def test_wrapper(self):
P = KroghInterpolator(self.xs, self.ys)
assert_almost_equal(P(self.test_xs), krogh_interpolate(self.xs, self.ys, self.test_xs))
assert_almost_equal(P.derivative(self.test_xs, 2), krogh_interpolate(self.xs, self.ys, self.test_xs, der=2))
assert_almost_equal(
P.derivatives(self.test_xs, 2), krogh_interpolate(self.xs, self.ys, self.test_xs, der=[0, 1])
)
def test_shapes_scalarvalue_derivative(self):
P = KroghInterpolator(self.xs,self.ys)
n = P.n
assert_array_equal(np.shape(P.derivatives(0)), (n,))
assert_array_equal(np.shape(P.derivatives(np.array(0))), (n,))
assert_array_equal(np.shape(P.derivatives([0])), (n,1))
assert_array_equal(np.shape(P.derivatives([0,1])), (n,2))
def test_derivatives_complex(self):
# regression test for gh-7381: krogh.derivatives(0) fails complex y
x, y = np.array([-1, -1, 0, 1, 1]), np.array([1, 1.0j, 0, -1, 1.0j])
func = KroghInterpolator(x, y)
cmplx = func.derivatives(0)
cmplx2 = (KroghInterpolator(x, y.real).derivatives(0) +
1j*KroghInterpolator(x, y.imag).derivatives(0))
assert_allclose(cmplx, cmplx2, atol=1e-15)
def test_int_inputs(self):
# Check input args are cast correctly to floats, gh-3669
x = [0, 234,468,702,936,1170,1404,2340,3744,6084,8424,13104,60000]
offset_cdf = np.array([-0.95, -0.86114777, -0.8147762, -0.64072425, -0.48002351,
-0.34925329, -0.26503107, -0.13148093, -0.12988833, -0.12979296,
-0.12973574, -0.08582937, 0.05])
f = KroghInterpolator(x, offset_cdf)
assert_allclose(abs((f(x) - offset_cdf) / f.derivative(x, 1)), 0, atol=1e-10)
def test_vector(self):
xs = [0, 1, 2]
ys = np.array([[0, 1], [1, 0], [2, 1]])
P = KroghInterpolator(xs, ys)
Pi = [KroghInterpolator(xs, ys[:, i]) for i in xrange(ys.shape[1])]
test_xs = np.linspace(-1, 3, 100)
assert_almost_equal(P(test_xs), np.rollaxis(np.asarray([p(test_xs) for p in Pi]), -1))
assert_almost_equal(
P.derivatives(test_xs), np.transpose(np.asarray([p.derivatives(test_xs) for p in Pi]), (1, 2, 0))
)
def krogh_deriv(x, y, axis=0):
return KroghInterpolator(x, y, axis).derivative
def derivatives(f, T0, dT=.001):
# evaluate function at Chebyshev zeros as suggested by docs
T = T0 + dT * np.cos(np.linspace(0, np.pi, nd))
return KroghInterpolator(T, f(T)).derivatives(T0)
def test_shapes_1d_vectorvalue(self):
P = KroghInterpolator(self.xs, np.outer(self.ys, [1]))
assert_array_equal(np.shape(P(0)), (1, ))
assert_array_equal(np.shape(P([0])), (1, 1))
assert_array_equal(np.shape(P([0, 1])), (2, 1))
def test_shapes_vectorvalue_derivative(self):
P = KroghInterpolator(self.xs, np.outer(self.ys, np.arange(3)))
n = P.n
assert_array_equal(np.shape(P.derivatives(0)), (n, 3))
assert_array_equal(np.shape(P.derivatives([0])), (n, 1, 3))
assert_array_equal(np.shape(P.derivatives([0, 1])), (n, 2, 3))
plt.plot(FaCO, e, 'o', x, f(x), '-') #courbe (FaCO,e) et (x,f(x))
plt.xlabel('FaCO') #nomme l'axe des x puis apres y
plt.ylabel('e')
plt.title('interpolation_lineaire,e=f(FaCO)') #on nomme le graphique
plt.show() #on affiche
g = interpolate.interp1d(FaCO, Y)
x = np.linspace(0.172, 6.89, 100)
y = g(x)
plt.plot(FaCO, Y, 'o', x, g(x), '-')
plt.xlabel('Y')
plt.ylabel('e')
plt.title('interpolation_lineaire,Y=f(Faco)')
plt.show()
f = KroghInterpolator(FaCO, e) #interpolation polynomiale
x = np.linspace(0.172, 6.89, 100)
y = f(x)
plt.plot(FaCO, e, 'o', x, f(x), '-')
plt.xlabel('FaCO')
plt.ylabel('e')
plt.title('interpolation_polynomiale,e=f(FaCO)')
plt.show()
g = KroghInterpolator(FaCO, Y)
x = np.linspace(0.172, 6.89, 100)
y = g(x)
plt.plot(FaCO, Y, 'o', x, g(x), '-')
plt.xlabel('Y')
plt.ylabel('e')
plt.title('interpolation_polynomiale,Y=f(Faco)')
# 埃尔米特插值
from scipy.interpolate import KroghInterpolator
import numpy as np
import matplotlib.pyplot as plt
def f(x):
return x**3 + 1
xs = np.linspace(-5, 5, 100)
nodes = np.array([0, 1, 2, 3])
xi = np.array([0, 1, 2, 3])
yi = np.array([1, 2, 9, 28])
interpolant = KroghInterpolator(xi, yi)
plt.figure()
plt.subplot(121)
plt.plot(xs, interpolant(xs), "b--", label="Hermite interpolation")
plt.plot(nodes, f(nodes), "ro", label="nodes")
# 显示图中的标签
plt.legend(loc=9)
# 设定坐标上下限
plt.xlim(-10.5, 10.5)
plt.title("$f(x) = x^3 + 1$")
# 1行 2列
plt.subplot(122)
plt.plot(xs, f(xs), "g--", label="original")
plt.plot(xi, f(xi), "ro", label="nodes")
plt.legend(loc=9)
def interpolate(x0, x1, m0, m1, t0, t1):
x = np.repeat([t0, t1], 2)
y = np.ravel(np.dstack(([x0, x1], [m0, m1])))
return KroghInterpolator(x, y)
def test_derivative(self):
P = KroghInterpolator(self.xs,self.ys)
m = 10
r = P.derivatives(self.test_xs,m)
for i in xrange(m):
assert_almost_equal(P.derivative(self.test_xs,i),r[i])
def test_lagrange(self):
P = KroghInterpolator(self.xs, self.ys)
assert_almost_equal(self.true_poly(self.test_xs), P(self.test_xs))
def test_low_derivatives(self):
P = KroghInterpolator(self.xs, self.ys)
D = P.derivatives(self.test_xs, len(self.xs) + 2)
for i in xrange(D.shape[0]):
assert_almost_equal(self.true_poly.deriv(i)(self.test_xs), D[i])
def main():
parser = argparse.ArgumentParser(
description=__doc__,
formatter_class=argparse.ArgumentDefaultsHelpFormatter)
parser.add_argument('--Tmin',
type=float,
default=.002,
help='minimum temperature')
parser.add_argument('--Ta',
type=float,
default=.165,
help='connection range minimum temperature')
parser.add_argument('--Tb',
type=float,
default=.200,
help='connection range maximum temperature')
parser.add_argument('--Tmax',
type=float,
default=.600,
help='maximum temperature')
parser.add_argument('--nsteps',
type=int,
default=10**5,
help='number of energy density steps')
parser.add_argument('--species',
choices=['all', 'urqmd', 'id'],
default='urqmd',
help='HRG species')
parser.add_argument('--res-width-off',
action='store_false',
dest='res_width',
help='do not account for finite width of resonances')
parser.add_argument('--plot',
nargs='?',
metavar='FILE',
const='<show>',
default=False,
help='plot EoS instead of printing table')
parser.add_argument('--write-bin',
metavar='FILE',
help='write binary file instead of printing table')
args = parser.parse_args()
hrg_kwargs = dict(species=args.species, res_width=args.res_width)
# split full temperature range into three parts:
# low-T (l): Tmin < T < Ta (HRG)
# mid-T (m): Ta < T < Tb (connection)
# high-T (h): Tb < T < Tmax (lattice)
# number of extra temperature points below Tmin and above Tmax
# (helps cubic interpolation)
nextrapts = 2
# compute low-T (HRG) trace anomaly
Tl = T_points(args.Tmin, args.Ta, 1000, extra_low=nextrapts)
e3p_T4_l = HRGEOS(Tl, **hrg_kwargs).e3p_T4()
# compute mid-T (connection) using an interpolating polynomial that
# matches the function values and first several derivatives at the
# connection endpoints (Ta, Tb)
nd = 5
# use Krogh interpolation near the endpoints to estimate derivatives
def derivatives(f, T0, dT=.001):
# evaluate function at Chebyshev zeros as suggested by docs
T = T0 + dT * np.cos(np.linspace(0, np.pi, nd))
return KroghInterpolator(T, f(T)).derivatives(T0)
# use another Krogh interpolation for the connection polynomial
# skip (Ta, Tb) endpoints in Tm since they are already in (Tl, Th)
Tm = T_points(args.Ta, args.Tb, 1000, extra_low=-1, extra_high=-1)
e3p_T4_m = KroghInterpolator(
nd * [args.Ta] + nd * [args.Tb],
np.concatenate([
derivatives(lambda T: HRGEOS(T, **hrg_kwargs).e3p_T4(), args.Ta),
derivatives(e3p_T4_lattice, args.Tb)
]))(Tm)
# compute high-T part (lattice)
Th = T_points(args.Tb, args.Tmax, 1000, extra_high=nextrapts)
e3p_T4_h = e3p_T4_lattice(Th)
# join temperature ranges together
T = np.concatenate([Tl, Tm, Th])
e3p_T4 = np.concatenate([e3p_T4_l, e3p_T4_m, e3p_T4_h])
# pressure is integral of trace anomaly over temperature starting from some
# reference temperature, Eq. (12) in HotQCD paper:
# p/T^4(T) = p/T^4(T_0) + \int_{T_0}^T dT (e - 3p)/T^5
delta_p_T4_spline = CubicSpline(T, e3p_T4 / T).antiderivative()
p_T4_0 = HRGEOS(T[:1], **hrg_kwargs).p_T4()[0]
def compute_p_T4(T):
p_T4 = delta_p_T4_spline(T)
p_T4 += p_T4_0
return p_T4
p_T4 = compute_p_T4(T)
e_T4 = e3p_T4 + 3 * p_T4
if args.plot:
plot(T, e3p_T4, p_T4, e_T4, args, hrg_kwargs)
return
# energy density at original temperature points
e_orig = e_T4 * T**4 / HBARC**3
T_orig = T
# compute thermodynamic quantities at evenly-spaced temperature points
# as required by osu-hydro-pce
T = np.linspace(T[nextrapts], T[-nextrapts - 1], args.nsteps)
e = CubicSpline(T_orig, e_orig)(T)
p = compute_p_T4(T) * T**4 / HBARC**3
# Nc = 3
# Nf = 3
# T = np.linspace(1e-10, 0.7, 10**5)
# e = np.pi**2 * 1/30. * (2*(Nc*Nc-1) + 3.5*Nc*Nf) * np.ones(T.shape) * T**4 / HBARC**3
# p = e/3.
s = (e + p) / T
cs2 = CubicSpline(e, p)(e, nu=1)
cstilde2 = p / e
if args.write_bin:
with open(args.write_bin, 'wb') as f:
# for x in [e[0], e[-1], p, s, T]:
for x in [T[0], T[-1], p, s, e, cs2, cstilde2]:
f.write(x.tobytes())
else:
# output table
fmt = 4 * '{:24.16e}'
def test_wrapper(self):
P = KroghInterpolator(self.xs,self.ys)
assert_almost_equal(P(self.test_xs),krogh_interpolate(self.xs,self.ys,self.test_xs))
assert_almost_equal(P.derivative(self.test_xs,2),krogh_interpolate(self.xs,self.ys,self.test_xs,der=2))
assert_almost_equal(P.derivatives(self.test_xs,2),krogh_interpolate(self.xs,self.ys,self.test_xs,der=[0,1]))
def main():
parser = argparse.ArgumentParser(
description=__doc__,
formatter_class=argparse.ArgumentDefaultsHelpFormatter
)
parser.add_argument(
'--Tmin', type=float, default=.050,
help='minimum temperature'
)
parser.add_argument(
'--Ta', type=float, default=.165,
help='connection range minimum temperature'
)
parser.add_argument(
'--Tb', type=float, default=.200,
help='connection range maximum temperature'
)
parser.add_argument(
'--Tmax', type=float, default=.500,
help='maximum temperature'
)
parser.add_argument(
'--nsteps', type=int, default=10**5,
help='number of energy density steps'
)
parser.add_argument(
'--species', choices=['all', 'urqmd', 'id'], default='urqmd',
help='HRG species'
)
parser.add_argument(
'--res-width-off', action='store_false', dest='res_width',
help='do not account for finite width of resonances'
)
parser.add_argument(
'--plot', nargs='?', metavar='FILE', const='<show>', default=False,
help='plot EoS instead of printing table'
)
parser.add_argument(
'--write-bin', metavar='FILE',
help='write binary file instead of printing table'
)
args = parser.parse_args()
hrg_kwargs = dict(species=args.species, res_width=args.res_width)
# split full temperature range into three parts:
# low-T (l): Tmin < T < Ta (HRG)
# mid-T (m): Ta < T < Tb (connection)
# high-T (h): Tb < T < Tmax (lattice)
# number of extra temperature points below Tmin and above Tmax
# (helps cubic interpolation)
nextrapts = 2
# compute low-T (HRG) trace anomaly
Tl = T_points(args.Tmin, args.Ta, 200, extra_low=nextrapts)
e3p_T4_l = HRGEOS(Tl, **hrg_kwargs).e3p_T4()
# compute mid-T (connection) using an interpolating polynomial that
# matches the function values and first several derivatives at the
# connection endpoints (Ta, Tb)
nd = 5
# use Krogh interpolation near the endpoints to estimate derivatives
def derivatives(f, T0, dT=.001):
# evaluate function at Chebyshev zeros as suggested by docs
T = T0 + dT*np.cos(np.linspace(0, np.pi, nd))
return KroghInterpolator(T, f(T)).derivatives(T0)
# use another Krogh interpolation for the connection polynomial
# skip (Ta, Tb) endpoints in Tm since they are already in (Tl, Th)
Tm = T_points(args.Ta, args.Tb, 100, extra_low=-1, extra_high=-1)
e3p_T4_m = KroghInterpolator(
nd*[args.Ta] + nd*[args.Tb],
np.concatenate([
derivatives(lambda T: HRGEOS(T, **hrg_kwargs).e3p_T4(), args.Ta),
derivatives(e3p_T4_lattice, args.Tb)
])
)(Tm)
# compute high-T part (lattice)
Th = T_points(args.Tb, args.Tmax, 1000, extra_high=nextrapts)
e3p_T4_h = e3p_T4_lattice(Th)
# join temperature ranges together
T = np.concatenate([Tl, Tm, Th])
e3p_T4 = np.concatenate([e3p_T4_l, e3p_T4_m, e3p_T4_h])
# pressure is integral of trace anomaly over temperature starting from some
# reference temperature, Eq. (12) in HotQCD paper:
# p/T^4(T) = p/T^4(T_0) + \int_{T_0}^T dT (e - 3p)/T^5
delta_p_T4_spline = CubicSpline(T, e3p_T4/T).antiderivative()
p_T4_0 = HRGEOS(T[:1], **hrg_kwargs).p_T4()[0]
def compute_p_T4(T):
p_T4 = delta_p_T4_spline(T)
p_T4 += p_T4_0
return p_T4
p_T4 = compute_p_T4(T)
e_T4 = e3p_T4 + 3*p_T4
# energy density at original temperature points
e_orig = e_T4 * T**4 / HBARC**3
# compute thermodynamic quantities at evenly-spaced energy density points
# as required by osu-hydro
e = np.linspace(e_orig[nextrapts], e_orig[-nextrapts - 1], args.nsteps)
T = CubicSpline(e_orig, T)(e)
p = compute_p_T4(T) * T**4 / HBARC**3
s = (e + p)/T
Tc = 0.270
tlat = Tc*np.array([0.7, 0.74, 0.78, 0.82, 0.86, 0.9, 0.94, 0.98, 1, 1.02, 1.06, 1.10, 1.14, 1.18, 1.22, 1.26, 1.30, 1.34, 1.38, 1.42, 1.46, 1.5, 2, 2.5, 3, 3.5, 4.0, 4.5, 5, 6, 7, 8, 9, 10, 20, 30, 40, 50, 60, 80, 100, 200, 300, 400, 500, 600, 800, 1000])
Ilat = np.array([.0104, .0162, .0232, .0318, .0433, .0594, .0859, .1433, 1.0008, 2.078, 2.4309, 2.4837, 2.4309, 2.3426, 2.2342, 2.1145, 1.998, 1.8867, 1.7809, 1.6810, 1.5872, 1.4995, 0.8038, 0.5057, 0.3589, 0.2736, 0.2207, 0.1855, 0.1606, 0.1266, 0.1050, 0.0903, 0.0798, 0.0720, 0.0375, 0.0265, 0.0216, 0.0191, 0.0174, 0.0154, 0.0142, 0.0112, 0.01, 0.0091, 0.0085, 0.0080, 0.0073, 0.0068])
plat = np.array([0.0015, 0.0023, 0.0033, 0.0046, 0.0064, 0.0087, 0.0118, 0.0164, 0.0222, 0.0571, 0.1455, 0.237, 0.325, 0.4074, 0.4837, 0.5539, 0.6181, 0.677, 0.7309, 0.7804, 0.8258, 0.8675, 1.189, 1.3319, 1.4098, 1.4582, 1.491, 1.5149, 1.533, 1.5591, 1.5768, 1.5898, 1.5998, 1.6078, 1.6444, 1.6572, 1.6641, 1.6686, 1.672, 1.6767, 1.68, 1.6887, 1.693, 1.6958, 1.6977, 1.6992, 1.7014, 1.703])
e3p_T4_gluon = CubicSpline(tlat,Ilat)
delta_p_T4_spline_gluon = CubicSpline(tlat, Ilat/tlat).antiderivative()
p_T4_0_gluon = HRGEOS(tlat[:1], **hrg_kwargs).p_T4()[0]
def compute_p_T4_gluon(T):
p_T4 = delta_p_T4_spline_gluon(T)
p_T4 += p_T4_0_gluon
return p_T4
p_T4_gluon = compute_p_T4_gluon(tlat)
e_T4_gluon = Ilat + 3*plat
# energy density at original temperature points
e_orig_gluon = e_T4_gluon * tlat**4 / HBARC**3
# compute thermodynamic quantities at evenly-spaced energy density points
# as required by osu-hydro
e_gluon = np.linspace(e_orig_gluon[nextrapts], e_orig_gluon[-nextrapts - 1], args.nsteps)
T_gluon = CubicSpline(e_orig_gluon, tlat)(e)
p_gluon = compute_p_T4_gluon(T) * T_gluon**4 / HBARC**3
s_gluon = (e_gluon + p_gluon)/T_gluon
plot(T, e3p_T4, p_T4, e_T4, T_gluon, e3p_T4_gluon, p_T4_gluon, e_T4_gluon, args, hrg_kwargs)
return
from scipy.interpolate import KroghInterpolator #from format_data import Formatter import matplotlib.pyplot as plt import time import numpy as np x_values = [1,2,3,4,5] y_values = [6,7,8,9,10] intra_x_values = [6,7,8] poly_func = KroghInterpolator(x_values,y_values) intra_x_values = np.asarray(intra_x_values) intra_y_values = poly_func.__call__(intra_x_values) print(intra_y_values)
def test_shapes_scalarvalue_derivative(self):
P = KroghInterpolator(self.xs, self.ys)
n = P.n
assert_array_equal(np.shape(P.derivatives(0)), (n, ))
assert_array_equal(np.shape(P.derivatives([0])), (n, 1))
assert_array_equal(np.shape(P.derivatives([0, 1])), (n, 2))
def test_scalar(self):
P = KroghInterpolator(self.xs, self.ys)
assert_almost_equal(self.true_poly(7), P(7))
assert_almost_equal(self.true_poly(np.array(7)), P(np.array(7)))
def test_derivative(self):
P = KroghInterpolator(self.xs, self.ys)
m = 10
r = P.derivatives(self.test_xs, m)
for i in range(m):
assert_almost_equal(P.derivative(self.test_xs, i), r[i])
def test_high_derivative(self):
P = KroghInterpolator(self.xs, self.ys)
for i in range(len(self.xs), 2 * len(self.xs)):
assert_almost_equal(P.derivative(self.test_xs, i),
np.zeros(len(self.test_xs)))
def test_empty(self):
P = KroghInterpolator(self.xs, self.ys)
assert_array_equal(P([]), [])
def test_low_derivatives(self):
P = KroghInterpolator(self.xs,self.ys)
D = P.derivatives(self.test_xs,len(self.xs)+2)
for i in xrange(D.shape[0]):
assert_almost_equal(self.true_poly.deriv(i)(self.test_xs),
D[i])
def test_shapes_scalarvalue(self):
P = KroghInterpolator(self.xs, self.ys)
assert_array_equal(np.shape(P(0)), ())
assert_array_equal(np.shape(P(np.array(0))), ())
assert_array_equal(np.shape(P([0])), (1, ))
assert_array_equal(np.shape(P([0, 1])), (2, ))
def test_high_derivative(self):
P = KroghInterpolator(self.xs,self.ys)
for i in xrange(len(self.xs),2*len(self.xs)):
assert_almost_equal(P.derivative(self.test_xs,i),
np.zeros(len(self.test_xs)))
def test_shapes_vectorvalue(self):
P = KroghInterpolator(self.xs, np.outer(self.ys, np.arange(3)))
assert_array_equal(np.shape(P(0)), (3, ))
assert_array_equal(np.shape(P([0])), (1, 3))
assert_array_equal(np.shape(P([0, 1])), (2, 3))
def test_shapes_vectorvalue_derivative(self):
P = KroghInterpolator(self.xs,np.outer(self.ys,np.arange(3)))
n = P.n
assert_array_equal(np.shape(P.derivatives(0)), (n,3))
assert_array_equal(np.shape(P.derivatives([0])), (n,1,3))
assert_array_equal(np.shape(P.derivatives([0,1])), (n,2,3))
def test_high_degree_warning(self):
with pytest.warns(UserWarning, match="40 degrees provided,"):
KroghInterpolator(np.arange(40), np.ones(40))
