def _initialize_pp_gm(self):
pp_gm_cent_array = numpy.zeros_like(self._ln_k_array)
for idx in xrange(self._ln_k_array.size):
pp_gm_cent = integrate.romberg(
self._pp_gm_integrand, numpy.log(self.mass.nu_min),
numpy.log(self.mass.nu_max), vec_func=True,
tol=defaults.default_precision["halo_precision"],
args=(self._ln_k_array[idx], "central_first_moment"))
pp_gm_cent_array[idx] = pp_gm_cent/self.n_bar
self._pp_gm_cent_spline = InterpolatedUnivariateSpline(
self._ln_k_array, pp_gm_cent_array)
pp_gm_sat_array = numpy.zeros_like(self._ln_k_array)
for idx in xrange(self._ln_k_array.size):
pp_gm_sat = integrate.romberg(
self._pp_gm_integrand, numpy.log(self.mass.nu_min),
numpy.log(self.mass.nu_max), vec_func=True,
tol=defaults.default_precision["halo_precision"],
args=(self._ln_k_array[idx], "satellite_first_moment"))
pp_gm_sat_array[idx] = pp_gm_sat/self.n_bar
self._pp_gm_sat_spline = InterpolatedUnivariateSpline(
self._ln_k_array, pp_gm_sat_array)
self._initialized_pp_gm = True
def dirichlet_integrate(alpha):
normalizer = exp(sum(gammaln(alpha)) - gammaln(sum(alpha)))
def f_recur(x, idx, upper, vals):
if idx == 1:
# base case.
# set values for last two components
vals[1] = x
vals[0] = 1.0 - sum(vals[1:])
# compute Dirichlet value
print(vals.T, prod(vals ** (alpha - 1)) , normalizer, alpha)
return prod(vals.T ** (alpha - 1)) / normalizer
else:
vals[idx] = x
split = alpha[idx-1] / sum(alpha)
if (split < upper - x):
return romberg(f_recur, 0, split, args=(idx - 1, upper - x, vals), vec_func=False) + \
romberg(f_recur, split, upper - x, args=(idx - 1, upper - x, vals), vec_func=False)
else:
return romberg(f_recur, 0, upper - x, args=(idx - 1, upper - x, vals), vec_func=False)
split = alpha[-1] / sum(alpha)
print(alpha / sum(alpha))
return romberg(f_recur, 0, split, args=(len(alpha) - 1, 1.0, zeros((len(alpha), 1), float64)), vec_func=False) + \
romberg(f_recur, split, 1, args=(len(alpha) - 1, 1.0, zeros((len(alpha), 1), float64)), vec_func=False)
def _normalize(self):
self.f_norm = 1.0
norm = integrate.romberg(
self.f_nu, self.nu_min, self.nu_max, vec_func=True,
tol=defaults.default_precision["global_precision"],
rtol=defaults.default_precision["mass_precision"],
divmax=defaults.default_precision["divmax"])
self.f_norm = 1.0/norm
self.bias_norm = 1.0
norm = integrate.romberg(
lambda x: self.f_nu(x)*self.bias_nu(x),
self.nu_min, self.nu_max, vec_func=True,
tol=defaults.default_precision["global_precision"],
rtol=defaults.default_precision["mass_precision"],
divmax=defaults.default_precision["divmax"])
self.bias_norm = 1.0/norm
self.bias_2_norm = 0.0
norm = integrate.romberg(
lambda x: self.f_nu(x)*self.bias_2_nu(x),
self.nu_min, self.nu_max, vec_func=True,
tol=defaults.default_precision["global_precision"],
rtol=defaults.default_precision["mass_precision"],
divmax=defaults.default_precision["divmax"])
self.bias_2_norm = -norm
def dirichlet_integrate(alpha):
normalizer = exp(sum(gammaln(alpha)) - gammaln(sum(alpha)))
def f_recur(x, idx, upper, vals):
if idx == 1:
# base case.
# set values for last two components
vals[1] = x
vals[0] = 1.0 - sum(vals[1:])
# compute Dirichlet value
print vals.T, prod(vals**(alpha - 1)), normalizer, alpha
return prod(vals.T**(alpha - 1)) / normalizer
else:
vals[idx] = x
split = alpha[idx - 1] / sum(alpha)
if (split < upper - x):
return romberg(f_recur, 0, split, args=(idx - 1, upper - x, vals), vec_func=False) + \
romberg(f_recur, split, upper - x, args=(idx - 1, upper - x, vals), vec_func=False)
else:
return romberg(f_recur,
0,
upper - x,
args=(idx - 1, upper - x, vals),
vec_func=False)
split = alpha[-1] / sum(alpha)
print alpha / sum(alpha)
return romberg(f_recur, 0, split, args=(len(alpha) - 1, 1.0, zeros((len(alpha), 1), float64)), vec_func=False) + \
romberg(f_recur, split, 1, args=(len(alpha) - 1, 1.0, zeros((len(alpha), 1), float64)), vec_func=False)
def raw_correlation(self, r):
"""
Compute the value of the correlation at array values r
Args:
r: float array of position values in Mpc/h
"""
try:
xi_out = numpy.empty(len(r))
for idx, value in enumerate(r):
xi_out[idx] = integrate.romberg(
self._correlation_integrand,
self._ln_k_min, self._ln_k_max, args=(value,),
vec_func=True,
tol=defaults.default_precision["global_precision"],
rtol=defaults.default_precision["corr_precision"],
divmax=defaults.default_precision["divmax"])
except TypeError:
xi_out = integrate.romberg(
self._correlation_integrand,
self._ln_k_min, self._ln_k_max, args=(r,), vec_func=True,
tol=defaults.default_precision["global_precision"],
rtol=defaults.default_precision["corr_precision"],
divmax=defaults.default_precision["divmax"])
return xi_out
def adj_term(self, N_bins, method):
# Area of ideal histogram bins.
A = 1. / N_bins
cum_areas = np.arange(N_bins) * A
ideal_bins = erfinv(
cum_areas) # Ideally the Nth bin has height 0 and infinite width
real_bins = np.append(ideal_bins, self.cutoff_val)
bin_widths = np.diff(real_bins)
# Heights of the ideal bins
id_bin_hs = A / bin_widths
#id_bin_hs[-1] = 0
# Calulate the adjustment to the error
if method == "sq":
adj_error = np.sqrt(2 / np.pi) - id_bin_hs.sum() * A
elif method == "abs":
adj_error = romberg(lambda z: np.square(id_bin_hs[np.digitize(
z, real_bins) - 1] - 2 / np.sqrt(np.pi) * np.exp(-z**2)),
0.,
self.cutoff_val * .999,
divmax=100,
tol=1.e-6,
vec_func=True)
elif method == "abs_sqrtM":
adj_error = np.sqrt(N_bins) * romberg(
lambda z: np.square(id_bin_hs[np.digitize(z, real_bins) - 1] -
2 / np.sqrt(np.pi) * np.exp(-z**2)),
0.,
self.cutoff_val * .999,
divmax=100,
tol=1.e-6,
vec_func=True)
return adj_error, real_bins, id_bin_hs
def raw_window_function(self, chi):
a = 1.0 / (1.0 + self.cosmo.redshift(chi))
try:
g_chi = numpy.empty(len(chi))
for idx, value in enumerate(chi):
chi_bound = value
if chi_bound < self._g_chi_min: chi_bound = self._g_chi_min
g_chi[idx] = integrate.romberg(
self._lensing_integrand,
chi_bound,
self.chi_max,
args=(value, ),
vec_func=True,
tol=defaults.default_precision["window_precision"])
except TypeError:
chi_bound = chi
if chi_bound < self._g_chi_min: chi_bound = self._g_chi_min
g_chi = integrate.romberg(
self._lensing_integrand,
chi_bound,
self.chi_max,
args=(chi, ),
vec_func=True,
tol=defaults.default_precision["window_precision"])
g_chi *= self.cosmo.H0 * self.cosmo.H0 * chi
return 3.0 / 2.0 * self.cosmo._omega_m0 * g_chi / a
def cont_est_logistic(ind, division, xs, coefs_p):
ncols = xs.shape[1]
#nrows=xs.shape[0]
x = np.zeros((ncols, 3))
x[:, 1] = xs.iloc[ind, :].values
x[:, 2] = (x[:, 1] - x[:, 0]) / division
y = np.zeros((division, ncols))
temp = np.zeros((ncols, 2))
for j in range(temp.shape[0]):
temp[j, 0] = x[j, 0]
temp[j, 1] = x[j, 0] + x[j, 2]
def f(x):
return pd_logistic(x, j, temp, coefs_p)
y[0, j] = sci.romberg(f, 0, 1)
for i in range(1, y.shape[0]):
for j in range(temp.shape[0]):
temp[j, 0] = temp[j, 1]
temp[j, 1] = temp[j, 0] + x[j, 2]
def f(x):
return pd_logistic(x, j, temp, coefs_p)
y[i, j] = sci.romberg(f, 0, 1)
return y.sum(axis=0)
def correlation(self, l):
"""
Compute the value of the correlation at array values theta
Args:
theta: float array of angular values in radians to compute the
correlation
"""
try:
power = numpy.empty(len(l))
for idx, value in enumerate(l):
power[idx] = integrate.romberg(
self._correlation_integrand,
self.kernel.chi_min,
self.kernel.chi_max,
args=(value, ),
vec_func=True,
tol=defaults.default_precision["global_precision"],
rtol=defaults.default_precision["corr_precision"],
divmax=defaults.default_precision["divmax"])
except TypeError:
power = integrate.romberg(
self._correlation_integrand,
self.kernel.chi_min,
self.kernel.chi_max,
args=(l, ),
vec_func=True,
tol=defaults.default_precision["global_precision"],
rtol=defaults.default_precision["corr_precision"],
divmax=defaults.default_precision["divmax"])
return power
def raw_correlation(self, r):
"""
Compute the value of the correlation at array values r
Args:
r: float array of position values in Mpc/h
"""
try:
xi_out = numpy.empty(len(r))
for idx, value in enumerate(r):
xi_out[idx] = integrate.romberg(
self._correlation_integrand,
self._ln_k_min,
self._ln_k_max,
args=(value, ),
vec_func=True,
tol=defaults.default_precision["global_precision"],
rtol=defaults.default_precision["corr_precision"],
divmax=defaults.default_precision["divmax"])
except TypeError:
xi_out = integrate.romberg(
self._correlation_integrand,
self._ln_k_min,
self._ln_k_max,
args=(r, ),
vec_func=True,
tol=defaults.default_precision["global_precision"],
rtol=defaults.default_precision["corr_precision"],
divmax=defaults.default_precision["divmax"])
return xi_out
def complex_quad(func, a, b, **kwargs):
real = lambda x: np.real(func(x))
imag = lambda x: np.imag(func(x))
real_i = integrate.romberg(real, a, b, **kwargs)
imag_i = integrate.romberg(imag, a, b, **kwargs)
return real_i + 1j * imag_i
def complex_integral(func,a,b,args,intype='stupid'):
real_func = lambda z: np.real(func(z,*args))
imag_func = lambda z: np.imag(func(z,*args))
if intype == 'quad':
real_int = integ.quad(real_func,a,b)
imag_int = integ.quad(imag_func,a,b)
# print(real_int)
# print(imag_int)
return real_int[0] + 1j * imag_int[0]
elif intype == 'quadrature':
real_int = integ.quadrature(real_func,a,b)
imag_int = integ.quadrature(imag_func,a,b)
# print(real_int)
# print(imag_int)
return real_int[0] + 1j * imag_int[0]
elif intype == 'romberg':
real_int = integ.romberg(real_func,a,b)
imag_int = integ.romberg(imag_func,a,b)
# print(real_int)
# print(imag_int)
return real_int + 1j * imag_int
elif intype == 'stupid':
Npoints = 500
z,dz = np.linspace(a,b,Npoints,retstep=True)
real_int = np.sum(real_func(z))*dz
imag_int = np.sum(imag_func(z))*dz
# print(real_int)
# print(imag_int)
return real_int + 1j*imag_int
def funct(A, r1, r2, r3, hcc, theor, bc0, cl0, k2, b1, b2, b3, b4):
ca = (pi *
(r1**2 * romberg(c, bc0, b1, args=(cl0, A, k2)) +
romberg(cpyra, b1, b2, args=(r1, hcc, b1, cl0, A, k2)) + r2**2 *
(romberg(c, b2, b3, args=(cl0, A, k2))) + r3**2 *
(romberg(c, b3, b4, args=(cl0, A, k2)))) - theor) * 100. / theor
return ca
def synphot (self, wlen, flam):
"""`wlen` and `flam` give a tabulated model spectrum in wavelength and f_λ
units. We interpolate linearly over both the model and the bandpass
since they're both discretely sampled.
Note that quadratic interpolation is both much slower and can blow up
fatally in some cases. The latter issue might have to do with really large
X values that aren't zero-centered, maybe?
I used to use the quadrature integrator, but Romberg doesn't issue
complaints the way quadrature did. I should probably acquire some idea
about what's going on under the hood.
"""
from scipy.interpolate import interp1d
from scipy.integrate import romberg
d = self._ensure_data ()
mflam = interp1d (wlen, flam,
kind='linear',
bounds_error=False, fill_value=0)
mresp = interp1d (d.wlen, d.resp,
kind='linear',
bounds_error=False, fill_value=0)
bmin = d.wlen.min ()
bmax = d.wlen.max ()
numer = romberg (lambda x: mresp (x) * mflam (x),
bmin, bmax, divmax=20)
denom = romberg (lambda x: mresp (x),
bmin, bmax, divmax=20)
return numer / denom
def raw_window_function(self, chi):
a = 1.0/(1.0 + self.cosmo.redshift(chi))
try:
g_chi = numpy.empty(len(chi))
for idx,value in enumerate(chi):
chi_bound = value
if chi_bound < self._g_chi_min: chi_bound = self._g_chi_min
g_chi[idx] = integrate.romberg(
self._lensing_integrand, chi_bound,
self.chi_max, args=(value,), vec_func=True,
tol=defaults.default_precision["window_precision"])
except TypeError:
chi_bound = chi
if chi_bound < self._g_chi_min: chi_bound = self._g_chi_min
g_chi = integrate.romberg(
self._lensing_integrand, chi_bound,
self.chi_max, args=(chi,), vec_func=True,
tol=defaults.default_precision["window_precision"])
g_chi *= self.cosmo.H0*self.cosmo.H0*chi
return 3.0/2.0*self.cosmo._omega_m0*g_chi/a
def pok(self): self.msfcbias() self.NFWpro() bias_interp = scipy.interpolate.interp1d(mass_arr, self.bias) nlnm_interp = scipy.interpolate.interp1d(mass_arr, self.ndlnm) pk_dat = np.loadtxt(self.name['pok_file']) pk_interp = scipy.interpolate.interp1d(pk_dat[:, 0], pk_dat[:, 1]) plk_arr = pk_interp(self.karr) p1h_arr = np.zeros(CP['num_k']) pok_arr = np.zeros(CP['num_k']) Ii_arr = np.zeros(CP['num_k']) m_min = 1.000001 * min(mass_arr) m_max = 0.999999 * max(mass_arr) for i in range(CP['num_k']): rho_interp = scipy.interpolate.interp1d(mass_arr, self.NFW[i, :]) p1h_arr[i] = romberg(pf,np.log(m_min),np.log(m_max),args=(nlnm_interp,rho_interp)) Ii_arr[i] = romberg(Itm,np.log(m_min),np.log(m_max),args=(nlnm_interp,bias_interp,rho_interp)) pok_arr[i] = plk_arr[i]*Ii_arr[i]**2.0 + p1h_arr[i] pkk_arr = np.zeros([CP['num_k'], 5]) pkk_arr[:, 0] = self.karr pkk_arr[:, 1] = p1h_arr pkk_arr[:, 2] = Ii_arr pkk_arr[:, 3] = plk_arr pkk_arr[:, 4] = plk_arr * (Ii_arr/max(Ii_arr))**2.0 + p1h_arr return pkk_arr
def _initialize_defaults(self):
if self._w0 != -1.0 or self._wa != 0.0:
a_array = numpy.logspace(
-16, 0,
defaults.default_precision["cosmo_npoints"])
self._de_presure_array = self._de_presure(1/a_array - 1.0)
self._de_presure_spline = InterpolatedUnivariateSpline(
numpy.log(a_array), self._de_presure_array)
self._chi = integrate.romberg(
self.E, 0.0, self._redshift, vec_func=True,
tol=defaults.default_precision["cosmo_precision"])
self.growth_norm = integrate.romberg(
self._growth_integrand, 1e-16, 1.0, vec_func=True,
tol=defaults.default_precision["cosmo_precision"])
self.growth_norm *= 2.5*self._omega_m0*numpy.sqrt(self.E0(0.0))
a = 1.0/(1.0 + self._redshift)
growth = integrate.romberg(
self._growth_integrand, 1e-16, a, vec_func=True,
tol=defaults.default_precision["cosmo_precision"])
growth *= 2.5*self._omega_m0*numpy.sqrt(self.E0(self._redshift))
self._growth = growth/self.growth_norm
self._sigma_norm = 1.0
self._sigma_norm = self._sigma_8*self._growth/self.sigma_r(8.0)
def correlation(self, l):
"""
Compute the value of the correlation at array values theta
Args:
theta: float array of angular values in radians to compute the
correlation
"""
try:
power = numpy.empty(len(l))
for idx, value in enumerate(l):
power[idx] = integrate.romberg(
self._correlation_integrand,
self.kernel.chi_min, self.kernel.chi_max, args=(value,),
vec_func=True,
tol=defaults.default_precision["global_precision"],
rtol=defaults.default_precision["corr_precision"],
divmax=defaults.default_precision["divmax"])
except TypeError:
power = integrate.romberg(
self._correlation_integrand,
self.kernel.chi_min, self.kernel.chi_max, args=(l,), vec_func=True,
tol=defaults.default_precision["global_precision"],
rtol=defaults.default_precision["corr_precision"],
divmax=defaults.default_precision["divmax"])
return power
def __call__(self,Phi_Mz,p_Mz,par): # integrand = lambda M,z: Phi_Mz(M,z,par) * p_Mz(M,z) * self.dVdzdO(z) inner = lambda z: romberg(integrand,*self.Mrange,args=(z,), **self.int_kwargs) outer = romberg(inner,*self.zrange,**self.int_kwargs) return outer
def ri(u):
''' Epälineaarisen osan integrointi Gaussin kaavalla '''
J = h/2 # Jakobiaani
I = lambda k: N(k)*np.cos(np.sum(N(k)*u))*J # Integrandi
I0 = lambda k: I(k)[0]
I1 = lambda k: I(k)[1]
#return 2.0*I(0.0)
#return quad(I,-1,1)[0]
return np.array([romberg(I0,-1,1),romberg(I1,-1,1)])
def pin(self,alpha, lmd):
# p(act -> inact)#
a = -lmd - alpha
b = lmd - alpha
c = -lmd + alpha
d = lmd + alpha
p1 = sci.romberg(self.f, a, b)
p2 = sci.romberg(self.f, c, d)
p = 0.5 * p1 + 0.5 * p2
return p
def P_2_numerical(a):
from scipy.integrate import romberg
from numpy import cos, sin, pi, exp
def num(beta):
return (1.5 * cos(beta)**2 - 0.5) * exp(a * cos(beta)**2) * sin(beta)
def denom(beta):
return exp(a * cos(beta)**2) * sin(beta)
return romberg(num, 0, pi / 2) / romberg(denom, 0, pi / 2)
def dcom(z, Om0, OmL, ninter=20):
"""
function computing comoving distance Dc for a given redshift and mean matter and vacuum energies Om0 and OmL
"""
Omk = 1. - Om0 - OmL
kwargs = {"Om0": Om0, "OmL": OmL, "Omk": Omk}
a = 1. / (1.0 + z)
nz = np.size(z)
if nz == 1:
if np.abs(a - 1.0) < 1.e-10:
dc = 0.
else:
dc = romberg(d_func,
a,
1.,
rtol=1.e-10,
mmax=16,
verbose=False,
**kwargs)[0]
elif nz > 1:
dc = np.zeros(nz)
if nz <= ninter:
for i, ad in enumerate(a):
if np.abs(ad - 1.0) < 1.e-10:
dc[i] = 0.
else:
dc[i] = romberg(d_func,
ad,
1.,
rtol=1.e-10,
mmax=16,
verbose=False,
**kwargs)[0]
else:
zmin = np.min(z)
zmax = np.max(z)
zi = np.linspace(zmin, zmax, num=ninter)
fi = np.zeros(ninter)
for i, zid in enumerate(zi):
aid = 1.0 / (1 + zid)
if np.abs(aid - 1.0) < 1.e-10:
fi[i] = 0.
else:
fi[i] = romberg(d_func,
aid,
1.,
rtol=1.e-10,
mmax=16,
verbose=False,
**kwargs)[0]
dsp = UnivariateSpline(zi, fi, s=0.)
dc = dsp(z)
return dc
def romberg_slit_1d(q, width, height, form, pars):
"""
Romberg integration for slit resolution.
This is an adaptive integration technique. It is called with settings
that make it slow to evaluate but give it good accuracy.
"""
from scipy.integrate import romberg, dblquad
if any(k not in form.info['defaults'] for k in pars.keys()):
keys = set(form.info['defaults'].keys())
extra = set(pars.keys()) - keys
raise ValueError("bad parameters: [%s] not in [%s]"%
(", ".join(sorted(extra)), ", ".join(sorted(keys))))
if np.isscalar(width):
width = [width]*len(q)
if np.isscalar(height):
height = [height]*len(q)
_int_w = lambda w, qi: eval_form(sqrt(qi**2 + w**2), form, pars)
_int_h = lambda h, qi: eval_form(abs(qi+h), form, pars)
# If both width and height are defined, then it is too slow to use dblquad.
# Instead use trapz on a fixed grid, interpolated into the I(Q) for
# the extended Q range.
#_int_wh = lambda w, h, qi: eval_form(sqrt((qi+h)**2 + w**2), form, pars)
q_calc = slit_extend_q(q, np.asarray(width), np.asarray(height))
Iq = eval_form(q_calc, form, pars)
result = np.empty(len(q))
for i, (qi, w, h) in enumerate(zip(q, width, height)):
if h == 0.:
r = romberg(_int_w, 0, w, args=(qi,),
divmax=100, vec_func=True, tol=0, rtol=1e-8)
result[i] = r/w
elif w == 0.:
r = romberg(_int_h, -h, h, args=(qi,),
divmax=100, vec_func=True, tol=0, rtol=1e-8)
result[i] = r/(2*h)
else:
w_grid = np.linspace(0, w, 21)[None,:]
h_grid = np.linspace(-h, h, 23)[:,None]
u = sqrt((qi+h_grid)**2 + w_grid**2)
Iu = np.interp(u, q_calc, Iq)
#print np.trapz(Iu, w_grid, axis=1)
Is = np.trapz(np.trapz(Iu, w_grid, axis=1), h_grid[:,0])
result[i] = Is / (2*h*w)
"""
r, err = dblquad(_int_wh, -h, h, lambda h: 0., lambda h: w,
args=(qi,))
result[i] = r/(w*2*h)
"""
# r should be [float, ...], but it is [array([float]), array([float]),...]
return result
def romberg_slit_1d(q, width, height, form, pars):
"""
Romberg integration for slit resolution.
This is an adaptive integration technique. It is called with settings
that make it slow to evaluate but give it good accuracy.
"""
from scipy.integrate import romberg # type: ignore
par_set = set([p.name for p in form.info.parameters.call_parameters])
if any(k not in par_set for k in pars.keys()):
extra = set(pars.keys()) - par_set
raise ValueError("bad parameters: [%s] not in [%s]"
% (", ".join(sorted(extra)),
", ".join(sorted(pars.keys()))))
if np.isscalar(width):
width = [width]*len(q)
if np.isscalar(height):
height = [height]*len(q)
_int_w = lambda w, qi: eval_form(sqrt(qi**2 + w**2), form, pars)
_int_h = lambda h, qi: eval_form(abs(qi+h), form, pars)
# If both width and height are defined, then it is too slow to use dblquad.
# Instead use trapz on a fixed grid, interpolated into the I(Q) for
# the extended Q range.
#_int_wh = lambda w, h, qi: eval_form(sqrt((qi+h)**2 + w**2), form, pars)
q_calc = slit_extend_q(q, np.asarray(width), np.asarray(height))
Iq = eval_form(q_calc, form, pars)
result = np.empty(len(q))
for i, (qi, w, h) in enumerate(zip(q, width, height)):
if h == 0.:
total = romberg(_int_w, 0, w, args=(qi,),
divmax=100, vec_func=True, tol=0, rtol=1e-8)
result[i] = total/w
elif w == 0.:
total = romberg(_int_h, -h, h, args=(qi,),
divmax=100, vec_func=True, tol=0, rtol=1e-8)
result[i] = total/(2*h)
else:
w_grid = np.linspace(0, w, 21)[None, :]
h_grid = np.linspace(-h, h, 23)[:, None]
u_sub = sqrt((qi+h_grid)**2 + w_grid**2)
f_at_u = np.interp(u_sub, q_calc, Iq)
#print(np.trapz(Iu, w_grid, axis=1))
total = np.trapz(np.trapz(f_at_u, w_grid, axis=1), h_grid[:, 0])
result[i] = total / (2*h*w)
# from scipy.integrate import dblquad
# r, err = dblquad(_int_wh, -h, h, lambda h: 0., lambda h: w,
# args=(qi,))
# result[i] = r/(w*2*h)
# r should be [float, ...], but it is [array([float]), array([float]),...]
return result
def _smith_parameters(self, a, **kwargs):
r""" Computes the non linear scale, effective spectral index
and spectral curvature"""
a = atleast_1d(a)
R_nl = zeros_like(a)
n = zeros_like(a)
C = zeros_like(a)
ksamp = logspace(log10(self._kmin), log10(self._kmax), 1024)
pklog = interp1d(
log(ksamp),
ksamp**3 * self.pk_lin(ksamp, **kwargs) / (2.0 * pi**2))
g = self.G(a)
def int_sigma(logk, r, _g):
y = exp(logk) * r
return pklog(logk) * _g**2 * exp(-y**2)
def int_neff(logk, r, _g):
y = exp(logk) * r
return pklog(logk) * _g**2 * y**2 * exp(-y**2)
def int_C(logk, r, _g):
y = exp(logk) * r
return pklog(logk) * _g**2 * (y**2 - y**4) * exp(-y**2)
for i in range(R_nl.size):
sigm = lambda r: romberg(int_sigma,
log(self._kmin),
log(self._kmax),
args=(exp(r), g[i]),
rtol=1e-4,
vec_func=True,
divmax=100) - 1
R_nl[i] = exp(brentq(sigm, -5, 1.5, rtol=1e-4))
n[i] = 2.0 * romberg(int_neff,
log(self._kmin),
log(self._kmax),
args=(R_nl[i], g[i]),
rtol=1e-4,
vec_func=True,
divmax=100) - 3
C[i] = (3 + n[i])**2 + 4 * romberg(int_C,
log(self._kmin),
log(self._kmax),
args=(R_nl[i], g[i]),
rtol=1e-4,
vec_func=True,
divmax=100)
k_nl = 1.0 / R_nl
return k_nl, n, C
def dL2(z,h):
c=3.e5
H0=100.*h
if type(z) == ndarray:
n=len(z)
DL=zeros(n,'d')
for i in range(n):
s=romberg(func,0.,z[i])#,tol=1.e-6)
DL[i]=c/H0*(1+z[i])*s #(Mpc/h)
else:
s=romberg(func,0.,z)#,tol=1.e-6)
DL=c/H0*(1+z)*s #(Mpc/h)
return DL
def dL2(z, h):
c = 3.e5
H0 = 100. * h
if type(z) == ndarray:
n = len(z)
DL = zeros(n, 'd')
for i in range(n):
s = romberg(func, 0., z[i]) #,tol=1.e-6)
DL[i] = c / H0 * (1 + z[i]) * s #(Mpc/h)
else:
s = romberg(func, 0., z) #,tol=1.e-6)
DL = c / H0 * (1 + z) * s #(Mpc/h)
return DL
def __massEjected(self, m_min):
"""
Return the mass integration of the mass ejected by the collapse of the
star
"""
if(self.imfType == "S"):
return self.__massEjectedSalpeter(m_min)
else:
mEject = (romberg(self.__mPhi, m_min, self.__amsup1, tol=1.48e-04)
- romberg(self.__mrPhi, m_min, self.__amsup1, tol=1.48e-04)
)
return mEject
def dL(z,h):
c=3.e5
H0=100.*h
try:#multiple objects
n=len(z)
DL=zeros(n,'d')
for i in range(n):
s=romberg(func,0.,z[i])#,tol=1.e-6)
DL[i]=c/H0*(1+z[i])*s #(Mpc/h)
except TypeError:
s=romberg(func,0.,z)#,tol=1.e-6)
DL=c/H0*(1+z)*s #(Mpc/h)
return DL
def dL(z, h):
c = 3.e5
H0 = 100. * h
try: #multiple objects
n = len(z)
DL = zeros(n, 'd')
for i in range(n):
s = romberg(func, 0., z[i]) #,tol=1.e-6)
DL[i] = c / H0 * (1 + z[i]) * s #(Mpc/h)
except TypeError:
s = romberg(func, 0., z) #,tol=1.e-6)
DL = c / H0 * (1 + z) * s #(Mpc/h)
return DL
def Lookback(z1,
z2,
Omega_m0,
Hz=H_LamCDM,
H0=H0_def): ### calculating the lookback time between z1 and z2
def h(z):
return 1 / ((1.0 + z) * Hz(Omega_m0, z, H0))
if z1 >= z2:
q = romberg(h, z2, z1) * H0
else:
q = romberg(h, z1, z2) * H0
time = q * (100 / H0) * 9.78 ### age is in Gyr
return time
def _de_presure(self, redshift):
dpresuredz = lambda z: (1 + self.w(z))/(1 + z)
try:
presure = numpy.empty(len(redshift))
for idx, z in enumerate(redshift):
presure[idx] = 3.0*integrate.romberg(
dpresuredz, 0, z, vec_func=True,
tol=defaults.default_precision["cosmo_precision"])
except TypeError:
presure = 3.0*integrate.romberg(
dpresuredz, 0, redshift, vec_func=True,
tol=defaults.default_precision["cosmo_precision"])
return presure
def _test_one(name, p, w, tol):
import scipy.integrate as sum
# Check that the pdf approximately matchs the numerical integral
# Check that integral(-inf,0) of pdf sums to 0.5
err = abs(sum.romberg(p.pdf, -20 * w, 0, tol=1e-15) - 0.5)
assert err < tol, "%s cdf(0) == 0.5 yields %g" % (name, err)
# Check that integral(-inf,x) of pdf sums to cdf when x != 0
err = abs(sum.romberg(p.pdf, -20 * w, w / 6, tol=1e-15) - p.cdf(w / 6))
assert err < tol, "%s cdf(w/6) == w/6 yields %g" % (name, err)
# Check that P = cdf(ppf(P))
P = 0.002
err = abs(p.cdf(p.ppf(P)) - P)
assert err < tol, "%s p(lo) = P yields %g" % (name, err)
def _test_one(name, p, w, tol):
import scipy.integrate as sum
# Check that the pdf approximately matchs the numerical integral
# Check that integral(-inf, 0) of pdf sums to 0.5
err = abs(sum.romberg(p.pdf, -20*w, 0, tol=1e-15) - 0.5)
assert err < tol, "%s cdf(0) == 0.5 yields %g"%(name, err)
# Check that integral(-inf, x) of pdf sums to cdf when x != 0
err = abs(sum.romberg(p.pdf, -20*w, w/6, tol=1e-15) - p.cdf(w/6))
assert err < tol, "%s cdf(w/6) == w/6 yields %g"%(name, err)
# Check that P = cdf(ppf(P))
P = 0.002
err = abs(p.cdf(p.ppf(P)) - P)
assert err < tol, "%s p(lo) = P yields %g"%(name, err)
def dL(z,
k,
Omega_m0,
H0=H0_def,
Hz=H_LamCDM): ### calculate luminosity distance in Mpc
def h(x):
return 1.0 / Hz(Omega_m0, x, H0)
p = H0 * sqrt(abs(Omega_k0(k, H0))) * romberg(h, 0, z)
if k:
q = (1.0 + z) * (c / H0) * (1.0 / sqrt(abs(Omega_k0(k, H0)))) * FF(
p, k)
else:
q = (1.0 + z) * (c / H0) * H0 * romberg(h, 0, z)
return q
def RhoCoul(r1,r2,theta,tau,xkappa,z,kmax,lmax,nmax,D):
uklim = sqrt(float(kmax)/(tau*xkappa))
rhocoul = 0.
if D == 2:
rhocoul = integrate.romberg(CGrand2D, 0., 2.*uklim, args=(r1,r2,theta,tau,xkappa,z,lmax))
elif D == 3:
rhocoul = integrate.romberg(CGrand3D, 0., 2.*uklim, args=(r1,r2,theta,tau,xkappa,z,lmax))
if (z > 0.):
return rhocoul
else:
if D == 2:
rhocoul += Bound2D(r1,r2,theta,tau,xkappa,z,nmax)
elif D == 3:
rhocoul += Bound3D(r1,r2,theta,tau,xkappa,z,nmax)
return rhocoul
def yieldcode(tmin, tmax, time, RBsol, fracinterp, polynomial, c, IMF):
"""
The yieldcode module will use the step solution from the rombergquad
module to calculate the yield in any time period, by calculating and
excess outside the step ranges.
"""
if tmin < time[-1]:
tmin = time[-1]
if tmax > time[0]:
tmax = time[0]
deriv = polynomial.deriv()
lowerindex = np.where(time<=tmax)[0][0]
upperindex = np.where(time>=tmin)[0][-1]
if lowerindex >= upperindex: # this is to avoid counting romberg intervals when the times don't actually cover any interval.
m = lambda t: np.exp(polynomial(np.log(t)))
y = lambda t: IMF(m(t))*c*deriv(np.log(t))*fracinterp(t)/t*m(t)
intsum = romberg(y, tmax, tmin)
else:
intervals = RBsol[lowerindex:upperindex]
intsum = np.sum(intervals)
# Finding the excess integrals
m = lambda t: np.exp(polynomial(np.log(t)))
y = lambda t: IMF(m(t))*c*deriv(np.log(t))*fracinterp(t)/t*m(t)
t1 = time[lowerindex] # corrected time is a decreasing vector
t2 = tmax
sol = romberg(y, t2, t1)
intsum += sol
t1 = tmin
t2 = time[upperindex]
sol = romberg(y, t2, t1)
intsum += sol
return intsum
def correl_swsw(self, angle, nz1, nz2, h1, h2, method='quad'):
"""Correlation coeficients between two spherical waves."""
alpha = angle * 4.85*1.e-6
R = self.pupil_diameter / 2.
R1 = (h1-self.h_profile) / h1 * R
R2 = (h2-self.h_profile) / h2 * R
zeta = alpha * self.h_profile / R1
w = R2 / R1
Lam = 2. * np.pi * R1 / self.large_scale
n1, m1 = zernike.noll2zern(nz1)
n2, m2 = zernike.noll2zern(nz2)
k1, k2 = _kvalues(n1, n2, m1, m2, nz1, nz2)
results = np.zeros(len(self.h_profile))
for idx in np.arange(len(self.h_profile)):
if method == 'quad':
result_quad = quad(_chassat_integral, 0, np.inf, args=(zeta[idx], Lam[idx], w[idx], k1, k2, n1, n2, m1, m2))
results[idx] = result_quad[0] * self.cn2[idx] * R1[idx]**(5./3.)
elif method == 'romberg':
result_quad = romberg(_modified_chassat, 1.e-26,np.pi/2., args=(zeta[idx], Lam[idx], w[idx], k1, k2, n1, n2, m1, m2), vec_func = False)
results[idx] = result_quad * self.cn2[idx] * R1[idx]**(5./3.)
if len(results) < 2:
final_integral = results / (self.cn2 * R1**(5./3.))
else:
final_integral = trapz(results, x=self.h_profile) / trapz(self.cn2 * R1**(5./3.), x=self.h_profile)
final_integral = 3.895 * (-1.)**((n1+n2-m1-m2)/2.) * np.sqrt((n1+1.)*(n2+1.)) * self.dr0**(5./3.) * final_integral
return final_integral
def romberg_pinhole_1d(q, q_width, form, pars, nsigma=5):
"""
Romberg integration for pinhole resolution.
This is an adaptive integration technique. It is called with settings
that make it slow to evaluate but give it good accuracy.
"""
from scipy.integrate import romberg
if any(k not in form.info['defaults'] for k in pars.keys()):
keys = set(form.info['defaults'].keys())
extra = set(pars.keys()) - keys
raise ValueError("bad parameters: [%s] not in [%s]" %
(", ".join(sorted(extra)), ", ".join(sorted(keys))))
_fn = lambda q, q0, dq: eval_form(q, form, pars) * gaussian(q, q0, dq)
r = [
romberg(_fn,
max(qi - nsigma * dqi, 1e-10 * q[0]),
qi + nsigma * dqi,
args=(qi, dqi),
divmax=100,
vec_func=True,
tol=0,
rtol=1e-8) for qi, dqi in zip(q, q_width)
]
return np.asarray(r).flatten()
def Dplus(self):
M, L = self.M, self.L
logx = numpy.linspace(logamin, 0, Np)
x = numpy.exp(logx)
def kernel(loga):
a = numpy.exp(loga)
return (a * self.Ea(a))**-3 * a # da = a * d loga
y = self.Ea(x) * numpy.array([
romberg(kernel, logx.min(), loga, vec_func=True) for loga in logx
])
def func(x,
nu=0,
intp=interp1d(logx,
y,
bounds_error=False,
fill_value=numpy.nan,
kind=5)):
return intp(numpy.log(x), nu=nu)
func.__doc__ = \
"""evaluates D+(a) There is no 2.5 factor neither.
(dimensionless, not multiplied by H0)
for OmegaM = %g, OmegaL = %g
""" % (M, L)
return func
def test_romberg_rtol(self):
# Typical function with two extra arguments:
def myfunc(x, n, z): # Bessel function integrand
return 1e19*cos(n*x-z*sin(x))/pi
val = romberg(myfunc, 0, pi, args=(2, 1.8), rtol=1e-10)
table_val = 1e19*0.30614353532540296487
assert_allclose(val, table_val, rtol=1e-10)
def test_romberg(self):
# Typical function with two extra arguments:
def myfunc(x, n, z): # Bessel function integrand
return cos(n*x-z*sin(x))/pi
val = romberg(myfunc, 0, pi, args=(2, 1.8))
table_val = 0.30614353532540296487
assert_almost_equal(val, table_val, decimal=7)
def DA(z,h):#uses numerical integration technique gaussian quadrature
c=3.e5
H0=100.*h
s=romberg(func,0.,z)
DA=c/H0/(1+z)*s #(Mpc/radian)
DA=DA*1000./206264 #(kpc/arcsec)
return DA
def match_abundance(self,
halomass,
bs,
mfunc='Mice',
aa=1,
Mmin=10.**11.,
Mmax=None):
'''Returns new halomasses by matching abundance to given mass function'''
if Mmax is None:
Mmax = halomass[0] * 1.01
marray = numpy.exp(numpy.arange(numpy.log(Mmin), numpy.log(Mmax),
0.01))
abund = []
l = marray.shape[0]
# f = lambda x:temp_mice(x, a)
f = lambda x: self.DnDlnm(numpy.exp(x), mfunc=mfunc, aa=aa)
for foo in range(0, marray.shape[0]):
abund.append(
romberg(f,
numpy.log(marray)[l - foo - 1],
numpy.log(marray)[-1]))
abund = numpy.array(abund)
nexpect = abund * bs**3
newmass = interpolate(nexpect, marray[::-1])
halomassnew = newmass(
numpy.linspace(1, len(halomass), num=len(halomass), endpoint=True))
return halomassnew
def _initialize_i_2_2(self):
_i_2_2_array = numpy.empty((len(self._ln_k_array),
len(self._ln_k_array)))
for idx1 in xrange(len(self._ln_k_array)):
for idx2 in xrange(idx1, len(self._ln_k_array)):
ln_k1 = self._ln_k_array[idx1]
ln_k2 = self._ln_k_array[idx2]
norm = 1.0/self._i_2_2_integrand(0.0, ln_k1, ln_k2, 1.0)
i_2_2 = integrate.romberg(
self._i_2_2_integrand, numpy.log(self.mass.nu_min),
numpy.log(self.mass.nu_max), vec_func=True,
args=(ln_k2, ln_k2, norm),
rtol=defaults.default_precision["halo_precision"],
tol=defaults.default_precision["global_precision"],
divmax=defaults.default_precision["divmax"])/(self.rho_bar)
if idx1 == idx2:
_i_2_2_array[idx1, idx2] = i_2_2/norm
else:
_i_2_2_array[idx1, idx2] = i_2_2/norm
_i_2_2_array[idx2, idx1] = i_2_2/norm
self._i_2_2_spline = RectBivariateSpline(self._ln_k_array,
self._ln_k_array,
_i_2_2_array)
print "Initialized::I_2_2"
self._initialized_i_2_2 = True
def _initialize_defaults(self):
self._initialized_growth_spline = False
if self._w0 != -1.0 or self._wa != 0.0:
# a_array = numpy.logspace(-4, 0,
# defaults.default_precision["cosmo_npoints"])
a_array = numpy.logspace(
numpy.log10(defaults.default_precision['cosmo_precision']),
0, defaults.default_precision["cosmo_npoints"])
self._de_pressure_array = self._de_pressure(1/a_array - 1.0)
self._de_pressure_spline = InterpolatedUnivariateSpline(
numpy.log(a_array), self._de_pressure_array)
self._chi = integrate.romberg(
self.E, 0.0, self._redshift, vec_func=True,
tol=defaults.default_precision["global_precision"],
rtol=defaults.default_precision["cosmo_precision"],
divmax=defaults.default_precision["divmax"])
self.growth_norm = self.growth_factor_eval(1.0)
a = 1.0 / (1.0 + self._redshift)
growth = self.growth_factor_eval(a)
self._growth = growth / self.growth_norm
self._sigma_norm = 1.0
self._sigma_norm = self._sigma_8*self._growth/self.sigma_r(8.0)
def _initialize_splines(self):
for idx in xrange(self._z_array.size):
dist = integrate.romberg(
self.epoch0.E, 0.0, self._z_array[idx], vec_func=True,
tol=defaults.default_precision["global_precision"],
rtol=defaults.default_precision["cosmo_precision"],
divmax=defaults.default_precision["divmax"])
self._chi_array[idx] = dist
self._chi_spline = InterpolatedUnivariateSpline(
self._z_array, self._chi_array)
self._z_spline = InterpolatedUnivariateSpline(
self._chi_array, self._z_array)
self.growth_norm = self.epoch0.growth_norm
# for idx in xrange(self._z_array.size):
# a = 1.0/(1.0 + self._z_array[idx])
# growth = integrate.romberg(
# self.epoch0._growth_integrand, 1e-16, a, vec_func=True,
# tol=defaults.default_precision["global_precision"],
# rtol=defaults.default_precision["cosmo_precision"],
# divmax=defaults.default_precision["divmax"])
# growth *= 2.5*self._omega_m0*numpy.sqrt(
# self.epoch0.E0(self._z_array[idx]))
# self._growth_array[idx] = growth/self.growth_norm
a = 1. / (1. + self._z_array)
self._growth_array = self.epoch0.growth_factor_eval(a)/self.growth_norm
self._growth_spline = InterpolatedUnivariateSpline(
self._z_array, self._growth_array)
def _growth_integrand_dynde(G, a, cosmo):
"""
Derivatives of normalized linear growth function G := D/a and dG/da.
For dynamical dark energy models, there is no closed integral
expression for the growth function and the defining differential
equation must be solved numerically.
This function is intended as input to scipy.integrate.odeint.
Args:
G: vector of the state variables (G,G')
a: scale factor
cosmo: Instance of cosmology.SingleEpoch
References:
Linder E. V., Jenkins A., 2003, MNRAS, 346, 573. (eq. 11)
"""
z = 1. / a - 1.
g1, g2 = G
Xde = lambda x: cosmo.w(1. / x - 1.) / x
X = numpy.exp(-3. * integrate.romberg(Xde, a, 1., vec_func=True,
tol=defaults.default_precision["global_precision"],
rtol=defaults.default_precision["cosmo_precision"],
divmax=defaults.default_precision["divmax"]))
X *= cosmo._omega_m0 / (1.0 - cosmo._omega_m0)
w = cosmo.w(z)
# create f = (x1', x2')
f = [(-(3.5 - 1.5 * w / (1. + X)) * g1 / a -
1.5 * ((1. - w) / (1. + X)) * g2 / a ** 2),
g1]
return f
def Beta_J_disk_BR(rdisks, mcold, mstar_disk, Jcold_molecular, vdisk_gas):
"""Calculate the angular momentum loading factor for the galaxy disk using the
empirical Blitz & Rosolowsky (2006) star formation law, assuming a flat rotation
profile for the disk.
Inputs:
rdisks = [gas, stellar] disk half mass radius in kpc
mcold = total gas mass in the disk in Msun
mstar_disk = total stellar mass in the disk in Msun
mgas_mol = total molecular gas mass in Msun Gyr^-1
Outputs:
Beta_J_disk = dJ_star/dt = rate of change of angular momentum for the stellar component of the disk in Msun kpc kms^-1 Gyr^-1"""
# Set integration limits so that 99.95% of the total gas mass is enclosed
rmin = 0.0
rmax = 10.0 * np.array(rdisks).max() / Constant.RDISK_HALF_SCALE
# Calculate the integral of r multiplied by the mass loading factor in annuli, weighted by molecular gas surface density in the disk
Sigma_Molecular_Weighted_Beta = integrate.romberg(Beta_J_integrand1, rmin, rmax, args = (rdisks, mcold, mstar_disk), tol=0.05,rtol=0.05) # Msun kpc
Norm = Jcold_molecular / vdisk_gas
# Renormalise to calculate the average mass loading factor of the entire disk
Beta_J_disk = Sigma_Molecular_Weighted_Beta / Norm
return Beta_J_disk
def test_romberg(self):
# Typical function with two extra arguments:
def myfunc(x, n, z): # Bessel function integrand
return cos(n*x-z*sin(x))/pi
val = romberg(myfunc, 0, pi, args=(2, 1.8))
table_val = 0.30614353532540296487
assert_almost_equal(val, table_val, decimal=7)
def _initialize_PT_averaged(self):
_pt_ave_array = numpy.empty((len(self._ln_k_array),
len(self._ln_k_array)))
_pt_norm_array = numpy.empty((len(self._ln_k_array),
len(self._ln_k_array)))
for idx1 in xrange(len(self._ln_k_array)):
for idx2 in xrange(idx1, len(self._ln_k_array)):
ln_k1 = self._ln_k_array[idx1]
ln_k2 = self._ln_k_array[idx2]
pt_ave_norm = 1.0/self._pt_ave_integrand(numpy.pi/2.0,
ln_k1, ln_k2, 1.0)
print "Norm", ln_k1, ln_k2, pt_ave_norm
pt = 1.0/numpy.pi*integrate.romberg(
self._pt_ave_integrand, 0.0, numpy.pi,
args=(ln_k1, ln_k2, pt_ave_norm), vec_func=True,
tol=defaults.default_precision["global_precision"],
rtol=defaults.default_precision["halo_precision"],
divmax=defaults.default_precision["divmax"])
if idx1 == idx2:
_pt_ave_array[idx1, idx2] = pt/pt_ave_norm
else:
_pt_ave_array[idx1, idx2] = pt/pt_ave_norm
_pt_ave_array[idx2, idx1] = pt/pt_ave_norm
self._min_pt_ave = numpy.min(_pt_ave_array)
self._t_PT_ave_spline = RectBivariateSpline(
self._ln_k_array, self._ln_k_array,
numpy.log(_pt_ave_array - self._min_pt_ave + 1.0))
self._initialzied_PT_averaged = True
def test_romberg_rtol(self):
# Typical function with two extra arguments:
def myfunc(x, n, z): # Bessel function integrand
return 1e19*cos(n*x-z*sin(x))/pi
val = romberg(myfunc, 0, pi, args=(2, 1.8), rtol=1e-10)
table_val = 1e19*0.30614353532540296487
assert_allclose(val, table_val, rtol=1e-10)
def test_non_dtype(self):
# Check that we work fine with functions returning float
import math
valmath = romberg(math.sin, 0, 1)
expected_val = 0.45969769413185085
assert_almost_equal(valmath, expected_val, decimal=7)