def test_chebyshev2_mpmath():
scheme = quadpy.line_segment.ChebyshevGauss2(4, mode="mpmath", decimal_places=51)
tol = 1.0e-50
x1 = mp.cos(2 * mp.pi / 5)
x2 = mp.cos(1 * mp.pi / 5)
assert (abs(scheme.points - [-x2, -x1, +x1, +x2]) < tol).all()
w1 = mp.pi / 5 * mp.sin(2 * mp.pi / 5) ** 2
w2 = mp.pi / 5 * mp.sin(1 * mp.pi / 5) ** 2
assert (abs(scheme.weights - [w2, w1, w1, w2]) < tol).all()
return
def dTk(x):
with mp.extradps(extradps):
x = clip(x, -1.0, 1.0)
if mp.almosteq(x, mp.one): return pos
if mp.almosteq(x, -mp.one): return neg
moredps = max(
0,
int(-math.log10(
min(abs(x - mp.one), abs(x + mp.one))) / 2))
moredps = min(moredps, 100)
with mp.extradps(moredps):
t = mp.acos(x)
return k * mp.sin(k * t) / mp.sin(t)
def test_chebyshev2_mpmath():
mp.dps = 51
scheme = quadpy.c1.chebyshev_gauss_2(4, mode="mpmath")
tol = 1.0e-50
x1 = mp.cos(2 * mp.pi / 5)
x2 = mp.cos(1 * mp.pi / 5)
assert (abs(scheme.points_symbolic - [+x2, +x1, -x1, -x2]) < tol).all()
w1 = mp.pi / 5 * mp.sin(2 * mp.pi / 5)**2
w2 = mp.pi / 5 * mp.sin(1 * mp.pi / 5)**2
assert (abs(scheme.weights_symbolic - [w2, w1, w1, w2]) < tol).all()
def __init__(self, domain=None, name='sin^2'):
super(SinSquaredExpression,
self).__init__(mp_terms=[
lambda x: mp.sin(x)**2, lambda x: mp.sin(2 * x),
lambda x: 2 * mp.cos(2 * x)
],
fp_terms=[
lambda x: math.sin(x)**2,
lambda x: math.sin(2 * x),
lambda x: 2 * math.cos(2 * x)
],
desc="sin^2(x)",
domain=domain,
name=name)
def mp_cyl(x):
f = mp.mpf
theta = f(THETA) * mp.pi / f(180)
qr = x * f(RADIUS) * mp.sin(theta)
qh = x * f(LENGTH) / f(2) * mp.cos(theta)
be = f(2) * mp.j1(qr) / qr
si = mp.sin(qh) / qh
background = f(0)
#background = f(1)/f(1000)
volume = mp.pi * f(RADIUS)**f(2) * f(LENGTH)
contrast = f(5)
units = f(1) / f(10000)
#return be
#return si
return units * (volume * contrast * be * si)**f(2) / volume + background
def ddTk(x):
with mp.extradps(extradps):
x = clip(x, -1.0, 1.0)
if mp.almosteq(x, mp.one): return pos
if mp.almosteq(x, -mp.one): return neg
moredps = max(
0,
int(-math.log10(
min(abs(x - mp.one), abs(x + mp.one))) * 1.5) +
2)
moredps = min(moredps, 100)
with mp.extradps(moredps):
t = mp.acos(x)
s = mp.sin(t)
return -k**2 * mp.cos(k * t) / s**2 + k * mp.cos(
t) * mp.sin(k * t) / s**3
def chebyshev_gauss_2(n, mode="numpy", decimal_places=None):
"""Chebyshev-Gauss quadrature for \\int_{-1}^1 f(x) * sqrt(1+x^2) dx."""
degree = n if n % 2 == 1 else n + 1
# TODO make explicit for all modes
if mode == "numpy":
points = numpy.cos(numpy.pi * numpy.arange(1, n + 1) / (n + 1))
weights = (
numpy.pi
/ (n + 1)
* (numpy.sin(numpy.pi * numpy.arange(1, n + 1) / (n + 1))) ** 2
)
elif mode == "sympy":
points = numpy.array(
[sympy.cos(sympy.Rational(k, n + 1) * sympy.pi) for k in range(1, n + 1)]
)
weights = numpy.array(
[
sympy.pi / (n + 1) * sympy.sin(sympy.pi * sympy.Rational(k, n + 1)) ** 2
for k in range(1, n + 1)
]
)
else:
assert mode == "mpmath"
points = numpy.array(
[mp.cos(mp.mpf(k) / (n + 1) * mp.pi) for k in range(1, n + 1)]
)
weights = numpy.array(
[
mp.pi / (n + 1) * mp.sin(mp.pi * mp.mpf(k) / (n + 1)) ** 2
for k in range(1, n + 1)
]
)
return C1Scheme("Chebyshev-Gauss 2", degree, weights, points)
def two_body_reference(self, t1, num_1 = 0, num_2 = 1):
""" reference notations are same as Yoel's notes
using a precision of 64 digits for max accuracy """
mp.dps = 64
self.load_data()
t1 = mp.mpf(t1)
m_1 = mp.mpf(self.planet_lst[num_1].mass)
m_2 = mp.mpf(self.planet_lst[num_2].mass)
x1_0 = mp.matrix(self.planet_lst[num_1].loc)
x2_0 = mp.matrix(self.planet_lst[num_2].loc)
v1_0 = mp.matrix(self.planet_lst[num_1].v)
v2_0 = mp.matrix(self.planet_lst[num_2].v)
M = m_1 + m_2
x_cm = (1/M)*((m_1*x1_0)+(m_2*x2_0))
v_cm = (1/M)*((m_1*v1_0)+(m_2*v2_0))
u1_0 = v1_0 - v_cm
w = x1_0 - x_cm
r_0 = mp.norm(w)
alpha = mp.acos((np.inner(u1_0,w))/(mp.norm(u1_0)*mp.norm(w)))
K = mp.mpf(self.G) * (m_2**3)/(M**2)
u1_0 = mp.norm(u1_0)
L = r_0 * u1_0 * mp.sin(alpha)
cosgamma = ((L**2)/(K*r_0)) - 1
singamma = -((L*u1_0*mp.cos(alpha))/K)
gamma = mp.atan2(singamma, cosgamma)
e = mp.sqrt(((r_0*(u1_0**2)*(mp.sin(alpha)**2)/K)-1)**2 + \
(((r_0**2)*(u1_0**4)*(mp.sin(alpha)**2)*(mp.cos(alpha)**2))/(K**2)) )
r_theta = lambda theta: (L**2)/(K*(1+(e*mp.cos(theta - gamma))))
f = lambda x: r_theta(x)**2/L
curr_theta = 0
max_it = 30
it = 1
converged = 0
while ((it < max_it) & (converged != 1)):
t_val = mp.quad(f,[0,curr_theta]) - t1
dt_val = f(curr_theta)
delta = t_val/dt_val
if (abs(delta)<1.0e-6):
converged = 1
curr_theta -= delta
it += 1
x1_new = mp.matrix([r_theta(curr_theta)*mp.cos(curr_theta), r_theta(curr_theta)*mp.sin(curr_theta)])
# x2_new = -(m_1/m_2)*(x1_new) +(x_cm + v_cm*t1)
x1_new = x1_new + (x_cm + v_cm*t1)
return x1_new
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 Construct_Dp(Phi, Psi, Omega, tt):
Cos = mp.matrix([[
mp.cos(eigvalst[i] * tt) if i == j else mp.mpf(0.0) for j in range(L)
] for i in range(L)])
Sin = mp.matrix([[
mp.sin(eigvalst[i] * tt) if i == j else mp.mpf(0.0) for j in range(L)
] for i in range(L)])
OmegaCosSin = Cos * Omega * Sin
Dp = -Phi * (OmegaCosSin.T - OmegaCosSin) * Phi.T
return (Dp)
def Construct_Dm(Phi, Psi, Omega, tt):
Cos = mp.matrix([[
mp.cos(eigvalst[i] * tt) if i == j else mp.mpf(0.0) for j in range(L)
] for i in range(L)])
Sin = mp.matrix([[
mp.sin(eigvalst[i] * tt) if i == j else mp.mpf(0.0) for j in range(L)
] for i in range(L)])
OmegaSinCos = Sin * Omega * Cos
Dm = -Psi * (OmegaSinCos - OmegaSinCos.T) * Psi.T
return (Dm)
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 Construct_K(Phi, Psi, Omega, tt):
Cos = mp.matrix([[
mp.cos(eigvalst[i] * tt) if i == j else mp.mpf(0.0) for j in range(L)
] for i in range(L)])
Sin = mp.matrix([[
mp.sin(eigvalst[i] * tt) if i == j else mp.mpf(0.0) for j in range(L)
] for i in range(L)])
OmegaCos = Cos * Omega * Cos
OmegaSin = Sin * Omega * Sin
Kmatrix = -Phi * (OmegaCos + OmegaSin.T) * Psi.T
return (Kmatrix)
def verify(self, x0, x1, y0, y1, s, t, x, y):
"Verify the transformation given by (s, t, x, y) on pentagon."
pentagon = [(x0, 0), (x1, 0), (0, y0), (0, y1), (self.side, self.side)]
# reference point in interior of pentagon
ref = matrix([x0 + x1 + self.side, y0 + y1 + self.side]) / 5
translation = matrix(
[s * ref[0] + t * ref[1] + x, -t * ref[0] + s * ref[1] + y])
# reconstruct rotation angle
norm = mp.sqrt(s * s + t * t)
if t < 0:
angle = -mp.acos(s / norm)
else:
angle = mp.acos(s / norm)
rotation = matrix([[mp.cos(angle), mp.sin(angle)],
[-mp.sin(angle), mp.cos(angle)]])
pentagon = list(map(lambda pt: matrix(pt), pentagon))
ipentagon = list(
map(lambda pt: rotation * (pt - ref) + translation, pentagon))
# Verify that ipentagon is a congruent image of pentagon
eps = mp.mpf(10)**-(mp.dps - 2) # accept error in last two digits
for i in range(5):
for j in range(i + 1, 5):
d1 = mp.norm(pentagon[j] - pentagon[i])
d2 = mp.norm(ipentagon[j] - ipentagon[i])
dd = abs(d1 - d2)
assert dd < eps, (pentagon, dd, eps)
dists = []
for p in ipentagon:
dists.append(self.nleft * p)
dists.append(self.ndown * p)
dists.append(self.nright * p - self.offset)
dist = max(map(lambda m: m[0], dists))
if dist > -self.distance:
sys.stderr.write("Pentagon failing slack test: %d %d %d %d\n" %
(x0, x1, y0, y1))
return False
return True
def dRdEe_noint(self, lx, ly, band, fdm, mchi_index):
if band == "pi":
local_mchi = ct.mchi
if mchi_index != 0:
local_mchi = ct.m_chi_list[mchi_index - 1]
qsqrt = lambda kr: mp.sqrt(local_mchi**2 * ct.vmax**2 - (
local_mchi / ct.me) * kr**2 - 2 * local_mchi *
(E_pi_minus(lx, ly) + ct.fi))
qmini = lambda kr: local_mchi * ct.vmax - qsqrt(kr)
qmaxi = lambda kr: local_mchi * ct.vmax + qsqrt(kr)
else:
local_mchi = ct.mchi
qsqrt = lambda kr: mp.sqrt(ct.mchi**2 * ct.vmax**2 - (
ct.mchi / ct.me) * kr**2 - 2 * ct.mchi *
(E_sigma(lx, ly, band) + ct.fi))
qmini = lambda kr: ct.mchi * ct.vmax - qsqrt(kr)
qmaxi = lambda kr: ct.mchi * ct.vmax + qsqrt(kr)
return lambda kr, kt, kp: (1 / 2) * mp.sin(kt) * self.dR(
lx, ly, kr, kt, kp, qmini(kr), qmaxi(kr), band, fdm, local_mchi)
def mp_factory(n):
return (mp.cos, lambda x: -mp.sin(x), lambda x: -mp.cos(x),
mp.sin)[n % 4]
#add_function(
# name="fnlibJ1",
# mp_function=mp.j1,
# np_function=fnlib.J1,
# ocl_function=make_ocl("return sas_J1(q);", "sas_J1", ["lib/polevl.c", "lib/sas_J1.c"]),
#)
add_function(
name="sin(x)",
mp_function=mp.sin,
np_function=np.sin,
#ocl_function=make_ocl("double sn, cn; SINCOS(q,sn,cn); return sn;", "sas_sin"),
ocl_function=make_ocl("return sin(q);", "sas_sin"),
)
add_function(
name="sin(x)/x",
mp_function=lambda x: mp.sin(x) / x if x != 0 else 1,
## scipy sinc function is inaccurate and has an implied pi*x term
#np_function=lambda x: scipy.special.sinc(x/pi),
## numpy sin(x)/x needs to check for x=0
np_function=lambda x: np.sin(x) / x,
ocl_function=make_ocl("return sas_sinx_x(q);", "sas_sinc"),
)
add_function(
name="cos(x)",
mp_function=mp.cos,
np_function=np.cos,
#ocl_function=make_ocl("double sn, cn; SINCOS(q,sn,cn); return cn;", "sas_cos"),
ocl_function=make_ocl("return cos(q);", "sas_cos"),
)
add_function(
name="gamma(x)",
def sin(z):
return mp.sin(z)
from mpmath import mp
mp.dps = 50
N = int(sys.argv[1]) if len(sys.argv) > 1 else 20000
inicio = -1 * mp.pi
fim = 1 * mp.pi
dx = (fim - inicio) / (N - 1)
idx = 1 / dx
Idx = mp.mpf(1 / dx)
with open('Sin.h', 'w') as f:
f.write("\n".join([
'#define SIN_N {N}',
'#define SIN_Idx {idx}',
'#define SIN_inicio {inicio}',
'#define SIN_fim {fim}',
'',
'',
]).format(**locals()))
f.write('const double Sin[] = {')
for i in range(N):
f.write('' + str(mp.sin(i * dx + inicio)) + ',\n')
f.write('\n};')
def sind(x):
try:
return mp.sin(x * mp.pi / 180)
except:
return math.sin(x * math.pi / 180)
def colourise(i):
var = float(mp.sin(mp.radians(i)))
return (var, var, var)
def log_cap(dim, theta):
return -0.5 * log2(2 * mp.pi * dim) - log2(
mp.cos(theta)) + (dim - 1) * log2(mp.sin(theta))
def dR(self, lx, ly, kr, kt, kp, qmin, qmax, band, fdm, local_mchi):
if band == "pi":
if not self.lastlx == lx or not self.lastly == ly:
self.P_pi = createP_pi(lx, ly)
self.lastlx = lx
self.lastly = ly
self.E_b = E_pi_minus(lx, ly)
sqP = lambda qr, qt, qp: mp.re(
self.P_pi(
qr * mp.sin(qt) * mp.cos(qp) - kr * mp.sin(kt) * mp.cos(kp
),
qr * mp.sin(qt) * mp.sin(qp) - kr * mp.sin(kt) * mp.sin(kp
),
qr * mp.cos(qt) - kr * mp.cos(kt)))**2 + mp.im(
self.P_pi(
qr * mp.sin(qt) * mp.cos(qp) - kr * mp.sin(kt) * mp
.cos(kp),
qr * mp.sin(qt) * mp.sin(qp) - kr * mp.sin(
kt) * mp.sin(kp),
qr * mp.cos(qt) - kr * mp.cos(kt)))**2
else:
if not self.lastlx == lx or not self.lastly == ly:
self.P_sigma = createP_sigma(lx, ly, band)
self.lastlx = lx
self.lastly = ly
self.E_b = E_sigma(lx, ly, band)
sqP = lambda qr, qt, qp: mp.re(
self.P_sigma(
qr * mp.sin(qt) * mp.cos(qp) - kr * mp.sin(kt) * mp.cos(kp
),
qr * mp.sin(qt) * mp.sin(qp) - kr * mp.sin(kt) * mp.sin(kp
),
qr * mp.cos(qt) - kr * mp.cos(kt)))**2 + mp.im(
self.P_sigma(
qr * mp.sin(qt) * mp.cos(qp) - kr * mp.sin(kt) * mp
.cos(kp),
qr * mp.sin(qt) * mp.sin(qp) - kr * mp.sin(
kt) * mp.sin(kp),
qr * mp.cos(qt) - kr * mp.cos(kt)))**2
integrand = lambda qr, qt, qp: (qr) * mp.sin(qt) / (
4 * mp.pi) * self.eta(qr, kr, self.E_b, local_mchi) * F_DM(
qr, fdm) * sqP(qr, qt, qp)
result = mint.integrate(integrand, [qmin, qmax], [0, mp.pi],
[0, 2 * mp.pi], self.method)
return result
def df3(x):
return mp.e**x - 2**(-x) * mp.log(2) - 2 * mp.sin(x)
# goji houteishiki solver from mpmath import mp mp.dps = 20 K = lambda k: mp.quad(lambda x: 1/(mp.sqrt(1 - (k**2)*(mp.sin(x)**2))), [0, mp.pi/2]) latparam = lambda k: mp.j*K(mp.sqrt(1-k**2))/K(k) J = lambda m, t: (phi5(t) + phi5m(t, 0))*(phi5m(t, 4) - phi5m(5, 1))*(phi5m(t, 3) - phi5m(t, 2)) inv_n = lambda n, t: -(1/(n + t)) phi = lambda t: kei = lambda R: (r*(5**(5/4)) + mp.sqrt((R**2)*mp.sqrt(5**5) - 16))/(r*(5**(5/4)) + mp.sqrt((R**2)*mp.sqrt(5**5) + 16))
dx = (fim-inicio)/(N-1)
idx = 1 / dx
Idx = mp.mpf(1/dx)
with open('Exponencial.h', 'w') as f:
f.write("\n".join([
'#include <complex>',
'',
'using namespace std;',
'',
'#define EXP_N {N}',
'#define EXP_Idx {idx}',
'#define EXP_inicio {inicio}',
'#define EXP_fim {fim}',
'',
'',
]).format(**locals()))
f.write('const complex<double> Exponencial[] = {')
for i in range(N):
value = mp.cos(i*dx + inicio)
value2 = mp.sin(i*dx + inicio)
v = "%.50e" % value
v2 = "%.50e" % value2
f.write('complex<double>(' + v + ',' + v2 + '),\n')
f.write('\n};')
from mpmath import mp
mp.dps = 50
N = int(sys.argv[1]) if len(sys.argv) > 1 else 20000
inicio = -1*mp.pi
fim = 1*mp.pi
dx = (fim-inicio)/(N-1)
idx = 1 / dx
Idx = mp.mpf(1/dx)
with open('Sin.h', 'w') as f:
f.write("\n".join([
'#define SIN_N {N}',
'#define SIN_Idx {idx}',
'#define SIN_inicio {inicio}',
'#define SIN_fim {fim}',
'',
'',
]).format(**locals()))
f.write('const double Sin[] = {')
for i in range(N):
f.write('' + str(mp.sin(i*dx + inicio)) + ',\n')
f.write('\n};')
N = long(sys.argv[1]) if len(sys.argv) > 1 else 20000
inicio = -1 * mp.pi
fim = 1 * mp.pi
dx = (fim - inicio) / (N - 1)
idx = 1 / dx
Idx = mp.mpf(1 / dx)
with open("Sin.h", "w") as f:
f.write(
"\n".join(
[
"#define SIN_N {N}",
"#define SIN_Idx {idx}",
"#define SIN_inicio {inicio}",
"#define SIN_fim {fim}",
"",
"",
]
).format(**locals())
)
f.write("const double Sin[] = {")
for i in range(N):
f.write("" + str(mp.sin(i * dx + inicio)) + ",\n")
f.write("\n};")
def sind(x):
try:
return mp.sin(x * mp.pi / 180)
except:
return math.sin(x * math.pi / 180)
def df1(x):
return mp.cos(x) * mp.sinh(x) - mp.sin(x) * mp.cosh(x)
dx = (fim - inicio) / (N - 1)
idx = 1 / dx
Idx = mp.mpf(1 / dx)
with open('Exponencial.h', 'w') as f:
f.write("\n".join([
'#include <complex>',
'',
'using namespace std;',
'',
'#define EXP_N {N}',
'#define EXP_Idx {idx}',
'#define EXP_inicio {inicio}',
'#define EXP_fim {fim}',
'',
'',
]).format(**locals()))
f.write('const complex<double> Exponencial[] = {')
for i in range(N):
value = mp.cos(i * dx + inicio)
value2 = mp.sin(i * dx + inicio)
v = "%.50e" % value
v2 = "%.50e" % value2
f.write('complex<double>(' + v + ',' + v2 + '),\n')
f.write('\n};')