 * rh / (2 * fp.pi * fp.sqrt(2) * l *
fp.sqrt(R**2 - rh**2)) * fp.exp(-tau**2 / 2 / sig**2)
return K * fp.exp(-(tau-s)**2/2/sig**2) * fp.exp(-fp.j*Om*s)\
* ( 1/fp.sqrt(Zm + mp.cosh(rh/l/mp.sqrt(R**2-rh**2) * s))\
- pm1*1/fp.sqrt(Zp + mp.cosh(rh/l/mp.sqrt(R**2-rh**2) * s)) )
def Ht_Effective(z, t):
"""
This uses the effective approximation of H_t from Terry's blog
:param z: point at which H_t is computed
:param t: the "time" parameter
:return: H_t as a sum of two terms that are analogous to A and B, but now also with an efffective error bound (returned as percentage of |H_t|
"""
z, t = mp.mpc(z), mp.mpc(t)
sigma = (1 - z.imag) / 2.0
T = (z.real) / 2.0
Tdash = T + t * mp.pi() / 8.0
s1 = sigma + 1j * T
s2 = 1 - sigma + 1j * T
N = int((mp.sqrt(Tdash / (2 * mp.pi()))).real)
alph1 = alpha1(s1)
alph2 = alpha1(s2).conjugate()
A0_expo = (t / 4.0) * alph1 * alph1
B0_expo = (t / 4.0) * alph2 * alph2
H01_est1 = H01(s1)
H01_est2 = H01(s2).conjugate()
#begin main estimate block
A0 = mp.exp(A0_expo) * H01_est1
B0 = mp.exp(B0_expo) * H01_est2
A_sum = 0.0
B_sum = 0.0
for n in range(1, N + 1):
A_sum += 1 / mp.power(n, s1 + (t / 2.0) * alph1 -
(t / 4.0) * mp.log(n))
B_sum += 1 / mp.power(
n, 1 - s1 + (t / 2.0) * alph2 - (t / 4.0) * mp.log(n))
A = A0 * A_sum
B = B0 * B_sum
H = (A + B) / 8.0
#end main estimate block
#begin error block
A0_err_expo = (t / 4.0) * (abs(alph1)**2) #A0_expo.real may also work
B0_err_expo = (t / 4.0) * (abs(alph2)**2) #B0_expo.real may also work
epserr_1 = mp.exp(A0_err_expo) * abs(H01_est1) * abs(eps_err(s1, t)) / (
(T - 3.33) * 8.0)
epserr_2 = mp.exp(B0_err_expo) * abs(H01_est2) * abs(eps_err(s2, t)) / (
(T - 3.33) * 8.0)
epserr = epserr_1 + epserr_2
C0 = mp.sqrt(mp.pi()) * mp.exp(-1 * (t / 64.0) * (mp.pi()**2)) * mp.power(
Tdash, 1.5) * mp.exp(-1 * mp.pi() * T / 4.0)
C = C0 * vwf_err(s1, t) / 8.0
toterr = epserr + C
#print(epserr_1, epserr_2, C0, vwf_err(s1, t), C, toterr.real)
#end error block
if z.imag == 0: return (H.real, toterr.real / abs(H.real))
else: return (H, toterr.real / abs(H))
def __init__(self, npts):
if not mp.isint(mp.sqrt(npts)):
raise ValueError('Invalid number of points for quad rule')
rulecls = subclass_where(BaseLineQuadRule, name=self.name)
rule = rulecls(int(mp.sqrt(npts)))
self.points = [(i, j) for j in rule.points for i in rule.points]
if hasattr(rule, 'weights'):
self.weights = [i*j for j in rule.weights for i in rule.weights]
def XGEON_integrand_nBA(y, n, RA, RB, rh, l, pm1, Om, lam, sig, deltaphi):
bA = mp.sqrt(RA**2 - rh**2) / l
bB = mp.sqrt(RB**2 - rh**2) / l
K = 1
alp2 = bA**2 * bB**2 / 2 / (bA**2 + bB**2) / sig**2
bet2 = (bA + bB) * bA * bB / (bA**2 + bB**2)
E = (bB - bA) / fp.sqrt(2) / fp.sqrt(bB**2 + bA**2) * (
(bB + bA) * y / 2 / sig + fp.j * sig * Om)
return K*mp.exp(-alp2*y**2)*mp.exp(-fp.j*bet2*Om*y) *fp.erfc(E) \
* XGEON_denoms_n(y,n,RA,RB,rh,l,pm1,Om,lam,sig,deltaphi)
def _CalculateLeastUpperBoundInoperativeInterval(x0, x1, v0, v1, vm, am):
# All input must already be of mp.mpf type
d = x1 - x0
temp1 = Prod([number('2'), Neg(Sqr(am)), Sub(Prod([number('2'), am, d]), Add(Sqr(v0), Sqr(v1)))])
if temp1 < zero:
T0 = number('-1')
T1 = number('-1')
else:
term1 = mp.fdiv(Add(v0, v1), am)
term2 = mp.fdiv(mp.sqrt(temp1), Sqr(am))
T0 = Add(term1, term2)
T1 = Sub(term1, term2)
temp2 = Prod([number('2'), Sqr(am), Add(Prod([number('2'), am, d]), Add(Sqr(v0), Sqr(v1)))])
if temp2 < zero:
T2 = number('-1')
T3 = number('-1')
else:
term1 = Neg(mp.fdiv(Add(v0, v1), am))
term2 = mp.fdiv(mp.sqrt(temp2), Sqr(am))
T2 = Add(term1, term2)
T3 = Sub(term1, term2)
newDuration = max(max(T0, T1), max(T2, T3))
if newDuration > zero:
dStraight = Prod([pointfive, Add(v0, v1), newDuration])
if Sub(d, dStraight) > 0:
amNew = am
vmNew = vm
else:
amNew = -am
vmNew = -vm
# import IPython; IPython.embed()
vp = Mul(pointfive, Sum([Mul(newDuration, amNew), v0, v1])) # the peak velocity
if (Abs(vp) > vm):
dExcess = mp.fdiv(Sqr(Sub(vp, vmNew)), am)
assert(dExcess > 0)
deltaTime = mp.fdiv(dExcess, vm)
newDuration = Add(newDuration, deltaTime)
log.debug('Calculation successful: T0 = {0}; T1 = {1}; T2 = {2}; T3 = {3}'.format(mp.nstr(T0, n=_prec), mp.nstr(T1, n=_prec), mp.nstr(T2, n=_prec), mp.nstr(T3, n=_prec)))
newDuration = Mul(newDuration, number('1.01')) # add 1% safety bound
return newDuration
else:
if (FuzzyEquals(x0, x1, epsilon) and FuzzyZero(v0, epsilon) and FuzzyZero(v1, epsilon)):
# t = 0 is actually a correct solution
newDuration = 0
return newDuration
log.debug('Unable to calculate the least upper bound: T0 = {0}; T1 = {1}; T2 = {2}; T3 = {3}'.\
format(mp.nstr(T0, n=_prec), mp.nstr(T1, n=_prec), mp.nstr(T2, n=_prec), mp.nstr(T3, n=_prec)))
return number('-1')
def w(sigma, s, t):
T = s.imag
T0 = T
T0dash = T0 + mp.pi() * t / 8.0
Tdash = T + mp.pi() * t / 8.0
wterm1 = 1 + (sigma**2) / (T0dash**2)
wterm2 = 1 + ((1 - sigma)**2) / (T0dash**2)
wterm3 = (sigma - 1) * mp.log(wterm1) / 4.0 + nonnegative(
(T0dash / 2.0) * mp.atan(sigma / T0dash) -
sigma / 2.0) + 1 / (12.0 * (Tdash - 0.33))
return mp.sqrt(wterm1) * mp.sqrt(wterm2) * mp.exp(wterm3)
def z_x123_frm_m(N, m):
"""Function to get x1, x2 and x3 (eq 3, 5 and 6, [McNamara93]_)."""
M = -ellipk(m) / N
snMM = ellipfun('sn', u= -M, m=m)
snM = ellipfun('sn', u=M, m=m)
cnM = ellipfun('cn', u=M, m=m)
dnM = ellipfun('dn', u=M, m=m)
znM = z_zn(M, m)
x3 = snMM
x1 = x3 * mp.sqrt(1 - m) / dnM
x2 = x3 * mp.sqrt(1 - (cnM * znM) / (snM * dnM))
return x1, x2, x3
def z_x123_frm_m(N, m):
"""Function to get x1, x2 and x3 (eq 3, 5 and 6, [McNamara93]_)."""
M = -ellipk(m) / N
snMM = ellipfun('sn', u=-M, m=m)
snM = ellipfun('sn', u=M, m=m)
cnM = ellipfun('cn', u=M, m=m)
dnM = ellipfun('dn', u=M, m=m)
znM = z_zn(M, m)
x3 = snMM
x1 = x3 * mp.sqrt(1 - m) / dnM
x2 = x3 * mp.sqrt(1 - (cnM * znM) / (snM * dnM))
return x1, x2, x3
def XGEON_denoms_n(y, n, RA, RB, rh, l, pm1, Om, lam, sig, deltaphi):
bA = mp.sqrt(RA**2 - rh**2) / l
bB = mp.sqrt(RB**2 - rh**2) / l
Zm = rh**2 / l**2 / bA / bB * (
RA * RB / rh**2 * mp.cosh(rh / l * (deltaphi - 2 * mp.pi *
(n + 1 / 2))) - 1)
# Z-minus
Zp = rh**2 / l**2 / bA / bB * (
RA * RB / rh**2 * mp.cosh(rh / l * (deltaphi - 2 * mp.pi *
(n + 1 / 2))) + 1)
# Z-plus
return 1 / mp.sqrt(Zm + mp.cosh(y)) - pm1 / mp.sqrt(Zp + mp.cosh(y))
def test_laguerre_mpmath():
mp.dps = 51
scheme = quadpy.e1r.gauss_laguerre(2, mode="mpmath")
tol = 1.0e-50
x1 = 2 - mp.sqrt(2)
x2 = 2 + mp.sqrt(2)
assert (abs(scheme.points - [x1, x2]) < tol).all()
w1 = (2 + mp.sqrt(2)) / 4
w2 = (2 - mp.sqrt(2)) / 4
assert (abs(scheme.weights - [w1, w2]) < tol).all()
def f02(y, n, R, rh, l, pm1, Om, lam, sig):
K = lam**2 * sig / 2 / fp.sqrt(2 * fp.pi)
a = (R**2 - rh**2) * l**2 / 4 / sig**2 / rh**2
b = fp.sqrt(R**2 - rh**2) * Om * l / rh
Zp = mp.mpf((R**2 + rh**2) / (R**2 - rh**2))
if Zp - mp.cosh(y) > 0:
return fp.mpf(K * mp.exp(-a * y**2) * mp.cos(b * y) /
mp.sqrt(Zp - mp.cosh(y)))
elif Zp - mp.cosh(y) < 0:
return fp.mpf(-K * mp.exp(-a * y**2) * mp.sin(b * y) /
mp.sqrt(mp.cosh(y) - Zp))
else:
return 0
def test_laguerre_mpmath():
scheme = quadpy.e1r.GaussLaguerre(2, mode="mpmath", decimal_places=51)
tol = 1.0e-50
x1 = 2 - mp.sqrt(2)
x2 = 2 + mp.sqrt(2)
assert (abs(scheme.points - [x1, x2]) < tol).all()
w1 = (2 + mp.sqrt(2)) / 4
w2 = (2 - mp.sqrt(2)) / 4
assert (abs(scheme.weights - [w1, w2]) < tol).all()
return
def fn1(y, n, R, rh, l, pm1, Om, lam, sig):
K = lam**2 * sig / 2 / fp.sqrt(2 * fp.pi)
a = (R**2 - rh**2) * l**2 / 4 / sig**2 / rh**2
b = fp.sqrt(R**2 - rh**2) * Om * l / rh
Zm = mp.mpf(rh**2 / (R**2 - rh**2) *
(R**2 / rh**2 * fp.cosh(2 * fp.pi * rh / l * n) - 1))
if Zm == mp.cosh(y):
return 0
elif Zm - fp.cosh(y) > 0:
return fp.mpf(K * mp.exp(-a * y**2) * mp.cos(b * y) /
mp.sqrt(Zm - mp.cosh(y)))
else:
return fp.mpf(-K * mp.exp(-a * y**2) * mp.sin(b * y) /
mp.sqrt(mp.cosh(y) - Zm))
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 _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 _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 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 z_x123_frm_m(N, m):
r"""
Function to get x1, x2 and x3 (eq 3, 5 and 6, [McNamara93]_).
:param N: Order of the Zolotarev polynomial
:param m: m is the elliptic parameter (not the modulus k and not the nome q)
:rtype: Returns [x1,x2,x3] ... where x1, x2, and x3 are defined in Fig. 1, [McNamara93]_
"""
M = -ellipk(m) / N
snMM = ellipfun('sn', u= -M, m=m)
snM = ellipfun('sn', u=M, m=m)
cnM = ellipfun('cn', u=M, m=m)
dnM = ellipfun('dn', u=M, m=m)
znM = z_zn(M, m)
x3 = snMM
x1 = x3 * mp.sqrt(1 - m) / dnM
x2 = x3 * mp.sqrt(1 - (cnM * znM) / (snM * dnM))
return x1, x2, x3
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 _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 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 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 _Interpolate1DNoVelocityLimit(x0, x1, v0, v1, am):
# Check types
if type(x0) is not mp.mpf:
x0 = mp.mpf("{:.15e}".format(x0))
if type(x1) is not mp.mpf:
x1 = mp.mpf("{:.15e}".format(x1))
if type(v0) is not mp.mpf:
v0 = mp.mpf("{:.15e}".format(v0))
if type(v1) is not mp.mpf:
v1 = mp.mpf("{:.15e}".format(v1))
if type(am) is not mp.mpf:
am = mp.mpf("{:.15e}".format(am))
# Check inputs
assert(am > zero)
# Check for an appropriate acceleration direction of the first ramp
d = Sub(x1, x0)
dv = Sub(v1, v0)
difVSqr = Sub(v1**2, v0**2)
if Abs(dv) < epsilon:
if Abs(d) < epsilon:
# Stationary ramp
ramp0 = Ramp(zero, zero, zero, x0)
return ParabolicCurve([ramp0])
else:
dStraight = zero
else:
dStraight = mp.fdiv(difVSqr, Prod([2, mp.sign(dv), am]))
if IsEqual(d, dStraight):
# With the given distance, v0 and v1 can be directly connected using max/min
# acceleration. Here the resulting profile has only one ramp.
a0 = mp.sign(dv) * am
ramp0 = Ramp(v0, a0, mp.fdiv(dv, a0), x0)
return ParabolicCurve([ramp0])
sumVSqr = Add(v0**2, v1**2)
sigma = mp.sign(Sub(d, dStraight))
a0 = sigma * am # acceleration of the first ramp
vp = sigma * mp.sqrt(Add(Mul(pointfive, sumVSqr), Mul(a0, d)))
t0 = mp.fdiv(Sub(vp, v0), a0)
t1 = mp.fdiv(Sub(vp, v1), a0)
ramp0 = Ramp(v0, a0, t0, x0)
assert(IsEqual(ramp0.v1, vp)) # check soundness
ramp1 = Ramp(vp, Neg(a0), t1)
curve = ParabolicCurve([ramp0, ramp1])
assert(IsEqual(curve.d, d)) # check soundness
return curve
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 _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 z_Zolotarev(N, x, m):
"""Function to evaluate the Zolotarev polynomial (eq 1, [McNamara93]_)."""
M = -ellipk(m) / N
x3 = ellipfun('sn', u= -M, m=m)
xbar = x3 * mp.sqrt((x ** 2 - 1) / (x ** 2 - x3 ** 2)) # rearranged eq 21, [Levy70]_
u = ellipf(mp.asin(xbar), m) # rearranged eq 20, [Levy70]_, asn(x) = F(asin(x)|m)
f = mp.cosh((N / 2) * mp.log(z_eta(M + u, m) / z_eta(M - u, m)))
if (f.imag / f.real > 1e-10):
print "imaginary part of the Zolotarev function is not negligible!"
print "f_imaginary = ", f.imag
else:
if (x > 0): # no idea why I am doing this ... anyhow, it seems working
f = -f.real
else:
f = f.real
return f
def SolveQuartic(a, b, c, d, e):
"""
SolveQuartic solves a quartic (fouth order) equation of the form
ax^4 + bx^3 + cx^2 + dx + e = 0.
For the detail of formulae presented here, see https://en.wikipedia.org/wiki/Quartic_function
"""
# Check types
if type(a) is not mp.mpf:
a = mp.mpf("{:.15e}".format(a))
if type(b) is not mp.mpf:
b = mp.mpf("{:.15e}".format(b))
if type(c) is not mp.mpf:
c = mp.mpf("{:.15e}".format(c))
if type(d) is not mp.mpf:
d = mp.mpf("{:.15e}".format(d))
if type(e) is not mp.mpf:
e = mp.mpf("{:.15e}".format(e))
"""
# Working code (more readable but probably less precise)
p = (8*a*c - 3*b*b)/(8*a*a)
q = (b**3 - 4*a*b*c + 8*a*a*d)/(8*a*a*a)
delta0 = c*c - 3*b*d + 12*a*e
delta1 = 2*(c**3) - 9*b*c*d + 27*b*b*e + 27*a*d*d - 72*a*c*e
Q = mp.nthroot(pointfive*(delta1 + mp.sqrt(delta1*delta1 - 4*mp.power(delta0, 3))), 3)
S = pointfive*mp.sqrt(-mp.fdiv(mp.mpf('2'), mp.mpf('3'))*p + (one/(3*a))*(Q + delta0/Q))
x1 = -b/(4*a) - S + pointfive*mp.sqrt(-4*S*S - 2*p + q/S)
x2 = -b/(4*a) - S - pointfive*mp.sqrt(-4*S*S - 2*p + q/S)
x3 = -b/(4*a) + S + pointfive*mp.sqrt(-4*S*S - 2*p - q/S)
x4 = -b/(4*a) + S - pointfive*mp.sqrt(-4*S*S - 2*p - q/S)
"""
p = mp.fdiv(Sub(Prod([number('8'), a, c]), Mul(number('3'), mp.power(b, 2))), Mul(number('8'), mp.power(a, 2)))
q = mp.fdiv(Sum([mp.power(b, 3), Prod([number('-4'), a, b, c]), Prod([number('8'), mp.power(a, 2), d])]), Mul(8, mp.power(a, 3)))
delta0 = Sum([mp.power(c, 2), Prod([number('-3'), b, d]), Prod([number('12'), a, e])])
delta1 = Sum([Mul(2, mp.power(c, 3)), Prod([number('-9'), b, c, d]), Prod([number('27'), mp.power(b, 2), e]), Prod([number('27'), a, mp.power(d, 2)]), Prod([number('-72'), a, c, e])])
Q = mp.nthroot(Mul(pointfive, Add(delta1, mp.sqrt(Add(mp.power(delta1, 2), Mul(number('-4'), mp.power(delta0, 3)))))), 3)
S = Mul(pointfive, mp.sqrt(Mul(mp.fdiv(mp.mpf('-2'), mp.mpf('3')), p) + Mul(mp.fdiv(one, Mul(number('3'), a)), Add(Q, mp.fdiv(delta0, Q)))))
# log.debug("p = {0}".format(mp.nstr(p, n=_prec)))
# log.debug("q = {0}".format(mp.nstr(q, n=_prec)))
# log.debug("delta0 = {0}".format(mp.nstr(delta0, n=_prec)))
# log.debug("delta1 = {0}".format(mp.nstr(delta1, n=_prec)))
# log.debug("Q = {0}".format(mp.nstr(Q, n=_prec)))
# log.debug("S = {0}".format(mp.nstr(S, n=_prec)))
x1 = Sum([mp.fdiv(b, Mul(number('-4'), a)), Neg(S), Mul(pointfive, mp.sqrt(Sum([Mul(number('-4'), mp.power(S, 2)), Mul(number('-2'), p), mp.fdiv(q, S)])))])
x2 = Sum([mp.fdiv(b, Mul(number('-4'), a)), Neg(S), Neg(Mul(pointfive, mp.sqrt(Sum([Mul(number('-4'), mp.power(S, 2)), Mul(number('-2'), p), mp.fdiv(q, S)]))))])
x3 = Sum([mp.fdiv(b, Mul(number('-4'), a)), S, Mul(pointfive, mp.sqrt(Sum([Mul(number('-4'), mp.power(S, 2)), Mul(number('-2'), p), Neg(mp.fdiv(q, S))])))])
x4 = Sum([mp.fdiv(b, Mul(number('-4'), a)), S, Neg(Mul(pointfive, mp.sqrt(Sum([Mul(number('-4'), mp.power(S, 2)), Mul(number('-2'), p), Neg(mp.fdiv(q, S))]))))])
return [x1, x2, x3, x4]
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 test_levin_3():
mp.dps = 17
z=mp.mpf(2)
eps = mp.mpf(mp.eps)
with mp.extraprec(7*mp.prec): # we need copious amount of precision to sum this highly divergent series
L = mp.levin(method = "levin", variant = "t")
n, s = 0, 0
while 1:
s += (-z)**n * mp.fac(4 * n) / (mp.fac(n) * mp.fac(2 * n) * (4 ** n))
n += 1
v, e = L.step_psum(s)
if e < eps:
break
if n > 1000: raise RuntimeError("iteration limit exceeded")
eps = mp.exp(0.8 * mp.log(eps))
exact = mp.quad(lambda x: mp.exp( -x * x / 2 - z * x ** 4), [0,mp.inf]) * 2 / mp.sqrt(2 * mp.pi)
# there is also a symbolic expression for the integral:
# exact = mp.exp(mp.one / (32 * z)) * mp.besselk(mp.one / 4, mp.one / (32 * z)) / (4 * mp.sqrt(z * mp.pi))
err = abs(v - exact)
assert err < eps
w = mp.nsum(lambda n: (-z)**n * mp.fac(4 * n) / (mp.fac(n) * mp.fac(2 * n) * (4 ** n)), [0, mp.inf], method = "levin", levin_variant = "t", workprec = 8*mp.prec, steps = [2] + [1 for x in xrange(1000)])
err = abs(v - w)
assert err < eps
def norm_q(prob):
r"""
A multi-precision calculation of the
standard normal quantile function:
.. math::
\int_{-\infty}^{q(p)} \frac{e^{-z^2/2}}{\sqrt{2\pi}} \; dz = p
where $p$ is `prob`.
Parameters
----------
prob : float
Returns
-------
quantile : float
"""
return np.array(mp.erfinv(2*prob-1)*mp.sqrt(2))
def z_Zolotarev(N, x, m):
r"""
Function to evaluate the Zolotarev polynomial (eq 1, [McNamara93]_).
:param N: Order of the Zolotarev polynomial
:param x: The argument at which one would like to evaluate the Zolotarev polynomial
:param m: m is the elliptic parameter (not the modulus k and not the nome q)
:rtype: Returns a float, the value of Zolotarev polynomial at x
"""
M = -ellipk(m) / N
x3 = ellipfun('sn', u= -M, m=m)
xbar = x3 * mp.sqrt((x ** 2 - 1) / (x ** 2 - x3 ** 2)) # rearranged eq 21, [Levy70]_
u = ellipf(mp.asin(xbar), m) # rearranged eq 20, [Levy70]_, asn(x) = F(asin(x)|m)
f = mp.cosh((N / 2) * mp.log(z_eta(M + u, m) / z_eta(M - u, m)))
if (f.imag / f.real > 1e-10):
print "imaginary part of the Zolotarev function is not negligible!"
print "f_imaginary = ", f.imag
else:
if (x > 0): # no idea why I am doing this ... anyhow, it seems working
f = -f.real
else:
f = f.real
return f
Yp = mp.matrix(f.size, 1)
Zc = mp.matrix(f.size, 1)
A = np.ndarray(f.size)
R = np.ndarray(f.size)
I = np.ndarray(f.size)
Gr = np.ndarray(f.size)
Gi = np.ndarray(f.size)
for n in range(f.size):
Zap[n] = ZapA[n]*( i0(gammac[n]*a) / i1(gammac[n]*a) )
Zbp[n] = ZbpA[n] * ((i0(gammac[n]*b) * k1(gammac[n]*c) + k0(gammac[n]*b)*i1(gammac[n]*c)) / ( i1(gammac[n]*c)*k1(gammac[n]*b) - i1(gammac[n]*b)*k1(gammac[n]*c) ) )
ZL[n] = 1j*omega[n]*Lp
YC[n] = 1j*omega[n]*Cp
Zp[n] = Zap[n] + Zbp[n] + ZL[n]
Yp[n] = YC[n] # + G[n]
Zc[n] = mp.sqrt(Zp[n]/Yp[n])
A[n] = mp.fabs(Zc[n])
R[n] = mp.re(Zc[n])
I[n] = mp.im(Zc[n])
Gr[n] = mp.re(mp.sqrt(Zp[n]*Yp[n]))
Gi[n] = mp.im(mp.sqrt(Zp[n]*Yp[n]))
fig = plt.figure()
plt.plot(f/10**6, A, label='Magnitude', color='red')
plt.xlabel('Frequency [MHz]')
plt.ylabel('Magnitude of Z$_{c}$ [$\Omega$]')
plt.legend()
plt.xlim([-0.3,100])
#plt.show()
fig2 = plt.figure()
plt.plot(f/10**6, R, label='Real Part', color='red')
def dcurve(a, b, c):
return a + b + c + 2 * mp.sqrt(a * b + b * c + c * a)
# * Description: AC.
# * http://en.wikipedia.org/wiki/Descartes%27_theorem
# ********************************/
from mpmath import mp
def dcurve(a, b, c):
return a + b + c + 2 * mp.sqrt(a * b + b * c + c * a)
def dfs(a, b, c, depth):
if depth == 10:
return 0
a, b, c = sorted([a, b, c])
d = dcurve(a, b, c)
if a == b == c:
return 1 / d**2 + 3 * dfs(a, a, d, depth + 1)
elif a == b != c:
return 1 / d**2 + dfs(a, a, d, depth + 1) + 2 * dfs(b, c, d, depth + 1)
elif a != b == c:
return 1 / d**2 + dfs(b, c, d, depth + 1) + 2 * dfs(a, b, d, depth + 1)
elif a != b != c:
return 1 / d**2 + dfs(a, b, d, depth + 1) + dfs(a, c, d, depth + 1) + dfs(b, c, d, depth + 1)
else:
print("ERROR")
k0 = 1 + 2 / mp.sqrt(3)
print("Answer is %.8f" % (1 - (dfs(k0, k0, k0, 0) + 3 * dfs(k0, k0, -1, 0) + 3 / k0**2)))
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]
