def _complexcn(z):
"""Jacobi elliptical function cn with parameter 1/2,
valid for complex numbers."""
x = z.real
y = z.imag
snx, cnx, dnx, phx = ellipj(x, 1 / 2)
sny, cny, dny, phy = ellipj(y, 1 / 2)
numer = cnx * cny - 1j * snx * dnx * sny * dny
denom = (1 - dnx**2 * sny**2)
return numer / denom
def complex_jacobi(u, k, tol=10.**(-10.)):
if abs(np.imag(u)) < tol:
return sp.ellipj(np.real(u), k**2)
real_ellip = sp.ellipj(np.real(u), k**2)
sn_r, cn_r, dn_r = real_ellip[0], real_ellip[1], real_ellip[2]
imag_ellip = sp.ellipj(np.imag(u), 1. - k**2)
sn_c, cn_c, dn_c = imag_ellip[0], imag_ellip[1], imag_ellip[2]
sn_c, cn_c, dn_c = 1j * sn_c / cn_c, 1. / cn_c, dn_c / cn_c
denom = 1. - (k * sn_r * sn_c)**2
sn = (sn_r * cn_c * dn_c + sn_c * cn_r * dn_r) / denom
cn = (cn_r * cn_c - sn_r * dn_r * sn_c * dn_c) / denom
dn = (dn_r * cn_r - k**2 * sn_r * cn_r * sn_c * dn_c) / denom
return sn, cn, dn
def into_jv(self, jgrid, vgrid):
sv = np.sqrt(vgrid)
jgrid = jgrid / pulsation_canon(jgrid, vgrid)
return np.select(
[vgrid < 1, vgrid > 1, True],
[2 * np.arcsin(sv * scsp.ellipj(jgrid, vgrid)[0]),
2 * np.arcsin(scsp.ellipj(sv * jgrid, 1/vgrid)[0]),
2 * np.arcsin(np.tanh(jgrid))
]
), np.select(
[vgrid < 1, vgrid > 1, True],
[2 * sv * scsp.ellipj(jgrid, vgrid)[1],
2 * sv * scsp.ellipj(sv * jgrid, 1/vgrid)[2],
2 / np.cosh(jgrid)
]
)
def cplx_cn( z, k):
z = numpy.asarray(z)
if z.dtype != complex:
return ellipj(z,k)[1]
# note, 'dn' result of ellipj is buggy, as of Mar 2015
# see https://github.com/scipy/scipy/issues/3904
ss,cc = ellipj( z.real, k )[:2]
dd = numpy.sqrt(1-k*ss**2) # make our own dn
s1,c1 = ellipj( z.imag, 1-k )[:2]
d1 = numpy.sqrt(1-k*s1**2)
ds1 = dd*s1
den = (1-ds1**2)
rx = cc*c1/den
ry = ss*ds1*d1/den
return rx - 1j*ry
def test_ellipfun_sn(self):
# Oscillating function --- limit range of first argument; the
# loss of precision there is an expected numerical feature
# rather than an actual bug
assert_mpmath_equal(lambda u, m: sc.ellipj(u, m)[0],
lambda u, m: mpmath.ellipfun("sn", u=u, m=m),
[Arg(-1e6, 1e6), Arg(a=0, b=1)],
atol=1e-20)
def test_ellipfun_sn(self):
# Oscillating function --- limit range of first argument; the
# loss of precision there is an expected numerical feature
# rather than an actual bug
assert_mpmath_equal(lambda u, m: sc.ellipj(u, m)[0],
lambda u, m: mpmath.ellipfun("sn", u=u, m=m),
[Arg(-1e6, 1e6), Arg(a=0, b=1)],
atol=1e-20)
def performPlot(self):
#Plotting waveform
plotxL = list(np.arange(-1, 1.001, 0.001))
plottheta = [2*self.plotDict["K"]*\
(i - (self.plotDict["time"]/self.plotDict["T"])) for i in plotxL]
pSN, pCN, pDN, pPH = sp.ellipj(plottheta, self.plotDict["m"])
pCSD = pSN * pCN * pDN
if self.plotDict["O"] == 1:
ploteta = self.plotDict["d"]*\
(self.plotDict["A0"] + self.plotDict["A1"]*pCN**2)
plotu = math.sqrt(self.g*self.plotDict["d"])*\
(self.plotDict["B00"] + self.plotDict["B10"]*pCN**2)
plotw = math.sqrt(self.g*self.plotDict["d"])*\
(4.0*self.plotDict["K"]*self.plotDict["d"]*pCSD/self.plotDict["L"])*\
((self.plotDict["z"] + self.plotDict["d"])/self.plotDict["d"])*self.plotDict["B10"]
else:
ploteta = self.plotDict["d"]*(self.plotDict["A0"] +\
self.plotDict["A1"]*pCN**2 + self.plotDict["A2"]*pCN**4)
plotu = math.sqrt(self.g*self.plotDict["d"])*\
((self.plotDict["B00"] + self.plotDict["B10"]*pCN**2 +\
self.plotDict["B20"]*pCN**4) -\
0.5*((self.plotDict["z"] + self.plotDict["d"]) / self.plotDict["d"])**2 *\
(self.plotDict["B01"] + self.plotDict["B11"]*pCN**2 + self.plotDict["B21"]*pCN**4))
pw1 = ((self.plotDict["z"] + self.plotDict["d"])/self.plotDict["d"])*\
(self.plotDict["B10"] + 2.0*self.plotDict["B20"]*pCN**2)
pw2 = (1.0/6.0)*(((self.plotDict["z"] + self.plotDict["d"])/self.plotDict["d"])**3)*\
(self.plotDict["B11"] + 2.0*self.plotDict["B21"]*pCN**2)
plotw = math.sqrt(self.g*self.plotDict["d"])*\
(4.0*self.plotDict["K"]*self.plotDict["d"]*pCSD/self.plotDict["L"])*(pw1 - pw2)
# end if
plt.figure(1, figsize=(8, 12), dpi=self.plotConfigDict["dpi"])
plt.subplot(3, 1, 1)
plt.plot(plotxL, ploteta)
plt.axhline(y=0.0, color="r", LineStyle="--")
plt.ylabel("Elevation [%s]" % self.labelUnitDist)
plt.subplot(3, 1, 2)
plt.plot(plotxL, plotu)
plt.axhline(y=0.0, color="r", LineStyle="--")
plt.ylabel("Velocity, u [%s/s]" % self.labelUnitDist)
plt.subplot(3, 1, 3)
plt.plot(plotxL, plotw)
plt.axhline(y=0.0, color="r", LineStyle="--")
plt.ylabel("Velocity, w [%s/s]" % self.labelUnitDist)
plt.xlabel("x/L")
plt.show()
self.plotDict["plotxL"] = plotxL
self.plotDict["ploteta"] = ploteta
self.plotDict["plotu"] = plotu
self.plotDict["plotw"] = plotw
self.fileOutputPlotWriteData()
def getnormal(a, b, t):
e = sl.ellipj(t, 1 - ((a**2 * (1 - b**2)) / (b**2 * (1 - a**2))))
s = e[0]
c = e[1]
d = e[2]
x = 1 / (math.sqrt(1 - a**2)) * s
y = 1 / (math.sqrt(1 - b**2)) * c
z = math.sqrt(b**2 / (1 - b**2)) * d
return [x, y, z]
def a_v(n, theta_deg):
'''
Calculate a[v] and delta values for both
odd and even orders of type 'a' cauer
lowpass filters. The equations are from
equations (37), (38), and (39) of Saal and Ulbrich
paper on page 294.
n = number of sections or lowpass order
p = reflection coefficient or % of reflection
theta_deg: theta in degrees such that sin(theta)=wc/ws. See page 292
'''
theta = theta_deg * np.pi / 180 #radians
#m calculation, see EQN(37) p294
if n % 2 == 0:
#n is even
m = n // 2
else:
#n is odd
m = (n - 1) // 2 #
#Complete elliptic integral of the 1st kind K with modulus k=sin(theta)
#In scipy library: ellipk(u,m) where m = k^2
K = ss.ellipk(np.sin(theta)**2)
a = np.array([1
]) #a[0] is filled with 1 so indices of a[] complies with SU
for v in range(1, n): #1 to n-1 inclusive
u = K * v / n
#EQN(38) notates sn(u,theta)
#In scipy library: sn(u,m) where m = sin^2(theta)
jac_ell = ss.ellipj(u, np.sin(theta)**2)
sn = jac_ell[0]
a = np.append(a, np.sqrt(np.sin(theta)) * sn)
a = np.append(a, np.sqrt(np.sin(theta))) #append a[n]
wc_pa = a[n]
#delta calculation, see EQN(39)
delta_a = 1
if n % 2 == 0:
m_end = m
scale = 1.0
else:
m_end = m + 1
scale = 1.0 / a[n]
for u in range(1, m_end + 1):
delta_a *= a[2 * u - 1]**2
delta_a = scale * delta_a
return a, delta_a, wc_pa
def cn_id(gridnum, kappa, k0, L):
xspace = np.linspace(-L / 2.0, L / 2.0 - L / gridnum, gridnum)
_, cn, _, asd = special.ellipj(kappa * (xspace), k0**2)
initialprofile = k0 * kappa / np.sqrt(2.0) * cn
k = kvec(gridnum)
expconsts = -1j * (2.0 * np.pi / L * k)**2
return (xspace, initialprofile, expconsts)
def __call__(self, t):
S = np.sin(0.5 * (self.theta0))
K_S = ellipk(S**2)
omega_0 = np.sqrt(9.81)
sn, cn, dn, ph = ellipj(K_S - omega_0 * t, S**2)
theta = 2.0 * np.arcsin(S * sn)
d_sn_du = cn * dn
d_sn_dt = -omega_0 * d_sn_du
d_theta_dt = 2.0 * S * d_sn_dt / np.sqrt(1.0 - (S * sn)**2)
return {'theta': theta, 'v': d_theta_dt}
def sol(t, theta0):
S = np.sin(0.5 * (theta0))
K_S = ellipk(S**2)
omega_0 = np.sqrt(9.81)
sn, cn, dn, ph = ellipj(K_S - omega_0 * t, S**2)
theta = 2.0 * np.arcsin(S * sn)
d_sn_du = cn * dn
d_sn_dt = -omega_0 * d_sn_du
d_theta_dt = 2.0 * S * d_sn_dt / np.sqrt(1.0 - (S * sn)**2)
return np.stack([theta, d_theta_dt], axis=1)
def compute_rational_approximation(self):
# Compute shifts and quadrature weights
m, M = 10e-6, 10e6
k2 = m / M
kp = special.ellipk(1.0 - k2)
t = 1j * np.arange(0.5, self.n_shift) * kp / self.n_shift
sn, cn, dn, ph = special.ellipj(t.imag, 1-k2)
cn = 1./cn
cn *= cn
shifts = -m*sn*sn*cn
weights = 2.*cn*dn*kp*np.sqrt(m) / (np.pi * self.n_shift)
return (shifts, weights)
def find_collision_pts(e, q):
""" Finds the collision points for a period q, given an eccentricity e
"""
collisions_dict = {}
a = semi_axes[e][0]
b = semi_axes[e][1]
l = period_lambda_dict[q][e]
k_l_sq = ((a**2) - (b**2)) / ((a**2) - (l**2))
for j in range(int(q)):
d_l_q = (4 * (special.ellipk(k_l_sq))) / int(q)
t_j = (special.ellipk(k_l_sq)) + j * d_l_q
collisions_dict[str(j).zfill(2)] = special.ellipj(t_j, k_l_sq)[3]
return (collisions_dict)
def analyticalSolution(x,t,k,a0,a1):
"""
Returns the cnoidal solution with parameters k,a0,a1
at (x,t) (possibly arrays)
"""
g = 9.81
kappa = np.sqrt(3*a1)/(2*np.sqrt(a0*(a0+a1)*(a0+(1-k*k)*a1)))
h0 = a0+ a1*special.ellipe(k)/special.ellipk(k)
c = np.sqrt(g*a0*(a0+a1)*(a0+(1.-k*k)*a1))/h0
sn,cn,dn,ph = special.ellipj(kappa*(x-c*t),k)
h = a0+a1*dn**2
u = c*(1-h0/h)
return h,u
def sol_an(theta_0, g, l, t, dt):
"""
Aquí se calcula la solución analítica de theta, donde tambien se calcula la velocidad
pero esta utilizando el método de Euler.
"""
f_sen = np.sin(theta_0 / 2)
k_ellip = special.ellipk(f_sen**2)
ellip_function = special.ellipj(k_ellip - np.sqrt(g / l) * t, f_sen**2)[0]
theta = 2 * np.arcsin(f_sen * ellip_function)
#Calcular v
F = lambda theta, g, l: -(g / l) * np.sin(theta)
v = np.zeros(len(theta))
for i in range(len(theta) - 1):
v[i + 1] = v[i] + dt * F(theta[i], g, l)
return theta, v
def calc_psi_z(qz, zp, zm, En, aa):
"""
angle used in polar geodesic calculations
Parameters:
qz (float)
zp (float): polar root
zm (float): polar root
En (float): energy
aa (float): spin
Returns:
psi_z (float)
"""
ktheta = (aa**2 * (1 - En**2) * zm**2) / zp**2
u = (2 * (pi / 2.0 + qz) * ellipk(ktheta)) / pi
m = ktheta
__, __, __, ph = ellipj(u, m)
return ph
def calc_psi_r(qr, r1, r2, r3, r4):
"""
radial geodesic angle
Parameters:
qr (float)
r1 (float): radial root
r2 (float): radial root
r3 (float): radial root
r4 (float): radial root
Returns:
psi_r (float)
"""
kr = ((r1 - r2) * (r3 - r4)) / ((r1 - r3) * (r2 - r4))
u = (qr * ellipk(kr)) / pi
m = kr
__, __, __, ph = ellipj(u, m)
return ph
def calc_zq(qz, zp, zm, En, aa):
"""
function used in computing polar geodesic coordinates
Parameters:
qz (float)
zp (float): polar root
zm (float): polar root
En (float): energy
aa (float): spin
Returns:
zq (float)
"""
ktheta = (aa**2 * (1 - En**2) * zm**2) / zp**2
u = (2 * (pi / 2.0 + qz) * ellipk(ktheta)) / pi
m = ktheta
sn, __, __, __ = ellipj(u, m)
return zm * sn
def calc_rq(qr, r1, r2, r3, r4):
"""
function used in computing radial geodesic coordinates
Parameters:
qr (float)
r1 (float): radial root
r2 (float): radial root
r3 (float): radial root
r4 (float): radial root
Returns:
rq (float)
"""
kr = ((r1 - r2) * (r3 - r4)) / ((r1 - r3) * (r2 - r4))
u = (qr * ellipk(kr)) / pi
m = kr
sn, __, __, __ = ellipj(u, m)
return (-(r2 * (r1 - r3)) + (r1 - r2) * r3 * sn**2) / (-r1 + r3 +
(r1 - r2) * sn**2)
def Test():
a = np.array([1,2,3,4])
b = np.array((5,6,7,8))
c = np.array([[1,2],[6,7]])
x = np.arange(0,2,0.1)
print(a,b,'\n',c)
print(a.shape, b.shape, c.shape)
print(x)
l=[]
for i in range(0, 10):
l.append(i)
y = l
y = np.logspace(0, 1, 12, base=2, endpoint=False)
print('y= ', y)
x = np.linspace(0, 2*np.pi, 10, endpoint=False)
y = np.sin(x)
x = special.ellipj(1,1)
print('x= ', x)
print('y= ', y)
print(linalg.det(c))
x = 0.1
y = 17.0 * 2.0 * np.pi / 60.0 + 0.1
z = 0.0
E = 0.5 * (Ix * x**2 + Iy * y**2 + Iz * z**2)
L = (Ix * x)**2 + (Iy * y)**2 + (Iz * z)**2
l = np.sqrt((Ix - Iy) * (2 * Iz * E - L) / Ix / Iy / Iz)
k = np.sqrt((Iz - Ix) / (Iy - Ix) * (2 * Iy * E - L) / (2 * Iz * E - L))
a = np.sqrt((2 * Iy * E - L) / Ix / (Iy - Ix))
b = np.sqrt((2 * Iz * E - L) / Iy / (Iz - Iy))
c = np.sqrt((2 * Iy * E - L) / Iz / (Iy - Iz))
K = ellipk(k)
m = k**2
ax131.plot(t, omega_x, color="c", linestyle="-", label="Numerical Solution")
ax131.plot(t,
a * ellipj(l * t, m)[1],
color="m",
linestyle="-.",
label="Exact Solution")
ax131.set_xlim(0, end_time)
ax131.set_ylim(-0.1, 0.1)
ax131.set_ylabel("$\omega_x\;\;$[rad/s]")
ax131.legend(loc="upper right").get_frame().set_alpha(1)
ax132.plot(t, omega_y, color="c", linestyle="-", label="Numerical Solution")
ax132.plot(t,
b * ellipj(l * t - K, m)[2],
color="m",
linestyle="-.",
label="Exact Solution")
ax132.set_xlim(0, end_time)
ax132.set_ylim(1.8800, 1.8812)
import numpy as np
from scipy import io as spio
from scipy import special as special
print("\n\r")
print("====================================")
print("特殊函数：scipy.special")
print("====================================")
print("=== 贝塞尔函数，如scipy.special.jn() (整数n阶贝塞尔函数) ===")
data = special.jn(22, 33)
print(data)
arrA = [11, 22, 33]
arrB = [44, 55, 66]
data = special.jn(arrA, arrB)
print(data)
print("=== 椭圆函数: scipy.special.ellipj() (雅可比椭圆函数，……) ===")
x1 = [0.4, 0.5, 0.3]
x2 = [0.6, 0.7, 0.8]
#x1, x2, out1=None, out2=None, out3=None, out4=None
data = special.ellipj(x1, x2)
print(data)
print("=== 伽马函数：scipy.special.gamma() ===")
#还要注意scipy.special.gammaln,这个函数给出对数坐标的伽马函数，因此有更高的数值精度
x = [0.4, 0.6, 0.8]
data = special.gammaln(x)
print(data)
def test_ellipfun_dn(self):
# see comment in ellipfun_sn
assert_mpmath_equal(lambda u, m: sc.ellipj(u, m)[2],
lambda u, m: mpmath.ellipfun("dn", u=u, m=m),
[Arg(-1e6, 1e6), Arg(a=0, b=1)],
atol=1e-20)
# MODULES
import numpy as np
import scipy.integrate as integrate
import scipy.special as sp
import cmath
import matplotlib.pyplot as plt
# PARAMETERS
q = 2 # coefficient of the linear operator
N = 64 # number of Fourier modes
D = 128 # number of Floquet modes
k = 0.9 # only used for elliptic functions!
L = 2 * sp.ellipk(k**2) # period
f1 = lambda x: -1
f2 = lambda x: 6 * (k**2) * (sp.ellipj(x, k**2)[0]
)**2 # sn(x|k); cf. docs for special.ellipj
# FUNCTIONS
def frange(start, stop, step):
i = start
while i < stop:
yield i
i += step
def fourier_coeffs(fun, modes, period):
cosines = np.zeros(modes + 1, dtype=np.float_)
sines = np.zeros(modes + 1, dtype=np.float)
output = np.zeros(2 * modes + 1, dtype=np.complex_)
def analytical_dtheta(t):
k = sin(theta0/2)
T = ellipk(k**2)
(sn, cn, dn, ph) = ellipj(T-t, k)
return -2*k*cn
def analytical_theta(t):
k = sin(theta0/2)
T = ellipk(k**2)
(sn, cn, dn, ph) = ellipj(T-t, k)
return 2 * arcsin(k*sn)
def test_ellipfun_dn(self):
assert_mpmath_equal(lambda u, m: sc.ellipj(u, m)[2],
lambda u, m: mpmath.ellipfun("dn", u=u, m=m),
[Arg(), Arg(a=0, b=1)])
P0 = 2 * np.pi / omega0
# eccentricity parameter for elliptic functions
k = np.sin(phi0 / 2)
m = k**2
# discrete times
t = np.linspace(0, 4 * P0, 4000)
# small angle approx
phi_approx = phi0 * np.cos(omega0 * t)
# exact solution
x = omega0 * t
P = (4 / omega0) * special.ellipk(m)
sn_x, cn_x, dn_x, ph_x = special.ellipj(x, m)
phi = 2 * np.arcsin(k * cn_x)
# define figure
fig = plt.figure()
ax = fig.add_subplot(111,
aspect='equal',
autoscale_on=False,
xlim=(-1.25, 1.25),
ylim=(-1.25, 1.25))
ax.grid()
line1, = ax.plot([], [], 'o--', color='grey', lw=2)
line2, = ax.plot([], [], 'o-', color='black', lw=2)
ax.text(-1, 0.5, 'grey, dashed: small-angle approx')
ax.text(-1, 0.75, 'black, solid: exact solution')
def dnsq(eta,m):
sn,cn,dn,ph = ellipj(eta,m)
return dn*dn
def ell(u,m): return ssp.ellipj(u,m)
"""
Copyright 2009-2012 Karsten Ahnert
Copyright 2009-2012 Mario Mulansky
Stochastic euler stepper example and Ornstein-Uhlenbeck process
Distributed under the Boost Software License, Version 1.0.
(See accompanying file LICENSE_1_0.txt or
copy at http://www.boost.org/LICENSE_1_0.txt)
"""
from pylab import *
from scipy import special
data1 = loadtxt("elliptic1.dat")
data2 = loadtxt("elliptic2.dat")
data3 = loadtxt("elliptic3.dat")
sn1, cn1, dn1, phi1 = special.ellipj(data1[:, 0], 0.51)
sn2, cn2, dn2, phi2 = special.ellipj(data2[:, 0], 0.51)
sn3, cn3, dn3, phi3 = special.ellipj(data3[:, 0], 0.51)
semilogy(data1[:, 0], abs(data1[:, 1] - sn1))
semilogy(data2[:, 0], abs(data2[:, 1] - sn2), 'ro')
semilogy(data3[:, 0], abs(data3[:, 1] - sn3), '--')
show()
C=I3 - I1
N=np.linspace(math.sqrt(I1*T), math.sqrt(I3*T), 20)
for (lnumber, L) in enumerate(N):
(ktype, k)=modulus(I, T, L)
outfile=open("L{}.dat".format(lnumber), "w")
outfile.write("t\t omega1 \t omega2 \t omega3\n")
print("#"*32)
print("Sequence={}, L={}, ktype={}".format(lnumber, L, ktype))
U=I3*T-L**2
V=L**2 - I2*T
if V > 0:
for t in np.linspace(0, 0.1, 50):
argument=-t*math.sqrt(C*V/I1*I2*I3)
if not int(k - 1)==1:
(sn, cn, dn, ph) = ellipj(argument, k)
else:
sn=np.tanh(argument)
cs=np.cosh(argument)
if int(cs - 1)==0:
print("********zero cosh for {}".format(argument))
cn=0
dn=0
else:
cn=1/cs
dn=1/cs
#print(U, I1, C)
omega1=math.sqrt(U/I1*C)*sn
omega2=math.sqrt(U/I2*B)*cn
omega3=math.sqrt(U/I3*B)*dn
print("{}\t{}".format(t, omega1))
"""
Copyright 2011 Mario Mulansky
Copyright 2012 Karsten Ahnert
Stochastic euler stepper example and Ornstein-Uhlenbeck process
Distributed under the Boost Software License, Version 1.0.
(See accompanying file LICENSE_1_0.txt or
copy at http://www.boost.org/LICENSE_1_0.txt)
"""
from pylab import *
from scipy import special
data1 = loadtxt("elliptic1.dat")
data2 = loadtxt("elliptic2.dat")
data3 = loadtxt("elliptic3.dat")
sn1,cn1,dn1,phi1 = special.ellipj( data1[:,0] , 0.51 )
sn2,cn2,dn2,phi2 = special.ellipj( data2[:,0] , 0.51 )
sn3,cn3,dn3,phi3 = special.ellipj( data3[:,0] , 0.51 )
semilogy( data1[:,0] , abs(data1[:,1]-sn1) )
semilogy( data2[:,0] , abs(data2[:,1]-sn2) , 'ro' )
semilogy( data3[:,0] , abs(data3[:,1]-sn3) , '--' )
show()
def test_ellipfun_dn(self):
# see comment in ellipfun_sn
assert_mpmath_equal(lambda u, m: sc.ellipj(u, m)[2],
lambda u, m: mpmath.ellipfun("dn", u=u, m=m),
[Arg(-1e6, 1e6), Arg(a=0, b=1)],
atol=1e-20)
def ellipj_(k):
return ellipj(k*k)
import numpy as np import scipy.special as sp import matplotlib.pyplot as plt #종속변수 설정하기 k = np.array([0.25, 0.5, 0.75, 0.9, 0.99]) u = np.linspace(-0.5, 0.5, 100) #타원적분 K(k)함수값 구하기 integralk = sp.ellipk(k) print(integralk) #타원함수 모음집에서 sn,cu,dn함수만 꺼내오기, 주기를 1로 normalize 하기 ellipf1 = np.array([sp.ellipj(u * 4 * integralk[0], k[0])]) ellipf2 = np.array([sp.ellipj(u * 4 * integralk[1], k[1])]) ellipf3 = np.array([sp.ellipj(u * 4 * integralk[2], k[2])]) ellipf4 = np.array([sp.ellipj(u * 4 * integralk[3], k[3])]) ellipf5 = np.array([sp.ellipj(u * 4 * integralk[4], k[4])]) y1 = ellipf1[0, 0, :] y2 = ellipf2[0, 0, :] y3 = ellipf3[0, 0, :] y4 = ellipf4[0, 0, :] y5 = ellipf5[0, 0, :] #그래프 그리기 plt.plot(u, y1) plt.plot(u, y2) plt.plot(u, y3) plt.plot(u, y4) plt.plot(u, y5) plt.show()
def JacobiAmplitude(q, k):
k2 = np.power(k, 2)
JA = sp.ellipj(q, k2)
return JA
def cnsoln(xspace,t,kappa,k0):
_,cn,_,asd=special.ellipj(kappa*(xspace),k0**2)
return (kappa*k0*cn/np.sqrt(2))*np.exp(1j*(kappa**2)*((2*k0**2)-1)*t)
def KdVcnSoln(x, t, kappa, k):
_, cn, _, _ = special.ellipj(
kappa * (x - 4.0 * kappa**2 * (2.0 * k**2 - 1.0) * t), k**2)
return 6.0 * kappa**2 * k**2 * cn**2
def ellipj_(k):
return ellipj(k*k)
def contour_integral_quad(
lazy_tensor,
rhs,
inverse=False,
weights=None,
shifts=None,
max_lanczos_iter=20,
num_contour_quadrature=None,
shift_offset=0,
):
r"""
Performs :math:`\mathbf K^{1/2} \mathbf b` or `\mathbf K^{-1/2} \mathbf b`
using contour integral quadrature.
:param gpytorch.lazy.LazyTensor lazy_tensor: LazyTensor representing :math:`\mathbf K`
:param torch.Tensor rhs: Right hand side tensor :math:`\mathbf b`
:param bool inverse: (default False) whether to compute :math:`\mathbf K^{1/2} \mathbf b` (if False)
or `\mathbf K^{-1/2} \mathbf b` (if True)
:param int max_lanczos_iter: (default 10) Number of Lanczos iterations to run (to estimate eigenvalues)
:param int num_contour_quadrature: How many quadrature samples to use for approximation. Default is in settings.
:rtype: torch.Tensor
:return: Approximation to :math:`\mathbf K^{1/2} \mathbf b` or :math:`\mathbf K^{-1/2} \mathbf b`.
"""
import numpy as np
from scipy.special import ellipj, ellipk
if num_contour_quadrature is None:
num_contour_quadrature = settings.num_contour_quadrature.value()
output_batch_shape = _mul_broadcast_shape(lazy_tensor.batch_shape,
rhs.shape[:-2])
preconditioner, preconditioner_lt, _ = lazy_tensor._preconditioner()
def sqrt_precond_matmul(rhs):
if preconditioner_lt is not None:
solves, weights, _, _ = contour_integral_quad(preconditioner_lt,
rhs,
inverse=False)
return (solves * weights).sum(0)
else:
return rhs
# if not inverse:
rhs = sqrt_precond_matmul(rhs)
if shifts is None:
# Determine if init_vecs has extra_dimensions
num_extra_dims = max(0, rhs.dim() - lazy_tensor.dim())
lanczos_init = rhs.__getitem__(
(*([0] * num_extra_dims), Ellipsis, slice(None, None, None),
slice(None, 1, None))).expand(*lazy_tensor.shape[:-1], 1)
with warnings.catch_warnings(), torch.no_grad():
warnings.simplefilter(
"ignore", NumericalWarning) # Supress CG stopping warning
_, lanczos_mat = linear_cg(
lambda v: lazy_tensor._matmul(v),
rhs=lanczos_init,
n_tridiag=1,
max_iter=max_lanczos_iter,
tolerance=1e-5,
max_tridiag_iter=max_lanczos_iter,
preconditioner=preconditioner,
)
lanczos_mat = lanczos_mat.squeeze(
0
) # We have an extra singleton batch dimension from the Lanczos init
"""
K^{-1/2} b = 2/pi \int_0^\infty (K - t^2 I)^{-1} dt
We'll approximate this integral as a sum using quadrature
We'll determine the appropriate values of t, as well as their weights using elliptical integrals
"""
# Compute an approximate condition number
# We'll do this with Lanczos
try:
if settings.verbose_linalg.on():
settings.verbose_linalg.logger.debug(
f"Running symeig on a matrix of size {lanczos_mat.shape}.")
approx_eigs = lanczos_mat.symeig()[0]
if approx_eigs.min() <= 0:
raise RuntimeError
except RuntimeError:
approx_eigs = lazy_tensor.diag()
max_eig = approx_eigs.max(dim=-1)[0]
min_eig = approx_eigs.min(dim=-1)[0]
k2 = min_eig / max_eig
# Compute the shifts needed for the contour
flat_shifts = torch.zeros(num_contour_quadrature + 1,
k2.numel(),
dtype=k2.dtype,
device=k2.device)
flat_weights = torch.zeros(num_contour_quadrature,
k2.numel(),
dtype=k2.dtype,
device=k2.device)
# For loop because numpy
for i, (sub_k2, sub_min_eig) in enumerate(
zip(k2.flatten().tolist(),
min_eig.flatten().tolist())):
# Compute shifts
Kp = ellipk(1 - sub_k2) # Elliptical integral of the first kind
N = num_contour_quadrature
t = 1j * (np.arange(1, N + 1) - 0.5) * Kp / N
sn, cn, dn, _ = ellipj(np.imag(t),
1 - sub_k2) # Jacobi elliptic functions
cn = 1.0 / cn
dn = dn * cn
sn = 1j * sn * cn
w = np.sqrt(sub_min_eig) * sn
w_pow2 = np.real(np.power(w, 2))
sub_shifts = torch.tensor(w_pow2,
dtype=rhs.dtype,
device=rhs.device)
# Compute weights
constant = -2 * Kp * np.sqrt(sub_min_eig) / (math.pi * N)
dzdt = torch.tensor(cn * dn, dtype=rhs.dtype, device=rhs.device)
dzdt.mul_(constant)
sub_weights = dzdt
# Store results
flat_shifts[1:, i].copy_(sub_shifts)
flat_weights[:, i].copy_(sub_weights)
weights = flat_weights.view(num_contour_quadrature, *k2.shape, 1, 1)
shifts = flat_shifts.view(num_contour_quadrature + 1, *k2.shape)
shifts.sub_(shift_offset)
# Make sure we have the right shape
if k2.shape != output_batch_shape:
weights = torch.stack(
[w.expand(*output_batch_shape, 1, 1) for w in weights], 0)
shifts = torch.stack(
[s.expand(output_batch_shape) for s in shifts], 0)
# Compute the solves at the given shifts
# Do one more matmul if we don't want to include the inverse
with torch.no_grad():
solves = minres(lambda v: lazy_tensor._matmul(v),
rhs,
value=-1,
shifts=shifts,
preconditioner=preconditioner)
no_shift_solves = solves[0]
solves = solves[1:]
if not inverse:
solves = lazy_tensor._matmul(solves)
return solves, weights, no_shift_solves, shifts
