def get_mete_rad(S, N, beta=None, beta_dict={}):
"""Use beta to generate SAD predicted by the METE
Keyword arguments:
S -- the number of species
N -- the total number of individuals
beta -- allows input of beta by user if it has already been calculated
"""
assert S > 1, "S must be greater than 1"
assert N > 0, "N must be greater than 0"
assert S/N < 1, "N must be greater than S"
if beta is None:
beta = get_beta(S, N, beta_dict=beta_dict)
p = e ** -beta
abundance = list(empty([S]))
rank = range(1, int(S)+1)
rank.reverse()
if p >= 1:
for i in range(0, int(S)):
y = lambda x: trunc_logser_cdf(x, p, N) - (rank[i]-0.5) / S
if y(1) > 0:
abundance[i] = 1
else:
abundance[i] = int(round(bisect(y,1,N)))
else:
for i in range(0, int(S)):
y = lambda x: logser.cdf(x,p) / logser.cdf(N,p) - (rank[i]-0.5) / S
abundance[i] = int(round(bisect(y, 0, N)))
return (abundance, p)
def compute_M(x, A):
'''Computes Mach number in the nozzle.
Parameters:
----------
x : 1D array of float
x grid
A : 1D array of float
nozzle area at each grid point
Returns:
-------
M : 1D array of float
Mach number at each grid point
'''
M = numpy.zeros_like(x)
for i, x_val in enumerate(x):
if x_val < 1.5:
M[i] = optimize.bisect(f, 1e-6, 1, args=(A[i],))
elif x_val == 1.5:
M[i] = 1
else:
M[i] = optimize.bisect(f, 1, 50, args=(A[i],))
return M
def choque(vn,yn,tn):
A=1
g=-1
n=1
t_0=0
n=0.15
w=1.66
e=0.0001
ys= lambda x: A*np.sin(w*(x))
yp= lambda x: (((1/2.)*g*(x-tn)**2)+vn*(x-tn)+yn)
y= lambda x: yp(x)-ys(x)
vs= lambda x: A*w*np.cos(w*(x))
vp= lambda x: (g*(x-tn)+vn)
das= lambda x: -a*w**3*np.cos(w*(x))
a=-vn/g+tn
b=(-vn-(vn**2-2*g*(yn+A))**(0.5))/g+tn
if vs(tn)<=vn:
s=y(tn)*y(a)
if s<0:
t=opt.bisect(y,yn+e,a)
else:
t=opt.bisect(y,a,b)
v=(1+n)*vs(t)-n*vp(t)
else:
c=tn
d=tn+np.pi/w
t=opt.bisect(das,c,d)
v=vs(t)
p=ys(t)
return v,p,t
def get_sunriseset(t,lat,lon,height,refalt):
loc=EarthLocation(lat=lat*u.deg,lon=lon*u.deg,height=height*u.m)
t0=Time(t,format='unix')
print(t0.isot)
# Compute solar position every 10 minutes
t=Time(t0.mjd+np.arange(160)/160.0,format='mjd',scale='utc')
psun=get_sun(t)
hor=psun.transform_to(AltAz(obstime=t,location=loc))
# Interpolating function
alt_diff=hor.alt.deg-refalt
f=interpolate.interp1d(t.mjd,alt_diff)
# Find sunrise/sunset
sign=alt_diff[1:]*alt_diff[:-1]
idx=np.where(sign<0.0)[0]
print(idx)
for i in idx:
# Set
if alt_diff[i]>alt_diff[i+1]:
tset=Time(optimize.bisect(f,t[i].mjd,t[i+1].mjd),format='mjd',scale='utc')
# Rise
else:
trise=Time(optimize.bisect(f,t[i].mjd,t[i+1].mjd),format='mjd',scale='utc')
return trise.unix,tset.unix
def hpd_beta(y, n, h=.1, a=1, b=1, plot=False, **plot_kwds):
apost = y + a
bpost = n - y + b
if apost > 1 and bpost > 1:
mode = (apost - 1)/(apost + bpost - 2)
else:
raise Exception("mode at 0 or 1: HPD not implemented yet")
post = stats.beta(apost, bpost)
dmode = post.pdf(mode)
lt = opt.bisect(lambda x: post.pdf(x) / dmode - h, 0, mode)
ut = opt.bisect(lambda x: post.pdf(x) / dmode - h, mode, 1)
coverage = post.cdf(ut) - post.cdf(lt)
if plot:
plt.figure()
plotf(post.pdf)
plt.axhline(h*dmode)
plt.plot([ut, ut], [0, post.pdf(ut)])
plt.plot([lt, lt], [0, post.pdf(lt)])
plt.title(r'$p(%s < \theta < %s | y)$' %
tuple(np.around([lt, ut], 2)))
return lt, ut, coverage, h
def calculate_all_tcrits(theta):
tau, area = theta_unsqueeze(theta)
tcrits = np.empty((3, len(tau)-1))
misclassified = np.empty((3, len(tau)-1, 4))
for i in range(len(tau)-1):
try:
tcrit = bisect(expPDF_tcrit_DC, tau[i], tau[i+1], args=(tau, area, i+1))
enf, ens, pf, ps = expPDF_misclassified(tcrit, tau, area, i+1)
except:
print('Bisection with DC criterion failed.\n')
tcrit = None
enf, ens, pf, ps = None, None, None, None
tcrits[0, i] = tcrit
misclassified[0, i] = np.array([enf, ens, pf, ps])
try:
tcrit = bisect(expPDF_tcrit_CN, tau[i], tau[i+1], args=(tau, area, i+1))
enf, ens, pf, ps = expPDF_misclassified(tcrit, tau, area, i+1)
except:
print('Bisection with Clapham & Neher criterion failed.\n')
tcrit = None
enf, ens, pf, ps = None, None, None, None
tcrits[1, i] = tcrit
misclassified[1, i] = np.array([enf, ens, pf, ps])
try:
tcrit = bisect(expPDF_tcrit_Jackson, tau[i], tau[i+1], args=(tau, area, i+1))
enf, ens, pf, ps = expPDF_misclassified(tcrit, tau, area, i+1)
except:
print('Bisection with Jackson criterion failed.\n')
tcrit = None
enf, ens, pf, ps = None, None, None, None
tcrits[2, i] = tcrit
misclassified[2, i] = np.array([enf, ens, pf, ps])
return tcrits, misclassified
def getSigmaFromChiSquared(chi_sq, dof):
"""
Compute the significance (in sigma) of a chi squared value and degrees
of freedom (from Numerical Recipes, section 6.2)
Parameters
----------
chi_sq : float
the chi squared value
dof : int
the degrees of freedom
Returns
-------
sigma : float
the significance in sigma
p-value : float
p_value is the probability of obtaining a test statistic at least as
extreme as the one that was actually observed
"""
def func1(x):
return x - 1.0 + gammaincc(dof/2.0, chi_sq/2.0)
# first calculate p, the confidence limit
p = sopt.bisect(func1, 0, 1.0)
def func2(x):
val = sint.quad(lambda y: 1.0/np.sqrt(2.0*np.pi)*np.exp(-y**2/2.0), -x, x)
return val[0] - p
# now calculate sigma
sigma = sopt.bisect(func2, 0, 100)
# p is probability that random variates could have this chi-squared value
return sigma, 1-p
def fwhm(rmesh, intensity, f_int):
"""Compute fwhm precisely from starting grid.
For extreme values of B0/eta2 (extremely smeared, no peak exists),
code prints error messages and returns FWHM = 1 (max possible value)"""
# Tolerance for finding intensity max position
eps2 = np.finfo(float).eps * 2 # Tolerance, 1 ~= 1+eps2/2.
# Compute half max from grid initial guess by bracketing intensity max
idxmax = np.argmax(intensity)
if idxmax == 0:
#print 'Warning: intensity max not found (rminarc too small)'
# No need to warn, nowadays
return 1 # Can't search r < rmin, no disttab/emisttab values computed
elif idxmax == len(intensity)-1:
#print 'Warning: intensity max not resolved in grid on right; searching'
rpk_a = rmesh[-2] # CANNOT be rmesh[-1], to find max
rpk_b = search_crossing(rmesh[-1], 1., lambda r:f_int(r)-intensity[-1],
eps2) # Find r s.t. f_int(r) < intensity[-1]
if rpk_b is None: # This should never happen, honestly
print 'ERROR: intensity max not found (stalled on right) (?!)'
return 1
else:
rpk_a = rmesh[idxmax - 1]
rpk_b = rmesh[idxmax + 1]
# option 'xatol' requires SciPy 0.14.0
# 'xatol' -- absolute error in res.x acceptable for convergence
# (as res.x is order 1, eps2 should be appropriate)
res = spopt.minimize_scalar(lambda x: -1*f_int(x), method='bounded',
bounds=(rpk_a, rpk_b), options={'xatol':eps2})
rpk = res.x
pk = f_int(rpk)
halfpk = 0.5 * pk
def f_thrsh(r): # For rootfinding
return f_int(r) - halfpk
# Right (upstream) FWHM crossing -- do not search on grid
# (grid is too coarse upstream of rim, will not find FWHM;
# spurious crossings occur if profile not monotonic, e.g. w/ damping)
rmax = spopt.bisect(f_thrsh, rpk, 1.)
# Left (downstream) FWHM crossing -- find bracketing r (position) values
# requires that rminarc is large enough so that grid contains crossing
cross = np.diff(np.sign(intensity - halfpk))
inds_rmin = np.where(cross > 0)[0] # Left (neg to pos)
if inds_rmin.size == 0: # No left crossing found
#print ('Warning: FWHM edge (rmin) not found '
# '(rminarc too small or peak FWHM cannot be resolved in profile)')
return 1. # Can't search r < rmin, no disttab/emisttab values computed
else:
rmin_a = rmesh[inds_rmin[-1]] # Crossing closest to peak (largest r)
rmin_b = rmesh[inds_rmin[-1] + 1]
rmin = spopt.bisect(f_thrsh, rmin_a, rmin_b)
return rmax - rmin
def stability_limits(S2, S1=S1, W=W):
func = lambda x: psi(x, S2, S1, W) + 1./x
rhocrit = critical_density(S2, S1, W)
if (psi(rhocrit, S2, S1, W) < -1./rhocrit):
rhomin = optimize.bisect(func, R2, rhocrit)
rhomax = optimize.bisect(func, rhocrit, 10.)
else:
rhomin = rhocrit
rhomax = rhocrit
return rhomin, rhomax
def hpd_unimodal(dist, mode_guess=0., lo=0., hi=1e6, alpha=0.10):
# TODO: fix this to work with unimodal but not symmetric dist'n
mode = opt.fmin(lambda x: -dist.pdf(x), 0)[0]
lt = opt.bisect(lambda x: dist.pdf(x) / mode - alpha, lo, mode)
ut = opt.bisect(lambda x: dist.pdf(x) / mode - alpha, mode, hi)
coverage = dist.cdf(ut) - dist.cdf(lt)
return lt, ut, coverage
def bisection(self):
def f(sigma_hat):
gamma = 1/(2*self.sigma**2) - 1 /(2*sigma_hat**2)
v = self.cumulants(gamma)[1] - (self.Y**2).sum()
print 'here', v, sigma_hat
print v
factor = 3
try:
return bisect(f, self.sigma/factor, factor*self.sigma, maxiter=20)
except ValueError:
factor *= 2
return bisect(f, self.sigma/factor, factor*self.sigma, maxiter=20)
def marginalRayAngle(self, fieldPoint, wavelength=None, eps=1e-5):
"""Compute the angle to graze the edge of the pupil."""
apInd = self.surfaces.index(self.apertureStop)
frontSys = System(self.surfaces[:apInd+1])
stopy = self.apertureStop.semidiam
def residual(theta, stopy):
"""Compute the residual of hitting the top edge of the aperture."""
if not np.isscalar(theta): theta = theta.flatten()[0] # Make theta be a scalar.
direction = np.zeros((self.ndim, 1));
direction[-2:,0] = [np.tan(theta), 1.0]
ray = Rays(np.reshape(fieldPoint, (-1,1)), direction, wavelength=wavelength)
pts, rayOut = frontSys.cast(ray, settings=RaytraceSettings(False))
#import pdb;pdb.set_trace()
#print '[{},{}],'.format(theta, rayOut.origins[-2,0] - stopy),
result = rayOut.origins[-2,0] - stopy
if np.isnan(result): result = np.inf
return result
def residFudged(theta, stopy):
return residual(theta, stopy) + eps * stopy
from scipy.optimize import fmin, bisect
theta0 = self.marginalRayAngleRTM(fieldPoint[-2:][::-1], wavelength)
results = [None, None]
for i, pm1 in enumerate((-1, 1)):
if True:
try:
results[i] = bisect(residFudged, pm1 * 0.5 * theta0, pm1 * 1.5 * theta0, args=(pm1 * stopy,))
except ValueError:
pass
if results[i] is None:
thetas = np.pi * 0.1 * np.linspace(-1, 1, 21)
resids = np.array([residFudged(theta, pm1*stopy) for theta in thetas])
th0 = thetas[np.argmin(resids**2)]
mask = np.isfinite(resids)
#print 'thetas', thetas
#print 'resids', resids
resids = resids[mask]
thetas = thetas[mask]
try:
pos = thetas[resids > 0][np.argmin(resids[resids > 0])]
neg = thetas[resids < 0][np.argmax(resids[resids < 0])]
except ValueError:
# We can get here if the lens focuses all points at this field point to the same point at the aperture plane.
raise
#print '------'
#print pos, residFudged(pos, pm1*stopy), resids[resids > 0][np.argmin(resids[resids > 0])]
#print neg, residFudged(neg, pm1*stopy), resids[resids < 0][np.argmax(resids[resids < 0])]
try:
results[i] = bisect(residFudged, pos, neg, args=(pm1 * stopy,))
except ValueError:
results[i] = fmin(lambda *a: residFudged(*a)**2, x0=np.mean(thetas), args=(pm1 * stopy,))[0]
print 'resid:',results[i], residFudged(results[i], pm1*stopy)
return results
def sunrise_sunset(date, longitude, latitude, solar_angular_radius=0.0):
"""Interface to the scipy.optimize.
Using the date (j2000 offset) and location, start by finding the local
midnights. the local mid-day is then (roughly) at the center of the two
midnights. Sunrise must occur between midnight and midday, sunset between
midday and midnight (except for polar night).
This method uses Ian's method, which is less annoying than my method that required
a conditional depending on whether 'date' was in the daytime or nighttime."""
mid1,mid2=midnight(date, longitude, latitude)
noon = 0.5*(mid1+mid2)
sunrise = so.bisect(solelev, mid1, noon, args=(longitude, latitude, solar_angular_radius))
sunset = so.bisect(solelev, noon, mid2, args=(longitude, latitude, solar_angular_radius))
return sunrise, sunset
def helper(key, value):
def f(v):
x = parameters.copy()
x[key] = v
return cost(hypothesis, x, data) - (min_S+1.)
# Find bracketing intervals for the roots of the S+1 objective function.
left, right = getBrackets(f, value)
# Use a bisection search to find the left and right roots within the
# intervals.
left = opt.bisect(f, *left, xtol=1.0e-5)
right = opt.bisect(f, *right, xtol=1.0e-5)
return (key, (abs(left - value), abs(right - value)))
def generate():
to_begin = Line(nearest, line.begin)
if to_begin.length() > 0:
def func(parameter):
return (self.position.distance(to_begin(parameter)) -
self.radius)
if sign(func(0)) != sign(func(1)):
yield to_begin(bisect(func, 0, 1))
to_end = Line(nearest, line.end)
if to_end.length() > 0:
def func(parameter):
return (self.position.distance(to_end(parameter)) -
self.radius)
if sign(func(0)) != sign(func(1)):
yield to_end(bisect(func, 0, 1))
def P(phi, phib, df): """ Numerically solve for partition coefficient as a function of \phi_s """ if f(0,phi,phib,df)*f(1,phi,phib,df) < 0: return opt.bisect(f, 0, 1, args=(phi,phib,df), maxiter=500) # Bisection method else: return opt.newton(f, 1.0, args=(phi,phib,df), maxiter=5000) # Newton-Raphson
def get_next_time(self, t_0, pixel_size, xtol=1e-12):
"""
Get next time at which the body will have traveled *pixel_size*, the starting time is *t_0*.
*xtol* is the absolute tolerance for bisection passed to :py:func:`scipy.optimize.bisect`.
"""
def func(t):
"""Objective function for time bisection. The
maximum dt obtained from individual trajectories might not be precise enough, but is
precise enough to deterimne when the *pixel_size* is overstepped. Thus, we compute the
time when the overall trajectory doesn't move more than *pixel_size*, then the time it
does move more than the *pixel_size* and bisect to obtain the movement by exactly
*pixel_size*.
"""
t = t * q.s
# scipy's bisection gets the root at 0, thus we need to shift by *pixel_size*
return (self.get_distance(t_0, t) - pixel_size).simplified.magnitude
self.bind_trajectory(pixel_size)
if self._dt is None:
self.get_maximum_dt(pixel_size)
if self._dt is None:
# All bodies are stationary
return np.inf * q.s
for current_time in np.arange(t_0.simplified.magnitude, self.time.simplified.magnitude,
self._dt.simplified.magnitude) * q.s:
if self.moved(t_0, current_time, pixel_size):
return bisect(func, t_0, current_time, xtol=xtol) * q.s
return np.inf * q.s
def binomial_ci(mle, N, alpha=0.05):
""" One sided confidence interval for a binomial test.
To find the two sided interval, call with (1-alpha/2) and alpha/2 as arguments
Parameters
----------
mle : float
Fraction of successes
N : int
Number of trials
If after N trials we obtain mle as the proportion of those
trials that resulted in success, find c such that
P(k/N < mle; theta = c) = alpha
where k/N is the proportion of successes in the set of trials,
and theta is the success probability for each trial.
"""
from scipy.stats import binom
from scipy.optimize import bisect
to_minimise = lambda c: binom.cdf(mle*N,N,c)-alpha
return bisect(to_minimise,0,1)
def solve_for_fermi_energy(self, temperature, chemical_potentials, bulk_dos):
"""
Solve for the Fermi energy self-consistently as a function of T
and p_O2
Observations are Defect concentrations, electron and hole conc
Args:
bulk_dos: bulk system dos (pymatgen Dos object)
gap: Can be used to specify experimental gap.
Will be useful if the self consistent Fermi level
is > DFT gap
Returns:
Fermi energy
"""
fdos = FermiDos(bulk_dos, bandgap=self.band_gap)
def _get_total_q(ef):
qd_tot = sum([
d['charge'] * d['conc']
for d in self.list_defect_concentrations(
chemical_potentials=chemical_potentials, temperature=temperature, fermi_level=ef)
])
qd_tot += fdos.get_doping(fermi=ef, T=temperature)
return qd_tot
return bisect(_get_total_q, -1., self.band_gap + 1.)
def velocity_law_U_of_R(R):
"Explicit velocity law U(R) requires numerically solving implicit equation"
def f(U):
"Function to be zeroed: f(U) = 0"
return R - velocity_law_R_of_U(U)
U1, U2 = 1.0, 10.0
return bisect(f, U1, U2)
def calc_dh(area, h0, r):
if _debug: print(area, h0)
def func(h):
return area_above_segment(r, h) - area_above_segment(r, h0) - area
return h0 - bisect(func, 0, 2*r)
def setup_smearing( eigs, n_electron, width = 0.1, exponent = 2.0 ):
def objective( e_f ):
r = nm.sum( smear( eigs, e_f, width, exponent ) ) - n_electron
# print e_f, r
return r
## import pylab
## x = nm.linspace(eigs[0], eigs[-1], 1000)
## pylab.plot( x, smear( x, -3, width, exponent ) )
## pylab.show()
## import pylab
## x = nm.linspace(eigs[0], eigs[-1], 1000)
## pylab.plot( x, [objective(y) for y in x] )
## pylab.show()
try:
e_f = bisect(objective, eigs[0], eigs[-1], xtol=1e-12)
except AssertionError:
e_f = None
## print eigs
## print e_f, e_f - width, e_f + width
## print objective(e_f)
## debug()
def smear_tuned( energies ):
return smear( energies, e_f, width, exponent )
## import pylab
## x = nm.linspace(eigs[0], eigs[-1], 1000)
## pylab.plot( x, smear_tuned( x ) )
## pylab.show()
return e_f, smear_tuned
def CalculatePTime(vels = [ 3500, 3500, 3500, 3500, 3500, 3500],
depths = [ 2000, 3000, 4000, 5000, 6000],
rhos = [2.32,2.55,2.75,2.32,2.55,2.75],
source_depth = 5000,
source_offset = 0,costFunc=costFunc) :
#vels =np.array([1550,3100, 6200])
#depths = np.array([2000, 4000])
#rhos = np.array([2.3,2.3, 2.7])
# Velocities for the segments v_j
# Thicknesses Hj
R=source_offset
Hi,Vi,Rhoi = GetHjVjRhoj(vels,rhos,depths,source_depth)
res,r = opt.bisect(f=costFunc,a=0,b=1E-3,args=(Hi,Vi,R),full_output=True,disp=True)
p=res
#import pdb; pdb.set_trace()
# create an array of cosines:
cosV = np.sqrt(1-(p**2)*Vi**2);
#print Hi,Vi,cosV
# create an array of times per segment:
t_int = Hi/(Vi*cosV);
t_total = np.sum(t_int);
return t_total,r
def roundCorner(curveXy, curveXyt, rReq):
def getOffsetPt(ptX,curveXy,rad):
tanDir1 = normalToXyCurve(curveXy,ptX)
curve = interp.interp1d(curveXy[:,0],curveXy[:,1],'cubic')
x = ptX + tanDir1[0]*rad
y = curve(ptX) + tanDir1[1]*rad
return [x,y]
class objFcn:
def __init__(self,curveXy,curveXyt,rad):
self.curveXy = curveXy
self.curveXyt = curveXyt
self.rad = rad
def __call__(self,ptX):
pt1 = getOffsetPt(ptX,self.curveXy,rReq)
return getDist(self.curveXyt, pt1)[1]-self.rad
def getPt(self,ptX):
pt1 = getOffsetPt(ptX,self.curveXy,rReq)
return getDist(self.curveXyt, pt1)[0]
objFcn1 = objFcn(curveXy, curveXyt, rReq)
xReq = bisect(objFcn1,curveXy[0,0]+.001,curveXy[-1,0]-.001)
center = getOffsetPt(xReq,curveXy,rReq)
curve1 = interp.interp1d(curveXy[:,0],curveXy[:,1])
yReq = curve1(xReq)
pt2 = objFcn1.getPt(xReq)
curve3 = xytCurveInterp(curveXyt)
pt3 = curve3(pt2)[0:2]
theta1 = getAngle(center,[xReq,yReq])
theta2 = getAngle(center,pt3)
circle1 = getCircle(center, rReq,theta1,theta2,3)
curveRes1 = xyCurveSplit(curveXy,[],xReq)
curveRes2 = xytCurveSplit(curveXyt,[],pt2)
return curveRes1, circle1, curveRes2
def meco_velocity(m1, m2, chi1, chi2):
"""
Returns the velocity of the minimum energy cutoff for 3.5pN (2.5pN spin)
Parameters
----------
m1 : float
First component mass in solar masses
m2 : float
Second component mass in solar masses
chi1 : float
First component dimensionless spin S_1/m_1^2 projected onto L
chi2 : float
Second component dimensionless spin S_2/m_2^2 projected onto L
Returns
-------
v : float
Velocity (dimensionless)
"""
energy0, energy2, energy3, energy4, energy5, energy6 = \
_energy_coeffs(m1, m2, chi1, chi2)
def eprime(v):
return 2. + v * v * (4.*energy2 + v * (5.*energy3 \
+ v * (6.*energy4
+ v * (7.*energy5 + 8.*energy6 * v))))
return bisect(eprime, 0.05, 1.0)
def gentab(N,amin):
" Generate the arrays x,y,d,delta, assuming a partition in n boxes for [0,+infty) "
area_low = 0.5
area_up = 0.7
area_zig = optimize.bisect(f,area_low,area_up,args=(N))
xpos = f(area_zig,N,stats.norm,return_x=True)
# don't store x_i such that x_i<a_min
j = 0
while(j<N) and (xpos[j]<-amin):
j += 1
x = concatenate((-xpos[j:0:-1],xpos))
nd = x.size-1 # number of intervals [xi,x(i+1)]
d = empty(nd,dtype="double")
delta = empty(nd,dtype="double")
yl = empty(nd,dtype="double")
yu = empty(nd,dtype="double")
for i in range(nd):
if x[i]<0.:
yl[i] = stats.norm.pdf(x[i])
yu[i] = stats.norm.pdf(x[i+1])
else:
yl[i] = stats.norm.pdf(x[i+1])
yu[i] = stats.norm.pdf(x[i])
d[i] = x[i+1] - x[i]
delta[i] = d[i]*yu[i]/yl[i]
return (x,yl,yu,d,delta)
def inverse(self, data, max_iter=100): # make sure data has right shape data = asarray(data).reshape(1, -1) # apply Gaussian CDF data = norm.cdf(data) # apply inverse model CDF val_max = mean(self.mog.means) + 1. val_min = mean(self.mog.means) - 1. for t in range(data.shape[1]): # make sure root lies between val_min and val_max while float(self.mog.cdf(val_min)) > data[0, t]: val_min -= 1. while float(self.mog.cdf(val_max)) < data[0, t]: val_max += 1. # find root numerically data[0, t] = bisect( f=lambda x: float(self.mog.cdf(x)) - data[0, t], a=val_min, b=val_max, maxiter=max_iter, disp=False) return data
def VariableHeight(Moment): #Returns the ideal height of the beam based on the input moment profile. Mass is not considered unless mass is included in the moment profile. h = np.zeros((50)) #Creating container for optimized beam heights #The optimize function loses its mind when the moment drops below 2980: for i in range(0,50): if abs(Moment[i]) < 2980: Moment[i] = -2980 Yield_steel = 415e6 #MPa Yield_iron = 98e6 #MPa Safety_factor = 5 #Adjust this value as needed Max_stress_steel = Yield_steel / Safety_factor Max_stress_iron = Yield_iron / Safety_factor def BeamHeight(h): x = 6*abs(Moment[i])*h/(fw*h**3-(fw-wt)*(h-2*ft)**3) - Max_stress_steel return x for i in range(0,50): h[i] = sp.bisect(BeamHeight, 0.02, 10,xtol = 0.001) if h[i] < 3*ft: h[i] = 3*ft #Preventing the I-beam from being unrealistically small. Minumum height is 3 times the flange thickness h = 1.01*h #The bisect function is not perfect. Small errors allow for under-designing for stress. Here the height is inflated by 2% to ensure the beam is sufficiently designed. return h
def lambdavalue(y,p,alpha,gamma,phi,NegativeDemands=True,ub=10,method='bisect'):
"""
Given income y, prices p and preference parameters
(alpha,gamma,phi), find the marginal utility of income lbda.
Elliott: Changed default method from 'root_with_precision' to 'bisect'.
I think Rw/P was hanging up somehow because the fake_data simulation freezes when direct==False
"""
n,alpha,gamma,phi = check_args(p,alpha,gamma,phi)
if NegativeDemands:
subsistence=sum([p[i]*phi[i] for i in range(n)])
else:
subsistence=sum([p[i]*phi[i] for i in range(n) if phi[i]<0])
if y+subsistence<0: # Income too low to satisfy subsistence demands
warnings.warn('Income too small to cover subsistence phis (%f < %f)' % (y,subsistence))
return nan
f = excess_expenditures(y,p,alpha,gamma,phi,NegativeDemands=NegativeDemands)
if method=='bisect':
try:
return optimize.bisect(f,1e-20,ub)
except ValueError:
return lambdavalue(y,p,alpha,gamma,phi,NegativeDemands=NegativeDemands,ub=ub*2.0)
elif method=='newton':
df = excess_expenditures_derivative(p,alpha,gamma,phi)
return optimize.newton(f,ub/2.,fprime=df)
elif method=='root_with_precision':
return root_with_precision(f,[0,ub,Inf],1e-12,open_interval=True)
else:
raise ValueError, "Method not defined."
def get_lambda_spatialdistrib(A, A0, n0):
"""Solve for lambda_Pi from Harte 2011 equ. 7.50 and 7.51
Arguments:
A = the spatial scale of interest
A0 = the maximum spatial scale under consideration
n0 = the number of individuals of the focal species at scale A0
"""
assert type(n0) is int, "n must be an integer"
assert A > 0 and A0 > 0, "A and A0 must be greater than 0"
assert A <= A0, "A must be less than or equal to A0"
y = lambda x: x / (1 - x) - (n0 + 1) * x ** (n0 + 1) / (1 - x ** (n0 + 1)) - n0 * A / A0
if A < A0 / 2:
# Set the distance from the undefined boundaries of the Lagrangian multipliers
# to set the upper and lower boundaries for the numerical root finders
BOUNDS = [0, 1]
DIST_FROM_BOUND = 10 ** -15
exp_neg_lambda = bisect(y, BOUNDS[0] + DIST_FROM_BOUND,
BOUNDS[1] - DIST_FROM_BOUND)
elif A == A0 / 2:
#Special case from Harte (2011). See text between Eq. 7.50 and 7.51
exp_neg_lambda = 1
else:
# x can potentially go up to infinity
# thus use solution of a logistic equation as the starting point
exp_neg_lambda = (fsolve(y, - log(A0 / A - 1)))[0]
lambda_spatialdistrib = -1 * log(exp_neg_lambda)
return lambda_spatialdistrib
def radiation(a):
global E
global L
print('PROCEED WITH CAUTION!')
print('Radiation in Progress')
Ra_array = np.zeros(1)
Rp_array = np.zeros(1)
t_gw_array = np.zeros(1)
t_array = np.zeros(1)
E_array = np.zeros(1)
E_array[0] = E
L_array = np.zeros(1)
L_array[0] = L
e_array = np.zeros(1)
a_array = np.zeros(1)
Q_array = np.zeros(1)
Sol_Array = np.zeros(shape=(3, 1))
XY = np.zeros(shape=(2, 1))
I = np.zeros(shape=(2, 1))
count = 0
# Fitting Numbers
A_E = -0.141421
B_E = 0.752091
C_E = -4.634643
A_L = -1.13137
B_L = 1.31899
C_L = -4.149103
# temp r for switching between orbits
temp_r = r_i
# Integration sequence to reach first apoapsis given that initial conditions start at periapsis
start_y0 = [temp_r, rdot_i, phi_i]
start_sol = solve_ivp(deriv,
y0=start_y0,
t_span=[t_array[0], 1000000000],
rtol=1e-8,
atol=1e-8,
events=apoapsis)
a_array[0] = (start_sol.y_events[0][0][0] - temp_r) / 2
e_array[0] = get_e(temp_r, start_sol.y_events[0][0][0])
start_y0 = [
start_sol.y_events[0][0][0], start_sol.y_events[0][0][1],
start_sol.y_events[0][0][2]
]
t_gw_array[0] = start_sol.t_events[0][0]
XY[0] = start_y0[0] * np.cos(start_y0[2])
XY[1] = start_y0[0] * np.sin(start_y0[2])
I[0] = m * (XY[0][-1]**2 - (1 / 3) * (start_y0[0]**2))
I[1] = m * (XY[0][-1]**2 + XY[1][-1]**2)
t_array[0] = start_sol.t_events[0][0]
Ra_array[0] = start_sol.y_events[0][0][0]
Rp_array[0] = temp_r
# Radiation integration sequence
try:
for i in range(a):
temp_sol_1 = solve_ivp(deriv,
y0=start_y0,
t_span=[t_array[-1], t_array[-1] + .001],
rtol=1e-8,
atol=1e-8)
temp_y0_2 = [
temp_sol_1.y[0][1], temp_sol_1.y[1][1], temp_sol_1.y[2][1]
]
temp_sol_2 = solve_ivp(deriv,
y0=temp_y0_2,
t_span=[temp_sol_1.t[-1], 1000000000],
rtol=1e-8,
atol=1e-8,
events=apoapsis)
s = temp_sol_2.t.size
# Plot of apoapsis to apoapsis
Sol_Array = np.append(
Sol_Array, [temp_sol_2.y[0], temp_sol_2.y[1], temp_sol_2.y[2]],
axis=1)
# Levin's Q
dPhi = np.abs((temp_sol_2.y_events[0][0][2] % (2 * np.pi)) -
(start_y0[2] % (2 * np.pi)))
w = int(
np.abs((temp_sol_2.y_events[0][0][2] - start_y0[2])) /
(2 * np.pi))
Q_array = np.append(Q_array, w + (dPhi / (2 * np.pi)))
# Gravitational Wave calculation
for j in range(s):
XY = np.append(
XY, [[temp_sol_2.y[0][j] * np.cos(temp_sol_2.y[2][j])],
[temp_sol_2.y[0][j] * np.sin(temp_sol_2.y[2][j])]],
axis=1)
I = np.append(
I,
[[m * (XY[0][-1]**2 - (1 / 3) * (temp_sol_2.y[0][j])**2)],
[m * (XY[0][-1]**2 + XY[1][-1]**2)]],
axis=1)
t_gw_array = np.append(t_gw_array, temp_sol_2.t[j])
t_array = np.append(t_array, temp_sol_2.t_events[0][0])
# Radiation calculations
E_Rad = \
(m / M) * (A_E * np.arccosh(1 + B_E * (4 * M / temp_r)**6 * (M / (temp_r - 4 * M))) + C_E * (temp_r / M - 4) * (M / temp_r)**(9/2))
L_Rad = m * (A_L * np.arccosh(1 + B_L * (4 * M / temp_r)**3 *
(M / (temp_r - 4 * M))) + C_L *
(temp_r / M - 4) * (M / temp_r)**3)
# Updates to global variables affected by radiation
E = E + E_Rad
E_array = np.append(E_array, E)
L = L + L_Rad
L_array = np.append(L_array, L)
global root1
root1 = sp.lambdify(
r, E + G * M / r - L**2 / (2 * r**2) + G * M * (L**2) / r**3 -
(1 / 2) * rdot_i**2)
global U_Eff
U_Eff = (-G * M / r + L**2 / (2 * r**2) - G * M * (L**2) / r**3)
global U_Eff_Func
U_Eff_Func = sp.lambdify(r, U_Eff)
global Eff_Force
Eff_Force = -sp.diff(U_Eff, r)
global Eff_Force_Func
Eff_Force_Func = sp.lambdify(r, Eff_Force)
global Phi_dot
Phi_dot = L / r**2
global Phi_dot_Func
Phi_dot_Func = sp.lambdify(r, Phi_dot)
global ISCO
ISCO = (6 * G * M) / (1 + np.sqrt(1 - 12 * (G * M / L)**2))
global IUCO
IUCO = (6 * G * M) / (1 - np.sqrt(1 - 12 * (G * M / L)**2))
global r_p
r_p = bisect(root1, a=IUCO, b=ISCO, disp=True)
# Update to initial conditions
start_y0 = [r_p, rdot_i, phi_i]
start_sol = solve_ivp(deriv,
y0=start_y0,
t_span=[0, 1000000000],
rtol=1e-8,
atol=1e-8,
events=apoapsis)
a_array = np.append(a_array,
(start_sol.y_events[0][0][0] - r_p) / 2)
e_array = np.append(e_array, get_e(r_p,
start_sol.y_events[0][0][0]))
start_y0 = [
start_sol.y_events[0][0][0], start_sol.y_events[0][0][1],
temp_sol_2.y_events[0][0][2]
]
Ra_array = np.append(Ra_array, start_sol.y_events[0][0][0])
Rp_array = np.append(Rp_array, r_p)
count = count + 1
print(str(count) + '/' + str(a))
I_dot = np.zeros(shape=(2, I[0].size - 2))
I_ddot = np.zeros(shape=(2, I_dot[0].size - 2))
for i in range(I_dot[0].size):
I_dot[0][i] = (I[0][i + 2] - I[0][i]) / (t_gw_array[i + 2] -
t_gw_array[i])
I_dot[1][i] = (I[1][i + 2] - I[1][i]) / (t_gw_array[i + 2] -
t_gw_array[i])
t_gw_array = np.delete(t_gw_array, 0)
t_gw_array = np.delete(t_gw_array, -1)
for i in range(I_ddot[0].size - 2):
I_ddot[0][i] = (I_dot[0][i + 2] -
I_dot[0][i]) / (t_gw_array[i + 2] - t_gw_array[i])
I_ddot[1][i] = (I_dot[1][i + 2] -
I_dot[1][i]) / (t_gw_array[i + 2] - t_gw_array[i])
t_gw_array = np.delete(t_gw_array, 0)
t_gw_array = np.delete(t_gw_array, -1)
dEdt_array = np.zeros(E_array.size - 2)
dLdt_array = np.zeros(L_array.size - 2)
for n in range(t_array.size - 2):
dEdt_array[n] = (E_array[n + 2] - E_array[n]) / (t_array[n + 2] -
t_array[n])
dLdt_array[n] = (L_array[n + 2] - L_array[n]) / (t_array[n + 2] -
t_array[n])
fig1 = plt.figure(figsize=(16, 9))
plt.subplots_adjust(bottom=.25)
spec = gridspec.GridSpec(nrows=2, ncols=2, figure=fig1)
ax1 = fig1.add_subplot(spec[0, 0])
# plt.scatter(abs(e_array[:]), a_array[:])
plt.scatter(Rp_array[1:a], np.abs(dEdt_array[:]), s=4)
plt.xlabel('Radius of Periapsis')
plt.ylabel('|dE/dt|')
ax2 = fig1.add_subplot(spec[1, 0])
# plt.plot(t_gw_array[:], 2 * M * I_ddot[0][:])
# plt.scatter(t_array[:], E_array[:])
# plt.scatter(Rp_array[1:a], np.abs(dEdt_array[:]))
plt.scatter(Rp_array[1:a], np.abs(dLdt_array[:]), s=4)
plt.xlabel('Radius of Periapsis')
plt.ylabel('|dL/dt|')
ax3 = fig1.add_subplot(spec[1, 1])
# plt.scatter(XY[0], XY[1])
# plt.scatter(Sol_Array[2][1:], Sol_Array[0][1:])
# plt.scatter(E_array[:-1], Q_array[1:])
plt.scatter(Rp_array[:], e_array[:], s=4)
plt.xlabel('r_p')
plt.ylabel('e')
ax4 = fig1.add_subplot(spec[0, 1])
# plt.polar(Sol_Array[2][1:], Sol_Array[0][1:])
plt.plot(E_array[:], L_array[:])
# plt.xlabel('Energy')
# plt.ylabel('Angular Momentum')
# plt.plot(t_gw_array[:], 2 * M * I_ddot[0][:], alpha=.7)
# plt.plot(sol.t[10: - 4], 2 * get_H()[0][0][10:], alpha=.7)
# plt.xlim(1000, 2000)
# plt.ylim(-.000025, .000025)
except (RuntimeError, ValueError):
I_dot = np.zeros(shape=(2, I[0].size - 2))
I_ddot = np.zeros(shape=(2, I_dot[0].size - 2))
for i in range(I_dot[0].size):
I_dot[0][i] = (I[0][i + 2] - I[0][i]) / (t_gw_array[i + 2] -
t_gw_array[i])
I_dot[1][i] = (I[1][i + 2] - I[1][i]) / (t_gw_array[i + 2] -
t_gw_array[i])
t_gw_array = np.delete(t_gw_array, 0)
t_gw_array = np.delete(t_gw_array, -1)
for i in range(I_ddot[0].size - 2):
I_ddot[0][i] = (I_dot[0][i + 2] -
I_dot[0][i]) / (t_gw_array[i + 2] - t_gw_array[i])
I_ddot[1][i] = (I_dot[1][i + 2] -
I_dot[1][i]) / (t_gw_array[i + 2] - t_gw_array[i])
t_gw_array = np.delete(t_gw_array, 0)
t_gw_array = np.delete(t_gw_array, -1)
dEdt_array = np.zeros(E_array.size - 2)
dLdt_array = np.zeros(L_array.size - 2)
for n in range(t_array.size - 2):
dEdt_array[n] = (E_array[n + 2] - E_array[n]) / (t_array[n + 2] -
t_array[n])
dLdt_array[n] = (L_array[n + 2] - L_array[n]) / (t_array[n + 2] -
t_array[n])
fig1 = plt.figure(figsize=(16, 9))
plt.subplots_adjust(bottom=.25)
spec = gridspec.GridSpec(nrows=2, ncols=2, figure=fig1)
ax1 = fig1.add_subplot(spec[0, 0])
# plt.scatter(abs(e_array[:]), a_array[:])
plt.scatter(Rp_array[1:a], np.abs(dEdt_array[:]), s=4)
plt.xlabel('Radius of Periapsis')
plt.ylabel('|dE/dt|')
ax2 = fig1.add_subplot(spec[1, 0])
# plt.plot(t_gw_array[:], 2 * M * I_ddot[0][:])
# plt.scatter(t_array[:], E_array[:])
# plt.scatter(Rp_array[1:a], np.abs(dEdt_array[:]))
plt.scatter(Rp_array[1:a], np.abs(dLdt_array[:]), s=4)
plt.xlabel('Radius of Periapsis')
plt.ylabel('|dL/dt|')
ax3 = fig1.add_subplot(spec[1, 1])
# plt.scatter(XY[0], XY[1])
# plt.scatter(Sol_Array[2][1:], Sol_Array[0][1:])
# plt.scatter(E_array[:-1], Q_array[1:])
plt.scatter(Rp_array[:], e_array[:], s=4)
plt.xlabel('r_p')
plt.ylabel('e')
ax4 = fig1.add_subplot(spec[0, 1])
# plt.polar(Sol_Array[2][1:], Sol_Array[0][1:])
plt.scatter(E_array[:], L_array[:], s=4)
plt.xlabel('Energy')
plt.ylabel('Angular Momentum')
gamma1 = entEn/elResEn
gamma = ((3/(16*np.pi))*deltV0*bendRad*beamPow/(radEl**2*massEl*(lght**2)*(lee)*betaE))**(1/3)
effCrossVol = np.pi*beamWid**3*np.cos(incRad)/(2*np.sin(incRad)) # m^3
lenInt = 2*beamWid*np.cos(incRad)/np.sin(incRad) #
kEn = entEn - elResEn*lght**2 # MeV
# solve for velocity of the particles
entEnBeam = entEn/2
veloc = np.arange(2.22e5,2.3e5,1) # range narrowed through experimentation
def func(veloc):
return np.sqrt(entEnBeam/((1/np.sqrt(1-veloc**2/lght**2))-1)/elResEn) - veloc
funcOut = np.array([func(veloc[i]) for i in range(len(veloc))])
plt.plot(veloc,funcOut)
solVeloc = bisect(func,226335,226336,maxiter = 10000)
# calculate momentum
loren = 1/np.sqrt(1-solVeloc**2/lght**2)
momPartMag = loren*elResEn*solVeloc
# the two particles have opp x momentums but similar y
momElX = momPartMag*np.cos(incRad)
momElY = momPartMag*np.sin(incRad)
momPosX = momPartMag*np.cos(-incRad)
momPosY = momPartMag*np.sin(incRad)
'''
def get_fdr_threshold_estimate(null,
nonnull,
region_list,
alpha,
maxima=False,
n_proc=None,
verbose=False):
'''
Calculate FDR-based detection threshold from given data for FDR=alpha.
Operates across regions and returns numpy array with threshold for each
region.
Uses direct estimation technique for FDR as in Storey (2002).
Estimates p-values using Gaussian KDE on null sample.
'''
# Setup pool if needed
if n_proc is not None:
pool = multiprocessing.Pool(processes=n_proc)
else:
pool = None
thresh_list = []
for region in region_list:
if verbose:
print >> sys.stderr, "Limits\t=\t%d\t%d" % (region[0], region[-1])
null_region = null[region]
if maxima:
subset = find_maxima(null_region)
if subset.size > 1:
null_region = null_region[subset]
else:
if verbose:
err_msg = "Region %d had only %d local maxima" % (
region, subset.size)
print >> sys.stderr, err_msg
null_region.sort()
nonnull_region = nonnull[region]
nonnull_max = nonnull.max()
if maxima:
nonnull_region = nonnull_region[find_maxima(nonnull_region)]
nonnull_region.sort()
# Handle case of no local maxima
if nonnull_region.size < 1:
thresh_list.append(nonnull_max * 2)
continue
# Transform data
log_null = np.log(null_region)
log_nonnull = np.log(nonnull_region)
# Calculate p-values
try:
p = estimate_pvalues(log_null, log_nonnull, pool, n_proc)
except:
# Empirical CDF fall-back for singular cases
p = np.searchsorted(log_null, log_nonnull).astype(np.float)
p /= log_null.size
# Check for pathological case
# Both cases have same sign -> give up
if estimate_fdr_direct(0.0, alpha, p) * estimate_fdr_direct(
1.0, alpha, p) > 0:
thresh_p = p.min() * (1 - np.sqrt(np.spacing(1)))
if verbose:
print >> sys.stderr, "Pathological case"
else:
# Otherwise, get threshold via bisection
thresh_p = optimize.bisect(estimate_fdr_direct,
0.0,
1.0,
args=(alpha, p))
thresh_ind = np.searchsorted(np.sort(p), thresh_p)
if thresh_ind > 0:
thresh_coef = np.sort(log_nonnull)[::-1][thresh_ind -
1:thresh_ind + 1]
thresh_coef = thresh_coef.mean()
else:
thresh_coef = log_nonnull.max() + np.log(2)
thresh = np.exp(thresh_coef)
thresh_list.append(thresh)
return np.array(thresh_list)
import numpy as np
import timeit
from scipy import optimize
start = timeit.default_timer()
# combined function
def f(x):
return (x**2 + (np.sqrt(3) * x)**2)**3 - 4 * x**2 * (np.sqrt(3) * x)**2
# roots are labeled from x_0 to x_2 from left to right
# root of x_0 using bisection method
x0_root = optimize.bisect(f, -1, -0.4)
x0_root = np.format_float_positional(x0_root,
precision=4,
unique=False,
fractional=False,
trim='k')
y = np.sqrt(3) * float(x0_root)
int0 = (x0_root, y)
print("Intersection 1 is {}".format(int0))
# root of x_1 using newton method
x1_root = optimize.newton(f, -0.1, tol=1.48e-06)
if (x1_root < 0.0001) | (x1_root > -0.0001):
x1_root = 0
y = np.sqrt(3) * float(x1_root)
int1 = (x1_root, y)
def solve_self_consistent_real_space\
(Nx, Ny, nOrb, nHole, invTemp, betaStart, betaSpeed, betaThreshold,\
anneal_or_not, t, U, itMax, dampFreq, dyn, singleExcitationFreq, osc,\
K, abs_t0, delta, nUp, nDown):
'''
Solves the self-consistent equation in real space.
'''
nUpBest, nDownBest = nUp, nDown
# Total number of sites + orbitals
N = nOrb * Nx * Ny
# Initialize deltas for the tolerance check (ensure that it does not stop
# at the first step)
deltaUp = delta + 1
deltaDown = delta + 1
# Initialize inverse temperature for the annealing
beta = betaStart
# Initialize energies
energies = np.zeros(itMax)
bestGrandpotential = 1e100
# Initialize iteration
it = 0
# Initialize iteration at which we finish annealing
itSwitch = 0
# How many iterations to wait between dynamic kicks
itWait = 3
# lbda is a parameter that reduces the weight
# on the density obtained in the previous iteration.
# the factor multiplied by itMax impedes that P ( I ) < delta
# initially, we give equal weights and then progressively more
# to the new configuration
factor = 1.2
lbda = 0.5 / (factor * itMax)
# This ensures that we do not stop at the first step
energies[-1] = 1e100
# Print frequency
printFreq = 10
if anneal_or_not == True:
print("Started annealing.\n")
while loopCondition(it, itMax, deltaUp, deltaDown,\
delta, beta, invTemp, betaThreshold):
# Annealing
if anneal_or_not == True:
beta = anneal(invTemp, betaStart, betaSpeed, beta,\
betaThreshold, it, osc)
else:
beta = noAnneal(invTemp, beta)
# Define the MF Hamiltonian for this iteration
C, Hup, Hdown = hamiltonian(nUp, nDown, K, U, N)
# Diagonalize
eUp, wUp = la.eigh(Hup)
eDown, wDown = la.eigh(Hdown)
# Save the previous fields to compare to test convergence
nUpOld = nUp.copy()
nDownOld = nDown.copy()
# Compute the chemical potential implicitly
interval_limits = 50
mu = bisect(rootToChem, -interval_limits, interval_limits, \
args = (eUp, eDown, beta, Nx, Ny, nHole) )
# Update fields
nUp, nDown = update(nUp, nDown, N, wUp, wDown, eUp, eDown, mu, beta)
# Grandpotential per site
energy = grandpotential(U, nUp, nDown, nUpOld, nDownOld, \
nOrb, Nx, Ny, mu, invTemp, eUp, eDown, abs_t0)
# Damping
nUp, nDown = damp(it, dampFreq, nUp, nDown, nUpOld, nDownOld, lbda)
# Relative difference between current and previous fields
deltaUp = np.dot(nUp - nUpOld, nUp - nUpOld) \
/ np.dot(nUp, nUp)
deltaDown = np.dot(nDown - nDownOld, nDown - nDownOld) \
/ np.dot(nDown, nDown)
if it % printFreq == 0:
print("\niteration: ", it)
print("deltaUp: ", deltaUp)
print("deltaDown: ", deltaDown, "\n")
if ( it + 1 ) % singleExcitationFreq == 0 :
if dyn == 'local' or dyn == 'wait':
for attempt in range(nOrb * N):
i = int( np.random.random() * N )
nUpTmp, nDownTmp = nUp[i], nDown[i]
nUp[i], nDown[i] = updateLocal(i, nUp, nDown, N, wUp, wDown,\
eUp, eDown, mu, beta)
# Define the MF Hamiltonian for this iteration
C, Hup, Hdown = hamiltonian(nUp, nDown, K, U, N)
# Diagonalize
eUp, wUp = la.eigh(Hup)
eDown, wDown = la.eigh(Hdown)
# Save the previous fields to compare to test convergence
nUpOld = nUp.copy()
nDownOld = nDown.copy()
# Compute the chemical potential implicitly
interval_limits = 50
mu = bisect(rootToChem, -interval_limits, interval_limits, \
args = (eUp, eDown, beta, Nx, Ny, nHole) )
# Update fields
nUp, nDown = update(nUp, nDown, N, wUp, wDown, eUp, eDown, mu, beta)
# Grandpotential per site
energyTmp = grandpotential(U, nUp, nDown, nUpOld, nDownOld, \
nOrb, Nx, Ny, mu, invTemp, eUp, eDown, abs_t0)
if energyTmp > energy:
nUp[i], nDown[i] = nUpTmp, nDownTmp
else:
energy = energyTmp
elif dyn == 'kick':
for attempt in range(nOrb * N):
i = int( np.random.random() * N )
nUpTmp, nDownTmp = nUp[i], nDown[i]
nUp[i], nDown[i] = singleExcitation(nUp, nDown, i)
# Define the MF Hamiltonian for this iteration
C, Hup, Hdown = hamiltonian(nUp, nDown, K, U, N)
# Diagonalize
eUp, wUp = la.eigh(Hup)
eDown, wDown = la.eigh(Hdown)
# Save the previous fields to compare to test convergence
nUpOld = nUp.copy()
nDownOld = nDown.copy()
# Compute the chemical potential implicitly
interval_limits = 50
mu = bisect(rootToChem, -interval_limits, interval_limits, \
args = (eUp, eDown, beta, Nx, Ny, nHole) )
# Update fields
nUp, nDown = update(nUp, nDown, N, wUp, wDown, eUp, eDown, mu, beta)
# Grandpotential per site
energyTmp = grandpotential(U, nUp, nDown, nUpOld, nDownOld, \
nOrb, Nx, Ny, mu, invTemp, eUp, eDown, abs_t0)
if energyTmp > energy:
nUp[i], nDown[i] = nUpTmp, nDownTmp
else:
energy = energyTmp
elif dyn == 'mixed':
for attempt in range(nOrb * N):
i = int( np.random.random() * N )
nUpTmp, nDownTmp = nUp[i], nDown[i]
nUp[i], nDown[i] = updateLocal(i, nUp, nDown, N, wUp, wDown,\
eUp, eDown, mu, beta)
nUp[i], nDown[i] = singleExcitation(nUp, nDown, i)
# Define the MF Hamiltonian for this iteration
C, Hup, Hdown = hamiltonian(nUp, nDown, K, U, N)
# Diagonalize
eUp, wUp = la.eigh(Hup)
eDown, wDown = la.eigh(Hdown)
# Save the previous fields to compare to test convergence
nUpOld = nUp.copy()
nDownOld = nDown.copy()
# Compute the chemical potential implicitly
interval_limits = 50
mu = bisect(rootToChem, -interval_limits, interval_limits, \
args = (eUp, eDown, beta, Nx, Ny, nHole) )
# Update fields
nUp, nDown = update(nUp, nDown, N, wUp, wDown, eUp, eDown, mu, beta)
# Grandpotential per site
energyTmp = grandpotential(U, nUp, nDown, nUpOld, nDownOld, \
nOrb, Nx, Ny, mu, invTemp, eUp, eDown, abs_t0)
if energyTmp > energy:
nUp[i], nDown[i] = nUpTmp, nDownTmp
else:
energy = energyTmp
if dyn == 'wait':
if ( it + 1 + itWait ) % singleExcitationFreq == 0 :
for attempt in range(nOrb * N):
i = int( np.random.random() * N )
nUpTmp, nDownTmp = nUp[i], nDown[i]
nUp[i], nDown[i] = singleExcitation(nUp, nDown, i)
# Define the MF Hamiltonian for this iteration
C, Hup, Hdown = hamiltonian(nUp, nDown, K, U, N)
# Diagonalize
eUp, wUp = la.eigh(Hup)
eDown, wDown = la.eigh(Hdown)
# Save the previous fields to compare to test convergence
nUpOld = nUp.copy()
nDownOld = nDown.copy()
# Compute the chemical potential implicitly
mu = -1
# interval_limits = 50
# mu = bisect(rootToChem, -interval_limits, interval_limits, \
# args = (eUp, eDown, beta, Nx, Ny, nHole) )
# Update fields
nUp, nDown = update(nUp, nDown, N, wUp, wDown, eUp, eDown, mu, beta)
# Grandpotential per site
energyTmp = grandpotential(U, nUp, nDown, nUpOld, nDownOld, \
nOrb, Nx, Ny, mu, invTemp, eUp, eDown, abs_t0)
if energyTmp > energy:
nUp[i], nDown[i] = nUpTmp, nDownTmp
else:
energy = energyTmp
energies[it] = energy
if invTemp == 'infty':
if energy < bestGrandpotential and beta >= betaThreshold:
bestGrandpotential = energy
nUpBest, nDownBest = nUp, nDown
eUpBest, eDownBest = eUp, eDown
wUpBest, wDownBest = wUp, wDown
else:
if energy < bestGrandpotential and beta == invTemp:
bestGrandpotential = energy
nUpBest, nDownBest = nUp, nDown
eUpBest, eDownBest = eUp, eDown
wUpBest, wDownBest = wUp, wDown
# Move to the next iteration
it += 1
# Save the last iteration
lastIt = it
print("\nTotal number of iterations: ", lastIt, "\n")
return nUpBest, nDownBest, energies, bestGrandpotential,\
itSwitch, lastIt, mu, eUpBest, eDownBest,\
np.absolute(wUpBest.flatten('C'))**2, np.absolute(wDownBest.flatten('C'))**2
while theta <= 2.0000001 * np.pi:
k = e0 / (h * vF)
#define the values of the limits for the root finding function.
valuea = 0.3 * k
valueb = 1.5 * k
#next use bisect on the Rotation value function to find the k values
#print(theta, Dispersion(valuea,args = ([a, t, vF, 1, theta, e0])), Dispersion(valueb,args = ([a, t, vF, -1, theta, e0])))
kx = k * np.cos(theta)
ky = k * np.sin(theta)
#We use bisect to determine the values of kplus around the fermi surface
kplus = so.bisect(Dispersion,
valuea,
valueb,
args=([a, t, vF, 1, theta, e0]))
#we then repeat the procedure for the negative spin values
kminus = so.bisect(Dispersion,
valuea,
valueb,
args=([a, t, vF, -1, theta, e0]))
###
#Next we determine what the x and y positions are.
kxp1, kyp1, kxm1, kym1 = kplus * np.cos(theta) / k, kplus * np.sin(
theta) / k, kminus * np.cos(theta) / k, kminus * np.sin(theta) / k
#then we print out the particular values of kplus and kminus to a file.
ustring = theta, kminus, kplus, k
file1.write(' '.join(map(str, ustring)))
file1.write('\n')
Eb = np.arange(0, 10, 0.1)
y = (np.sqrt(10 - Eb) * np.tan(np.sqrt(10 - Eb)) - np.sqrt(Eb))
plt.plot(Eb, y)
plt.ylim(-50, 50)
plt.title("Even function with binding potential of 10")
plt.grid()
plt.show()
#Now use the Bisection method to determine where F(Eb) = 0
def funct(Eb):
return (np.sqrt(10 - Eb) * np.tan(np.sqrt(10 - Eb)) - np.sqrt(Eb))
guess = optimize.bisect(funct, 8, 9)
print("Guess for the Bisection Method:", guess)
#Now use the Newton / Raphson Method to determine where F(Eb) = 0
x0 = 8.5
newguess = optimize.newton(funct, x0)
print("Guess for the Newton/Raphson Method:", newguess)
#When looking at the two results, there are quite similar to 10**-8 decimal place.
#Then the question would be which is more accurate, both are precise to 10**-8.
#Now we can check the values returned for each method within our function
test1 = funct(guess)
test2 = funct(newguess)
result = (np.newton(f, 3.0, tol=0.001, rtol=0.01, maxiter=100))
print(math.isclose(result, ref_result, rel_tol=1E-6, abs_tol=1E-6))
ref_result = 1.4142135623715149
result = (np.bisect(f, 1.0, 3.0))
print(math.isclose(result, ref_result, rel_tol=1E-6, abs_tol=1E-6))
ref_result = -7.105427357601002e-15
result = np.fmin(f, 3.0, fatol=1e-15)
print(math.isclose(result, ref_result, rel_tol=1E-6, abs_tol=1E-6))
ref_result = -7.105427357601002e-15
result = np.fmin(f, 3.0, xatol=1e-8, fatol=1e-15, maxiter=500)
print(math.isclose(result, ref_result, rel_tol=1E-6, abs_tol=1E-6))
else:
ref_result = 1.41421826342255
result = optimize.newton(f, 3., tol=0.001, rtol=0.01)
print(math.isclose(result, ref_result, rel_tol=1E-9, abs_tol=1E-9))
result = optimize.newton(f, 3., tol=0.001, rtol=0.01, maxiter=100)
print(math.isclose(result, ref_result, rel_tol=1E-9, abs_tol=1E-9))
ref_result = 1.4142135623715149
result = optimize.bisect(f, 1.0, 3.0)
print(math.isclose(result, ref_result, rel_tol=1E-9, abs_tol=1E-9))
ref_result = -7.105427357601002e-15
result = optimize.fmin(f, 3.0, disp=0)
print(math.isclose(result[0], ref_result, rel_tol=1E-9, abs_tol=1E-9))
result = optimize.fmin(f, 3.0, xtol=0.0001, ftol=0.0001, disp=0)
print(math.isclose(result[0], ref_result, rel_tol=1E-9, abs_tol=1E-9))
return ["Wynik: ", x, "Iteracje: ", iteracje]
x = x - f(x) / pochodna_f(x)
return ["Wynik: ", x, "Iteracje: ", iteracje]
def secant(x0, x1, n):
iteracje = 0
for i in range(n):
iteracje += 1
if f(x1) - f(x0) == 0:
return ["Wynik: ", x1, "Iteracje: ", iteracje]
x_temp = x1 - (f(x1) * (x1 - x0) * 1.0) / (f(x1) - f(x0))
x0 = x1
x1 = x_temp
#print(x1)
return ["Wynik: ", x1, "Iteracje: ", iteracje]
print("Metoda Brenta")
m_brent = optimize.brenth(f, a, b, full_output=True)
print(m_brent)
print("\n Metoda Bisekscji")
m_bisekcja = optimize.bisect(f, a, b, full_output=True)
print(m_bisekcja)
print("\n Metoda Siecznych")
m_sieczne = secant(a, b, 100)
print(m_sieczne)
def printout_tcrit(mec):
"""
Output calculations based on division into bursts by critical time (tcrit).
Parameters
----------
mec : dcpyps.Mechanism
The mechanism to be analysed.
output : output device
Default device: sys.stdout
"""
str = ('\n\n*******************************************\n' +
'CALCULATIONS BASED ON DIVISION INTO BURSTS BY' +
' tcrit- CRITICAL TIME.\n')
# Ideal shut time pdf
eigs, w = ideal_dwell_time_pdf_components(mec.QII, qml.phiF(mec))
str += ('\nIDEAL SHUT TIME DISTRIBUTION\n')
str += pdfs.expPDF_printout(eigs, w)
taus = 1 / eigs
areas = w / eigs
taus, areas = sortShell2(taus, areas)
comps = taus.shape[0] - 1
tcrits = np.empty((3, comps))
for i in range(comps):
str += ('\nCritical time between components {0:d} and {1:d}\n'.format(
i + 1, i + 2) + '\nEqual % misclassified (DC criterion)\n')
try:
tcrit = so.bisect(pdfs.expPDF_tcrit_DC,
taus[i],
taus[i + 1],
args=(taus, areas, i + 1))
enf, ens, pf, ps = pdfs.expPDF_misclassified(
tcrit, taus, areas, i + 1)
str += pdfs.expPDF_misclassified_printout(tcrit, enf, ens, pf, ps)
except:
str += ('Bisection with DC criterion failed.\n')
tcrit = None
tcrits[0, i] = tcrit
str += ('\nEqual # misclassified (Clapham & Neher criterion)\n')
try:
tcrit = so.bisect(pdfs.expPDF_tcrit_CN,
taus[i],
taus[i + 1],
args=(taus, areas, i + 1))
enf, ens, pf, ps = pdfs.expPDF_misclassified(
tcrit, taus, areas, i + 1)
str += pdfs.expPDF_misclassified_printout(tcrit, enf, ens, pf, ps)
except:
str += ('Bisection with Clapham & Neher criterion failed.\n')
tcrit = None
tcrits[1, i] = tcrit
str += ('\nMinimum total # misclassified (Jackson et al criterion)')
try:
tcrit = so.bisect(pdfs.expPDF_tcrit_Jackson,
taus[i],
taus[i + 1],
args=(taus, areas, i + 1))
enf, ens, pf, ps = pdfs.expPDF_misclassified(
tcrit, taus, areas, i + 1)
str += pdfs.expPDF_misclassified_printout(tcrit, enf, ens, pf, ps)
except:
str += ('\nBisection with Jackson et al criterion failed.')
tcrit = None
tcrits[2, i] = tcrit
str += ('\nSUMMARY of tcrit values:\n' + 'Components DC\tC&N\tJackson\n')
for i in range(comps):
str += ('{0:d} to {1:d} '.format(i + 1, i + 2) +
'\t{0:.5g}'.format(tcrits[0, i] * 1000) +
'\t{0:.5g}'.format(tcrits[1, i] * 1000) +
'\t{0:.5g}\n'.format(tcrits[2, i] * 1000))
return str
def _solve_electroneutrality_equation(self, equation, T) -> float:
return bisect(equation, -self.Eg, 2 * self.Eg, T,
xtol=1e-6 * self.Eg) # noqa
transitABlist = []
transitBAlist = []
oldt = time[0]
for i in range(len(time)):
t = time[i]
sim.integrate(t)
p = sim.particles
olddyAB = DyAB(oldt)
dyAB = DyAB(t)
#if i%10==0:
# pl.scatter(-p[0].x,-p[0].z,s=1,c='k')
# pl.scatter(-p[1].x,-p[1].z,s=1,c='k')
# pl.scatter(-p[2].x,-p[2].z,s=1,c='k')
if dyAB > 0 and olddyAB < 0 and p[
0].z < 0: # and np.abs(p[0].x)<0.05:
transitAB = optimize.bisect(DyAB, oldt, t)
transitABlist.append(transitAB)
sim.integrate(transitAB)
p = sim.particles
#pl.scatter(-p[0].x,-p[0].z,s=50,c='b')
#pl.scatter(-p[1].x,-p[1].z,s=50,c='b')
if dyAB > 0 and olddyAB < 0 and p[
0].z > 0: # and np.abs(p[0].x)<0.1:
transitBA = optimize.bisect(DyAB, oldt, t)
transitBAlist.append(transitBA)
sim.integrate(transitBA)
p = sim.particles
#pl.scatter(-p[0].x,-p[0].z,s=50,c='r')
#pl.scatter(-p[1].x,-p[1].z,s=50,c='r')
oldt = t
#pl.xlim([-1.5,1.5])
def _likelihood_ratio_confint(self, alpha: float) -> List[float]:
"""Compute the likelihood ratio confidence interval for the MLE of the previous run.
Args:
alpha: Specifies the (1 - alpha) confidence level (0 < alpha < 1).
Returns:
The likelihood ratio confidence interval.
"""
# Compute the two intervals in which we the look for values above
# the likelihood ratio: the two bubbles next to the QAE estimate
M = 2**self._m
qae = self._ret['value']
y = int(np.round(M * np.arcsin(np.sqrt(qae)) / np.pi))
if y == 0:
right_of_qae = np.sin(np.pi * (y + 1) / M)**2
bubbles = [qae, right_of_qae]
elif y == int(M / 2): # remember, M = 2^m is a power of 2
left_of_qae = np.sin(np.pi * (y - 1) / M)**2
bubbles = [left_of_qae, qae]
else:
left_of_qae = np.sin(np.pi * (y - 1) / M)**2
right_of_qae = np.sin(np.pi * (y + 1) / M)**2
bubbles = [left_of_qae, qae, right_of_qae]
# likelihood function
ai = np.asarray(self._ret['values'])
pi = np.asarray(self._ret['probabilities'])
m = self._m
shots = self._ret['shots']
def loglikelihood(a):
return np.sum(shots * pi * np.log(pdf_a(ai, a, m)))
# The threshold above which the likelihoods are in the
# confidence interval
loglik_mle = loglikelihood(self._ret['ml_value'])
thres = loglik_mle - chi2.ppf(1 - alpha, df=1) / 2
def cut(x):
return loglikelihood(x) - thres
# Store the boundaries of the confidence interval
# It's valid to start off with the zero-width confidence interval, since the maximum
# of the likelihood function is guaranteed to be over the threshold, and if alpha = 0
# that's the valid interval
lower = upper = self._ret['ml_value']
# Check the two intervals/bubbles: check if they surpass the
# threshold and if yes add the part that does to the CI
for a, b in zip(bubbles[:-1], bubbles[1:]):
# Compute local maximum and perform a bisect search between
# the local maximum and the bubble boundaries
locmax, val = bisect_max(loglikelihood, a, b, retval=True)
if val >= thres:
# Bisect pre-condition is that the function has different
# signs at the boundaries of the interval we search in
if cut(a) * cut(locmax) < 0:
left = bisect(cut, a, locmax)
lower = np.minimum(lower, left)
if cut(locmax) * cut(b) < 0:
right = bisect(cut, locmax, b)
upper = np.maximum(upper, right)
# Put together CI
ci = [lower, upper]
return [self.a_factory.value_to_estimation(bound) for bound in ci]
def teardrop_angle(a1, a2, y):
a = optimize.bisect(angle_func, a1, a2, args=(y), maxiter=1000)
return a
print(newton2(5, 1e-5))
def newton3(f, fp, a, eps):
x = a - f(a) / fp(a)
x2 = a
while abs(f(x) - f(x2)) > eps:
x2 = x
x = x - f(x) / fp(x)
return x
f = lambda x: (x - pi)
print(opt.bisect(f, 0, 4))
f = lambda x: x**2 - 2
print(opt.newton(f, 2))
print(np.roots([5, 6, 0, 0, 2]))
def D0(f, x, h=1e-3):
return (f(x + h) - f(x)) / h
def D1(f, x, h=1e-3):
return (f(x + h) - f(x - h)) / (2 * h)
print(D0(f, 5), D1(f, 5))
argument. It scales the rate with the size of the inhibitory
population and configures the inhibitory Poisson generator
(``noise[1]``) accordingly. Then, the spike counter of the
`spike_detector` is reset to zero. The network is simulated using
`Simulate`, which takes the desired simulation time in milliseconds
and advances the network state by this amount of time. During
simulation, the `spike_detector` counts the spikes of the target
neuron and the total number is read out at the end of the simulation
period. The return value of ``output_rate()`` is the firing rate of
the target neuron in Hz.
Second, the scipy function ``bisect`` is used to determine the optimal
firing rate of the neurons of the inhibitory population.
'''
in_rate = bisect(lambda x: output_rate(x) - r_ex, lower, upper, xtol=prec)
print("Optimal rate for the inhibitory population: %.2f Hz" % in_rate)
'''
The function ``bisect`` takes four arguments: first a function whose
zero crossing is to be determined. Here, the firing rate of the target
neuron should equal the firing rate of the neurons of the excitatory
population. Thus we define an anonymous function (using ``lambda``)
that returns the difference between the actual rate of the target
neuron and the rate of the excitatory Poisson generator, given a rate
for the inhibitory neurons. The next two arguments are the lower and
upper bound of the interval in which to search for the zero
crossing. The fourth argument of ``bisect`` is the desired relative
precision of the zero crossing.
Finally, we plot the target neuron's membrane potential as a function
of time.
tau = kap * (rho * dr).sum()
return tau - tau0
sigma_smalldust = sigma_dust_2d[0]
astr = ("{0:" + sizeformat + "}").format(a_grains[0] * 1e4)
osmall = readOpac(ext=astr, scatmat=True)
lamstar = 0.45 # Representative wavelength for stellar radiation
kappa = np.interp(lamstar, osmall.wav[0], osmall.kabs[0] + osmall.ksca[0])
thetaupp = np.zeros(nphi)
for iphi in range(nphi):
args = (0.01, ri, sigma_smalldust[:, iphi], hpr, kappa)
thetaupp[iphi] = bisect(ftauroot,
0.2,
np.pi / 2,
args=args,
xtol=1e-6,
rtol=1e-6)
thetaup = thetaupp.min() # Min, because pi/2-thetaup must be max
#
# Make the theta coordinate, and refine near the midplane
# (to resolve the dust layer)
#
# NOTE: If your grains have not-so-large Stokes numbers,
# you can save computing time by reducing nlev_zr.
# Just make sure that the vertical structure of the
# distribution of the largest grains remains resolved
# by the theta-grid.
#
ntheta = 32
import timeit
import numpy as np
import scipy.optimize as optimize
import time
def f(x):
return np.cos(x) / (1. + x**2)
def fprime(x):
return (-(x**2 + 1.) * np.sin(x) - 2. * x * np.cos(x)) / (x**2 + 1)**2
start_time = time.time()
brentq_x = optimize.brentq(f, 0.1, 2.4)
print("--- %s seconds ---" % (time.time() - start_time))
start_time = time.time()
bisect_x = optimize.bisect(f, 0.1, 2.4)
print("--- %s seconds ---" % (time.time() - start_time))
start_time = time.time()
newton_x = optimize.newton(f, 1.3)
print("--- %s seconds ---" % (time.time() - start_time))
start_time = time.time()
newtonx2_x = optimize.newton(f, 1.3, fprime)
print("--- %s seconds ---" % (time.time() - start_time))
def solve_model(rho,
gam,
T,
phi1,
phi2,
verbose=False,
transform=False,
empirical=1,
calibrated=True,
risk_free_adj=1,
shock=1,
verbose_ss=False):
#######################################################
# Section 1: Calibration #
#######################################################
if empirical == 0 or empirical == 0.5:
beta2 = 0.0022
zeta = 0.014
# Calibrated params old model
# a_k = 0.03191501061172916
# phi = 13.353240248981844
# A = 0.26314399999999993
# delta = 0.007 * risk_free_adj - alpha_c * rho
# Original, randomly selected params
# a_k = 0.017
# phi = 13.807 / 2
# A = 0.052
# delta = 0.025
if empirical == 0:
# Eberly Wang annual params
# a_k = .1
# phi2 = 100
# phi1 = .05
# # A = 0.1 + .004
# A = 0.1 + .042
# delta = .02
# Eberly Wang quarterly params
a_k = .1 / 4
# phi2 = 100 * 4
# phi1 = .05 / 4
A = (.1 + .042) / 4
delta = .02 / 4
if empirical == 0.5:
# Low adjustment cost model annual params
# a_k = .05
# phi2 = 3.
# phi1 = 1. / phi1
# A = .14
# delta = .05
# Low adjustment cost model quarterly params
a_k = .05 / 4
# phi2 = 3. * 4
# phi1 = 1. / phi1
A = .14 / 4
delta = .05 / 4
def f(c):
Phi = (1 + phi2 * (A - np.exp(c)))**(phi1)
Phiprime = phi2 * phi1 * (1 + phi2 * (A - np.exp(c)))**(phi1 - 1)
k = np.log(Phi) - a_k
if rho == 1:
v = c + k * np.exp(-delta) / (1 - np.exp(-delta))
else:
v = np.log((1 - np.exp(-delta)) * np.exp(c * (1 - rho)) /
(1 - np.exp(-delta + k * (1 - rho)))) / (1 - rho)
r1 = Phiprime - (np.exp(delta) - 1) * Phi * np.exp(c * -rho +
(v + k) *
(rho - 1))
return r1
sol = opt.bisect(f, -10, np.log(A), disp=True)
cstar = sol
# print(f(cstar))
def v(c):
Phi = (1 + phi2 * (A - np.exp(c)))**(phi1)
# Phiprime = phi2 * phi1 * (1 + phi2 * (A - np.exp(c)))**(phi1 - 1)
k = np.log(Phi) - a_k
if rho == 1:
v = c + k * np.exp(-delta) / (1 - np.exp(-delta))
else:
v = np.log((1 - np.exp(-delta)) * np.exp(c * (1 - rho)) /
(1 - np.exp(-delta + k * (1 - rho)))) / (1 - rho)
# r1 = Phiprime - (np.exp(delta) - 1) * Phi * np.exp(c * -rho + (v + k) * (rho - 1))
return v
def denom(c):
Phi = (1 + phi2 * (A - np.exp(c)))**(phi1)
# Phiprime = phi2 * phi1 * (1 + phi2 * (A - np.exp(c)))**(phi1 - 1)
k = np.log(Phi) - a_k
return (1 - np.exp(-delta + k * (1 - rho)))
def k(c):
Phi = (1 + phi2 * (A - np.exp(c)))**(phi1)
# Phiprime = phi2 * phi1 * (1 + phi2 * (A - np.exp(c)))**(phi1 - 1)
k = np.log(Phi) - a_k
return k
dom = np.linspace(-10, np.log(A), 500)
# plt.plot(dom, f(dom))
# plt.plot(dom, v(dom))
# plt.subplot(1,2,1)
# plt.plot(dom, denom(dom), label='denominator')
# plt.plot(dom, k(dom), label='k')
# plt.plot(dom, np.zeros_like(dom))
# plt.plot(dom, np.ones_like(dom) * delta / (1 - rho), label = r'$\frac{\delta}{(1 - \rho)}$')
# plt.legend()
# plt.subplot(1,2,2)
# plt.plot(dom, f(dom), label = 'Root function')
# plt.legend()
# plt.show()
if np.min(np.abs(cstar - np.array([-10, np.log(A)]))) < 1e-8:
raise ValueError("Not actually solved")
Phi = (1 + phi2 * (A - np.exp(cstar)))**(phi1)
Phiprime = phi2 * phi1 * (1 + phi2 * (A - np.exp(cstar)))**(phi1 - 1)
kstar = np.log(Phi) - a_k
istar = np.log(A - np.exp(cstar))
if rho == 1:
vstar = cstar + kstar * np.exp(-delta) / (1 - np.exp(-delta))
else:
vstar = np.log(
(1 - np.exp(-delta)) * np.exp(cstar * (1 - rho)) /
(1 - np.exp(-delta + kstar * (1 - rho)))) / (1 - rho)
istar = np.log(A - np.exp(cstar))
zstar = 0
dstar = 0
# Calculate parameters using empirical targets
elif empirical == 1:
# Use all empirical targets with all parameters free
istar = np.log(IoverK)
cstar = np.log(CoverI * np.exp(istar))
delta0 = 0.007 * risk_free_adj - alpha_c * rho
A0 = np.exp(istar) + np.exp(cstar)
kstar = alpha_c
def f(v0):
if rho != 1:
r3 = np.exp(v0) ** (1 - rho) - (1 - np.exp(-delta0)) \
* np.exp(cstar) ** (1 - rho) - np.exp(-delta0) \
* (np.exp(v0) * np.exp(kstar)) ** (1 - rho)
else:
r3 = np.exp(v0) ** (1 - np.exp(-delta0)) \
- np.exp(cstar) ** (1 - np.exp(-delta0)) \
* np.exp(kstar) ** (np.exp(-delta0))
return r3
vstar = opt.root(f, -3.2).x[0]
def g(phi):
Phi = (1. + phi * np.exp(istar))**(1. / phi)
PhiPrime = (1. + phi * np.exp(istar))**(1. / phi - 1)
return np.exp(-rho * cstar + (rho - 1) * (vstar + kstar)) \
* (np.exp(delta0) - 1) * (Phi) \
- PhiPrime
phi0 = opt.root(g, 700).x[0]
a_k0 = np.log((1. + phi0 * np.exp(istar))**(1. / phi0)) - kstar
zstar = 0
A = A0
delta = delta0
a_k = a_k0
phi = phi0
elif empirical == 2:
raise ValueError(
"The specifications for C and V are not yet developed for this empirical case."
)
# Fix phi = 0 and free C/I
I = IoverK
istar = np.log(I)
phi = 0
delta = 0.007 * risk_free_adj - alpha_c * rho
kstar = alpha_c
G = np.exp(kstar)
a_k = np.log(1 + I) - kstar
if (-delta + (1 - rho) * kstar) >= 0:
raise ValueError(("The constraint to solve for V is not"
"satisfied for rho = {}").format(rho))
if rho != 1:
C = (np.exp(delta) - 1) * (G ** (rho - 1) - np.exp(-delta)) \
* (1 + I) / (1 - np.exp(-delta))
V = ((1 - np.exp(-delta)) * C ** (1 - rho) \
/ (1 - np.exp(-delta) * G ** (1 - rho))) ** (1 / (1 - rho))
else:
C = (np.exp(delta) - 1) * G ** ((rho - 1) / (1 - np.exp(-delta))) \
* (1 + I)
V = C * G**(np.exp(-delta) / (1 - np.exp(-delta)))
cstar = np.log(C)
vstar = np.log(V)
A = np.exp(cstar) + np.exp(istar)
zstar = 0
else:
raise ValueError("'Empirical' must be 1 or 2.")
#######################################################
# Section 2: Model Solution #
#######################################################
#######################################################
# Section 2.1: Symbolic Model Declaration #
#######################################################
# Declare necessary symbols in sympy notation
# Note that the p represents time, so k = k_t and kp = k_{t+1}
k, kp, c, cp, v, vp, zo, zop, zt, ztp = symbols(
"k kp c cp v vp zo zop zt ztp")
# k = log(K_t / K_{t-1})
# c = log(C_t / K_t)
# v = log(V_t / K_t)
# zo = Z_{t,1}
# zt = Z_{t,2}
# d = log D_t
# Set up the equations from the model in sympy
# The equations come from subtracting the right side from the left of the
# log linearized governing equations
I = A - exp(c) # I_t / K_t
# NEW FUNCTION
# i = 1 + I - phi / 2 * I ** 2 # quadratic version of I; also I^*/K_t
i = (1. + phi2 * I)**(phi1)
# i = 1 + log(phi * I + 1)/phi
# phip = 1 - phi * I
phip = phi2 * phi1 * (1. + phi2 * I)**(phi1 - 1)
# phip = 1 / (phi * I + 1)
r = vp + kp + zt
# Equation 1: Capital Evolution
eq1 = kp - log(i) + a_k - zo
# Equation 2: First Order Conditions on Consumption
eq2 = log(exp(delta) - 1) - rho * c + (rho - 1) * r + \
log(i) - log(phip)
# Equation 3: Value Function Evolution: rho == 1 is separate case
if rho != 1:
eq3 = exp(v * (1 - rho)) - ((1 - exp(-delta)) * exp((1 - rho) * c) \
+ exp(-delta) * exp((1 - rho) * r))
else:
eq3 = v - (1 - exp(-delta)) * c - exp(-delta) * r
# Equations 4 and 5: Shock Processes Evolution
eq4 = zop - exp(-zeta) * zo
eq5 = ztp - exp(-beta2) * zt
eqs = [eq1, eq2, eq3, eq4, eq5]
lead_vars = [kp, cp, vp, zop, ztp]
current_vars = [k, c, v, zo, zt]
substitutions = {
k: kstar,
kp: kstar,
c: cstar,
cp: cstar,
v: vstar,
vp: vstar,
zo: zstar,
zop: zstar,
zt: zstar,
ztp: zstar
}
# print(substitutions)
#######################################################
#Section 2.2: Generalized Schur Decomposition Solution#
#######################################################
# Take the appropriate derivatives and evaluate at steady state
Amat = np.array([[eq.diff(var).evalf(subs=substitutions) for \
var in lead_vars] for eq in eqs]).astype(np.float)
B = -np.array([[eq.diff(var).evalf(subs=substitutions) for var in \
current_vars] for eq in eqs]).astype(np.float)
# print(Amat)
# print(B)
# print(la.inv(Amat[:3, :3]))
# Substitute for k and c to reduce A and B to 2x2 matrices, noting that:
# A[0,0]kp - B[0,1]c = zo
# A[1,0]kp - B[1,1]c = B[1,4]zt - A[1,2]vp
M = np.array([[Amat[0, 0], -B[0, 1]], [Amat[1, 0], -B[1, 1]]])
Minv = la.inv(M)
# kp = Minv[0,0] * zo + Minv[0,1] * (B[1,4]zt - A[1,2]vp) (1)
# c = Minv[1,0] * zo + Minv[1,1] * (B[1,4]zt - A[1,2]vp) (2)
# So the system can be reduced in the following way:
Anew = np.copy(Amat[2:, 2:])
Bnew = np.copy(B[2:, 2:])
# Update the column of Anew corresponding to vp, subbing in with (1)
Anew[:, 0] += Minv[0, 1] * Amat[2:, 0] * (-Amat[1, 2])
# Update the column of Bnew corresponding to zo, subbing in with (1)
Bnew[:, 1] -= Minv[0, 0] * Amat[2:, 0]
# Update the column of Bnew corresponding to zt, subbing in with (1)
Bnew[:, 2] -= Minv[0, 1] * Amat[2:, 0] * B[1, 4]
# Update the column of Anew corresponding to vp, subbing in with (2)
Anew[:, 0] -= Minv[1, 1] * B[2:, 1] * (-Amat[1, 2])
# Update the column of Bnew corresponding to zo, subbing in with (2)
Bnew[:, 1] += Minv[1, 0] * B[2:, 1]
# Update the column of Bnew corresponding to zt, subbing in with (2)
Bnew[:, 2] += Minv[1, 1] * B[2:, 1] * B[1, 4]
# Compute the generalized Schur decomposition of the reduced A and B,
# sorting so that the explosive eigenvalues are in the bottom right
BB, AA, a, b, Q, Z = la.ordqz(Bnew, Anew, sort='iuc')
total_dim = len(Anew)
# a/b is a vector of the generalized eiganvals
exp_dim = len(a[np.abs(a / b) > 1])
stable_dim = total_dim - exp_dim
if verbose:
print("Rho = {}".format(rho))
# print(-delta + (1 - rho) * kstar)
print(("{} out of {} eigenvalues were found to be"
" unstable.").format(exp_dim, total_dim))
J1 = Z.T[stable_dim:, :exp_dim][0][0]
J2 = Z.T[stable_dim:, exp_dim:][0]
# J1v = J2 @ [zo, zt]
v_loading = -(J2 / J1)
# Recall the following identities:
# kp = Minv[0,0] * zo + Minv[0,1] * (B[1,4]zt - A[1,2]vp) (1)
# c = Minv[1,0] * zo + Minv[1,1] * (B[1,4]zt - A[1,2]vp) (2)
# Rewrite as
# kp = -Minv[0,1]*A[1,2]vp + Minv[0,0]zo + Minv[0,1]*B[1,4]zt (1)
# c = -Minv[1,1]*A[1,2]vp + Minv[1,0]zo + Minv[1,1]*B[1,4]zt (2)
k_loading = -Minv[0, 1] * Amat[1, 2] * v_loading
c_loading = -Minv[1, 1] * Amat[1, 2] * v_loading * np.exp(-zeta)
# Add the zo and zt specific dependencies to each entry of each vector
k_loading += np.array([Minv[0, 0], Minv[0, 1] * B[1, 4]]) * np.exp(zeta)
c_loading += np.array([Minv[1, 0], Minv[1, 1]])
istar = np.log(A - np.exp(cstar))
i_loading = (-exp(cstar) * c_loading / exp(istar)).astype(np.float)
slopes1 = [k_loading[0], c_loading[0], i_loading[0], v_loading[0]]
slopes2 = [k_loading[1], c_loading[1], i_loading[1], v_loading[1]]
#######################################################
# Section 3: First Order Adjustments #
#######################################################
# sigz = np.array([.00011, .00025, 0.0])
# scriptB = np.array([[0.012, 0.027, 0, 0],[0, 0, 0.132, 0]]) * 0.01
# sigk = np.array([.00477, 0.0, 0.0, 0.0])
# scriptK = np.array([0.481, 0, 0, 0]) * 0.01
# Create first order adjustments on constant terms
# if verbose:
# print("Making first order adjustments to constants.")
if transform:
sig = np.vstack((sigk, sigz))
s1 = sigk + sigz / (1 - np.exp(-zeta))
s1 = s1 / la.norm(s1)
s2 = s1[::-1] * np.array([-1, 1])
s = np.column_stack((s2, s1))
snew = sig @ s
sigk = snew[0][::-1] * np.array([1, -1])
sigz = snew[1][::-1] * np.array([1, -1])
#print(np.array([sigk,sigz]) * 100)
selector = np.zeros(4)
selector[shock - 1] = 1
B = np.array([[0.011, 0.025, 0, 0], [0, 0, 0.119, 0]])
sigk = np.array([0.477, 0, 0, 0])
# sigk = np.array([0.477, 0, 0, 0]) * 0.01
sigd = np.array([0, 0, 0, 0])
# adjustment = - (1 - gam) / 2 * la.norm(v_loading * sigz + sigk) ** 2
# adjustment = - (1 - gam) / 2 * la.norm(v_loading @ scriptB + scriptK) ** 2
# adjustments = la.solve((Amat - B)[:3,:3], np.array([0,0,adjustment]))
# print(adjustments)
# kstar += adjustments[0]
# cstar += adjustments[1]
# vstar += adjustments[2]
# istar = log(A - np.exp(cstar))
levels = [kstar, cstar, istar, vstar]
if verbose:
print("Log Levels: k, c, i, v")
print(levels)
print("Log slopes, growth shock: k, c, i, v")
print(slopes1)
print("Log slopes, preference shock: k, c, i, v")
print(slopes2)
print("\n")
if verbose_ss:
print("Rho = {}:".format(rho))
print("\tLog capital growth: \t\t{0:.3f}".format(levels[0], 3))
print("\tConsumption to capital: \t{0:.3f}".format(
np.exp(levels[1]), 3))
print("\tInvestment to capital: \t\t{0:.3f}".format(
np.exp(levels[2]), 3))
print("\tQ: \t\t\t\t{0:.3f}".format(
(1 - np.exp(-delta)) * np.exp(levels[3] * (rho - 1)) *
np.exp(levels[1] * -rho), 3))
Atransition = np.array([[np.exp(-zeta), 0], [0, np.exp(-beta2)]])
#######################################################
# Section 4: Impulse Response Generation #
#######################################################
Z = np.zeros((2, T))
Z[:, 0] = B @ selector
for i in range(1, T):
Z[:, i] = Atransition @ Z[:, i - 1]
if shock == 1 or shock == 2:
pass
# bool_marker = np.array([1,0])
elif shock == 3:
pass
# bool_marker = np.array([0, 1])
else:
raise ValueError("'shock' parameter must be set to 1, 2, or 3.")
K = np.zeros(T)
S = np.zeros(T)
C = np.zeros(T)
I = np.zeros(T)
K[0] = sigk @ selector
for p in range(1, T):
K[p] = K[p - 1] + k_loading @ Z[:, p]
S[0] = -rho * c_loading @ Z[:,0] + (1 - rho) * Z[1,0] + \
(rho - gam) * ((v_loading @ B) + sigk + sigd)[shock - 1] \
- rho * K[0]
for p in range(1, T):
S[p] = S[p-1] - rho * c_loading @ (Z[:,p] - Z[:,p-1]) \
- rho * k_loading @ Z[:,p] + (1 - rho) * Z[1,p]
C = c_loading @ Z + K
I = i_loading @ Z + K
return levels, slopes1, slopes2, np.array([
-S.astype(np.float),
K.astype(np.float),
C.astype(np.float),
I.astype(np.float)
])
def power_law_graph(self, N, exp, dmin, expect_n_edges,
num_trial=100, tol=100):
'''
generate power law [connected] graph by given number of nodes, edges and
exponent.
:param N:
:param exp: [float] negative exponent in power low graph, typical value is -2.0 to -3.0
:param dmin: minimum degree of a node.
:param expect_n_edges: expected number of edges in graph
:param num_trial: number of trial to try find a connected graph
:param tol: the tolerance number of edges versus the expected given number of edges.
:return: networkx graph
'''
def tune_dmin(x, G, N, exp, expect_n_edges, max_num_try=50):
max_try = 50
for i in range(max_num_try):
sequence = generate_power_law_dist(N, exp, x, N-1)
G = nx.configuration_model(sequence)
G.remove_edges_from(G.selfloop_edges())
G = nx.Graph(G)
simple_seq = np.asarray([deg for (node, deg) in G.degree()])
num_edges = int(np.sum(simple_seq)/2.0)
if nx.is_connected(G):
# print "n = ", num_edges
break
if i == (max_num_try-1):
print("G is not connected")
exit(0)
return num_edges - expect_n_edges
G = nx.Graph()
dmax = N-1
for itr in range(num_trial):
# tune minimum degree of the node
d_tuned = bisect(tune_dmin,
dmin-3, dmin+10, # sweep the interval around dmin
args=(G, N, exp, expect_n_edges),
xtol=0.01)
simple_seq = np.asarray([deg for (node, deg) in G.degree()])
num_edges = np.sum(simple_seq)/2.0
error = np.abs(num_edges - expect_n_edges)
if verbocity:
print ("dmin = %.2f, error = %d, n_edges = %d " % (
d_tuned, error, num_edges))
# check if number of edges is close to the expected number of edges
if error < tol:
return G
if itr == num_trial-1:
print("could not find a proper graph with given properties!")
exit(0)
import matplotlib.pyplot as plt
import numpy as np
from scipy import optimize as op
def f(x):
y = ((np.exp(-x)) * (x + 1)) - 0.5
return y
x1 = 0.0
x2 = 2.0
print(f(x1), f(x2))
root = op.bisect(f, x1, x2)
print(root)
#3c
h = 5.4
I0 = 1.38e8
def L(x):
y = 2 * np.pi * I0 * h * (h - (np.exp(-x / h) * (x + h)))
return y
r = np.arange(0, 100, 0.01)
fa.set_xlabel(r'$k\alpha$',fontsize=20)
fa.set_ylabel(r'$m$',fontsize=20)
fa.text(2.5,0.1,'a',fontsize=24,backgroundcolor='w')
fb.set_xlabel(r'$k\alpha$',fontsize=20)
fb.set_ylabel(r'$v_m$',fontsize=20)
fb.text(2.5,0.025,'b',fontsize=24,backgroundcolor='w')
#***********************************************************************
fc = fig.add_subplot(2,2,3)
delta=arange(0.,1.01,0.01)
kalpha=[]
k0=20.
for d in delta:
kn=optimize.bisect((lambda x:mr(x,d)[0][0]-0.0001),0.,k0)
kalpha.append(kn)
k0=kn
#print d,kn
fc.plot(delta,kalpha)
fc.set_ylabel(r'$k\alpha$',fontsize=20)
fc.set_xlabel(r'$\delta$',fontsize=20)
fc.set_xlim((0.,1.))
fc.set_ylim((0.,20.))
fc.grid(True)
fc.text(.1,2.,'c',fontsize=24,backgroundcolor='w')
fc.text(.3,2.5,r'$m=0$',fontsize=24,backgroundcolor='w')
fc.text(.5,12,r'$m>0$',fontsize=24,backgroundcolor='w')
#***********************************************************************
fd = fig.add_subplot(2,2,4)
def write_layered_dx(x):
'''dxx = write_layered_dx(x)
Returns dxx - a list of grid distance increments for a series of layers with set thicknesses, desired nodes, and bias flag (see layer class).
Requires x - a list of layer classes with defined thickness number of nodes and bias type. Layers should be ordered in pflotran natural ordering - (i.e. starting a lower southwest corner.'''
dx = n.array([])
l_count = 0
txx = 0.
for l in x:
txx = txx + l.h
bias_l = l.bias
nx = l.n_step
x1 = l.h / l.n_step # just a guess for now
if l.bias == 0:
px = sum(dx)
tx = px + l.h
for i in xrange(nx):
dx = n.append(dx, x1)
#hit end of layer on the nose
dx[-1] = tx - sum(dx[0:-1])
if l.bias == 1: #single bias decreasing in positive direction
if x[l_count + 1].bias == 0:
x1 = x[l_count + 1].h / x[
l_count +
1].n_step # want to end with the next grid spacing
else:
print 'not sure what do with bias min x distance, figure it out'
args = (l.h, nx, x1)
bx = optimize.bisect(txb_diff_single_bias, 1., 2.5, args=args)
px = sum(dx)
tx = px + l.h
#construct the biased vector
for i in xrange(1, int(nx)):
dx = n.append(dx, x1 * bx**(nx - (i + 1)))
#hit end of bias on the nose
dx[-1] = tx - sum(dx[0:-1])
if l.bias == 2: #single bias increasing in positive direction
if l.num >= 1:
x1 = dx[-1] #start with last grid spacing
else:
print 'not sure what to do about min x distance, figure it out'
bx = optimize.bisect(txb_diff_single_bias,
1.,
2.5,
args=(l.h, nx, x1))
px = sum(dx)
tx = px + l.h
#construct the biased vector
for i in xrange(1, int(nx)):
dx = n.append(dx, x1 * bx**(i - 1))
#hit end of domain on the nose
dx[-1] = tx - sum(dx[0:-1])
if l.bias == 3: #dual bias
if l.num >= 1:
x1 = dx[-1] #start with last grid spacing
else:
print 'not sure what to do about min x distance, figure it out'
# now to figure out the correct bias (for a dual bias)
nnx = nx / 2 - 1
px = sum(dx)
tx = px + l.h
bx = optimize.bisect(txb_diff_dual_bias,
1.,
2.5,
args=(l.h, nx, x1))
#first half of the biased vector
for i in xrange(1, int(nx / 2)):
dx = n.append(dx, x1 * bx**(i - 1))
#second half of the biased x vector
for i in xrange(1, int(nx / 2)):
dx = nappend(dx, x1 * bx**(nnx - (i)))
#hit end of bias on the nose
dx[-1] = tx - sum(dx[0:-1])
tl = n.sum(dx) - px
print 'Total thickness of layer ' + l.name + ' = %1.2g units' % tl
l_count += 1
#make the ending length get us to a the desired total length.
print "layer discretization error %1.5g" % (txx - tx)
dx[-1] = txx - sum(dx[0:-1])
dxx = list(dx)
print 'Total layered thickness' + l.name + ' = %1.2g units' % (sum(dxx))
return dxx
from scipy.optimize import bisect
from numpy import sin, cos, exp
def f(x):
return sin(cos(exp(x)))
b = bisect(f, -1, 1)
print("zero of sin(cos(exp(x)))=0 is", b, ".")
q = sin(cos(exp(b)))
print("value of function at found zero is", q, ".")
def getLambert(r):
R2 = R1 + R20 * r
a = NAN
try:
lambert_result = lambert_problem(R1, R2, dt)
if lambert_result.is_reliable():
a = lambert_result.get_a()[0]
except:
a = NAN
#print " r,a ", r,a
return a
# ============ Pre optimize
x0 = bisect(getLambert, rmin, rmax, xtol=0.00001)
# print " x0 = ", x0
print " bisect DONE "
def fitFractionalFunc(x0, A, B):
a = (A[1] - B[1]) / (1 / (x0 - A[0]) - 1 / (x0 - B[0]))
b = A[1] - a / (x0 - A[0])
return a, b
ysc, y0 = fitFractionalFunc(x0, (rmin, getLambert(rmin)),
(x0 * 0.9, getLambert(x0 * 0.9)))
'''
# NOT GOOD
Gen = (y0, ysc, 0, 0, 0, 0, 0); print " Gen0 = ",Gen
def solve_self_consistent_k_space_2sublat\
(abs_t0, E0, E1, E2, E3, E4,\
Nk, Ny, nOrb, nHole, invTemp, betaStart, betaSpeed, betaThreshold,anneal_or_not,\
U, itMax, dampFreq, dyn, singleExcitationFreq, osc, delta, nUp, nDown):
'''
Solves the self-consistent equation in momentum space.
'''
# Initialize deltas for the tolerance check and
# beta for the annealing.
deltaUp = delta + 1
deltaDown = delta + 1
beta = betaStart
# Initialize energies
energies = np.zeros(itMax)
# Initialize iteration
it = 0
# Initialize iteration at which we finish annealing
itSwitch = 0
# How many iterations to wait between kicks
itWait = 3
# lbda is a parameter that reduces weight
# on the density obtained in the previous iteration.
# the factor multiplied by itMax impedes that P ( I ) < delta
# initially, we give equal weights and then progressively more
# to the new configuration
factor = 1.2
lbda = 0.5 / (factor * itMax)
# This ensures that we do not stop at the first step
energies[-1] = 1e100
# Print frequency
printFreq = 10
# Initialize arrays to store energies and eigenstates
eUp = np.zeros((Nk, 2 * nOrb * Ny))
wUp = np.zeros((Nk, 2 * nOrb * Ny, 2 * nOrb * Ny), dtype=np.complex64)
eDown = np.zeros((Nk, 2 * nOrb * Ny))
wDown = np.zeros((Nk, 2 * nOrb * Ny, 2 * nOrb * Ny), dtype=np.complex64)
eUpBest = np.zeros((Nk, 2 * nOrb * Ny))
wUpBest = np.zeros((Nk, 2 * nOrb * Ny, 2 * nOrb * Ny), dtype=np.complex64)
eDownBest = np.zeros((Nk, 2 * nOrb * Ny))
wDownBest = np.zeros((Nk, 2 * nOrb * Ny, 2 * nOrb * Ny), dtype=np.complex64)
ks = np.linspace(-np.pi / 2,np.pi / 2, num=Nk, endpoint=False)
if anneal_or_not == True:
print("Started annealing.\n")
nUpBest, nDownBest = nUp, nDown
bestGrandpotential = 1e100
while loopCondition(it, itMax, deltaUp, deltaDown,\
delta, beta, invTemp, betaThreshold):
# Annealing
if anneal_or_not == True:
beta = anneal(invTemp, betaStart, betaSpeed, beta,\
betaThreshold, it, osc)
else:
beta = noAnneal(invTemp, beta)
for kCount, k in enumerate(ks):
# Define the MF Hamiltonian for this iteration and k-point
K = HribbonKSpace2sublat(k, nOrb, Ny, E0, E1, E2, E3, E4)
Hup = K + U * np.eye(2 * nOrb * Ny) * nDown
Hdown = K + U * np.eye(2 * nOrb * Ny) * nUp
# Diagonalize
eUp[kCount, :], wUp[kCount, :, :] = la.eigh(Hup)
eDown[kCount, :], wDown[kCount, :, :] = la.eigh(Hdown)
# Save the previous fields to compare to test convergence
nUpOld = nUp.copy()
nDownOld = nDown.copy()
# Compute the chemical potential implicitly
mu = bisect(rootToChem2sublat, -50, 50,\
args = (eUp, eDown, beta, Nk, Ny, nHole) )
# Update fields
nUp, nDown = update2sublat(nUp, nDown, Nk, Ny,\
wUp, wDown, eUp, eDown, mu, beta, nOrb)
# Damping
nUp, nDown = damp(it, dampFreq, nUp, nDown, nUpOld, nDownOld, lbda)
# Relative difference between current and previous fields
deltaUp = np.dot(nUp - nUpOld, nUp - nUpOld)\
/ np.dot(nUpOld, nUpOld)
deltaDown = np.dot(nDown - nDownOld, nDown - nDownOld)\
/ np.dot(nDownOld, nDownOld)
if it % printFreq == 0:
print("iteration: ", it)
print("deltaUp: ", deltaUp)
print("deltaDown: ", deltaDown)
# Grandpotential per site
energy = grandpotentialKSpace2sublat(U, nUp, nDown, nUpOld, nDownOld,\
invTemp, Nk, Ny, eUp, eDown, mu, abs_t0)
if it % singleExcitationFreq == 0 :
for attempt in range(2 * nOrb * Ny):
i = int( np.random.random() * 2 * nOrb * Ny )
nUpTmp, nDownTmp = nUp[i], nDown[i]
nUp[i], nDown[i] = updateLocal2sublat(i, nUp, nDown, Nk, Ny,\
wUp, wDown, eUp, eDown, mu, beta, nOrb)
energyTmp = grandpotentialKSpace2sublat(U, nUp, nDown,\
nUpOld, nDownOld, invTemp, Nk, Ny, eUp, eDown, mu, abs_t0)
if energyTmp > energy:
nUp[i], nDown[i] = nUpTmp, nDownTmp
else:
energy = energyTmp
if ( it + 1 ) % singleExcitationFreq == 0 :
for attempt in range(nOrb * 2 * Ny):
i = int( np.random.random() * 2 * nOrb * Ny )
nUpTmp, nDownTmp = nUp[i], nDown[i]
if dyn == 'local' or dyn == 'wait':
nUp[i], nDown[i] = updateLocalKSpace(i, nUp, nDown, Nk, Ny,\
wUp, wDown, eUp, eDown, mu, beta, nOrb)
# We do not take steps that increase energy
energyTmp = grandpotentialKSpace2sublat(U, nUp, nDown,\
nUpOld, nDownOld, invTemp, Nk, Ny, eUp, eDown, mu, abs_t0)
if energyTmp > energy:
nUp[i], nDown[i] = nUpTmp, nDownTmp
else:
energy = energyTmp
if dyn == 'kick':
nUp[i], nDown[i] = singleExcitation(nUp, nDown, i)
# We do not take steps that increase energy
energyTmp = grandpotentialKSpace2sublat(U, nUp, nDown,\
nUpOld, nDownOld, invTemp, Nk, Ny, eUp, eDown, mu, abs_t0)
if energyTmp > energy:
nUp[i], nDown[i] = nUpTmp, nDownTmp
else:
energy = energyTmp
if dyn == 'mixed':
nUp[i], nDown[i] = updateLocalKSpace(i, nUp, nDown, Nk, Ny,\
wUp, wDown, eUp, eDown, mu, beta, nOrb)
# We do not take steps that increase energy
energyTmp = grandpotentialKSpace2sublat(U, nUp, nDown,\
nUpOld, nDownOld, invTemp, Nk, Ny, eUp, eDown, mu, abs_t0)
if energyTmp > energy:
nUp[i], nDown[i] = nUpTmp, nDownTmp
else:
energy = energyTmp
nUp[i], nDown[i] = singleExcitation(nUp, nDown, i)
# We do not take steps that increase energy
energyTmp = grandpotentialKSpace2sublat(U, nUp, nDown,\
nUpOld, nDownOld, invTemp, Nk, Ny, eUp, eDown, mu, abs_t0)
if energyTmp > energy:
nUp[i], nDown[i] = nUpTmp, nDownTmp
else:
energy = energyTmp
if dyn == 'wait':
if ( it + 1 + itWait ) % singleExcitationFreq == 0 :
for attempt in range(nOrb * 2 * Ny):
i = int( np.random.random() * 2 * nOrb * Ny )
nUpTmp, nDownTmp = nUp[i], nDown[i]
nUp[i], nDown[i] = singleExcitation(nUp, nDown, i)
# We do not take steps that increase energy
energyTmp = grandpotentialKSpace2sublat(U, nUp, nDown,\
nUpOld, nDownOld, invTemp, Nk, Ny, eUp, eDown, mu, abs_t0)
if energyTmp > energy:
nUp[i], nDown[i] = nUpTmp, nDownTmp
else:
energy = energyTmp
## HERE
if ( it + 1 ) % singleExcitationFreq == 0 :
if dyn == 'local' or dyn == 'wait':
for attempt in range(2 * nOrb * Ny):
i = int( np.random.random() * 2 * nOrb * Ny )
nUpTmp, nDownTmp = nUp[i], nDown[i]
nUp[i], nDown[i] = updateLocal2sublat(i, nUp, nDown, Nk, Ny,\
wUp, wDown, eUp, eDown, mu, beta, nOrb)
for kCount, k in enumerate(ks):
# Define the MF Hamiltonian for this iteration and k-point
K = HribbonKSpace2sublat(k, nOrb, Ny, E0, E1, E2, E3, E4)
Hup = K + U * np.eye(2 * nOrb * Ny) * nDown
Hdown = K + U * np.eye(2 * nOrb * Ny) * nUp
# Diagonalize
eUp[kCount, :], wUp[kCount, :, :] = la.eigh(Hup)
eDown[kCount, :], wDown[kCount, :, :] = la.eigh(Hdown)
# Save the previous fields to compare to test convergence
nUpOld = nUp.copy()
nDownOld = nDown.copy()
# Compute the chemical potential implicitly
mu = bisect(rootToChem2sublat, -50, 50,\
args = (eUp, eDown, beta, Nk, Ny, nHole) )
# Update fields
nUp, nDown = update2sublat(nUp, nDown, Nk, Ny,\
wUp, wDown, eUp, eDown, mu, beta, nOrb)
# Grandpotential per site
energyTmp = grandpotentialKSpace2sublat(U, nUp, nDown,\
nUpOld, nDownOld, invTemp, Nk, Ny, eUp, eDown, mu, abs_t0)
if energyTmp > energy:
nUp[i], nDown[i] = nUpTmp, nDownTmp
else:
energy = energyTmp
elif dyn == 'kick':
for attempt in range(2 * nOrb * Ny):
i = int( np.random.random() * 2 * nOrb * Ny )
nUpTmp, nDownTmp = nUp[i], nDown[i]
nUp[i], nDown[i] = singleExcitation(nUp, nDown, i)
for kCount, k in enumerate(ks):
# Define the MF Hamiltonian for this iteration and k-point
K = HribbonKSpace2sublat(k, nOrb, Ny, E0, E1, E2, E3, E4)
Hup = K + U * np.eye(2 * nOrb * Ny) * nDown
Hdown = K + U * np.eye(2 * nOrb * Ny) * nUp
# Diagonalize
eUp[kCount, :], wUp[kCount, :, :] = la.eigh(Hup)
eDown[kCount, :], wDown[kCount, :, :] = la.eigh(Hdown)
# Save the previous fields to compare to test convergence
nUpOld = nUp.copy()
nDownOld = nDown.copy()
# Compute the chemical potential implicitly
mu = bisect(rootToChem2sublat, -50, 50,\
args = (eUp, eDown, beta, Nk, Ny, nHole) )
# Update fields
nUp, nDown = update2sublat(nUp, nDown, Nk, Ny,\
wUp, wDown, eUp, eDown, mu, beta, nOrb)
# Grandpotential per site
energyTmp = grandpotentialKSpace2sublat(U, nUp, nDown,\
nUpOld, nDownOld, invTemp, Nk, Ny, eUp, eDown, mu, abs_t0)
if energyTmp > energy:
nUp[i], nDown[i] = nUpTmp, nDownTmp
else:
energy = energyTmp
elif dyn == 'mixed':
for attempt in range(2 * nOrb * Ny):
i = int( np.random.random() * 2 * nOrb * Ny )
nUpTmp, nDownTmp = nUp[i], nDown[i]
nUp[i], nDown[i] = updateLocal2sublat(i, nUp, nDown, Nk, Ny,\
wUp, wDown, eUp, eDown, mu, beta, nOrb)
nUp[i], nDown[i] = singleExcitation(nUp, nDown, i)
for kCount, k in enumerate(ks):
# Define the MF Hamiltonian for this iteration and k-point
K = HribbonKSpace2sublat(k, nOrb, Ny, E0, E1, E2, E3, E4)
Hup = K + U * np.eye(2 * nOrb * Ny) * nDown
Hdown = K + U * np.eye(2 * nOrb * Ny) * nUp
# Diagonalize
eUp[kCount, :], wUp[kCount, :, :] = la.eigh(Hup)
eDown[kCount, :], wDown[kCount, :, :] = la.eigh(Hdown)
# Save the previous fields to compare to test convergence
nUpOld = nUp.copy()
nDownOld = nDown.copy()
# Compute the chemical potential implicitly
mu = bisect(rootToChem2sublat, -50, 50,\
args = (eUp, eDown, beta, Nk, Ny, nHole) )
# Update fields
nUp, nDown = update2sublat(nUp, nDown, Nk, Ny,\
wUp, wDown, eUp, eDown, mu, beta, nOrb)
# Grandpotential per site
energyTmp = grandpotentialKSpace2sublat(U, nUp, nDown,\
nUpOld, nDownOld, invTemp, Nk, Ny, eUp, eDown, mu, abs_t0)
if energyTmp > energy:
nUp[i], nDown[i] = nUpTmp, nDownTmp
else:
energy = energyTmp
if dyn == 'wait':
if ( it + 1 + itWait ) % singleExcitationFreq == 0 :
for attempt in range(nOrb * Ny * 2):
i = int( np.random.random() * nOrb * Ny * 2 )
nUpTmp, nDownTmp = nUp[i], nDown[i]
nUp[i], nDown[i] = singleExcitation(nUp, nDown, i)
for kCount, k in enumerate(ks):
# Define the MF Hamiltonian for this iteration and k-point
K = HribbonKSpace2sublat(k, nOrb, Ny, E0, E1, E2, E3, E4)
Hup = K + U * np.eye(2 * nOrb * Ny) * nDown
Hdown = K + U * np.eye(2 * nOrb * Ny) * nUp
# Diagonalize
eUp[kCount, :], wUp[kCount, :, :] = la.eigh(Hup)
eDown[kCount, :], wDown[kCount, :, :] = la.eigh(Hdown)
# Save the previous fields to compare to test convergence
nUpOld = nUp.copy()
nDownOld = nDown.copy()
# Compute the chemical potential implicitly
mu = bisect(rootToChem2sublat, -50, 50,\
args = (eUp, eDown, beta, Nk, Ny, nHole) )
# Update fields
nUp, nDown = update2sublat(nUp, nDown, Nk, Ny,\
wUp, wDown, eUp, eDown, mu, beta, nOrb)
# Grandpotential per site
energyTmp = grandpotentialKSpace2sublat(U, nUp, nDown,\
nUpOld, nDownOld, invTemp, Nk, Ny, eUp, eDown, mu, abs_t0)
if energyTmp > energy:
nUp[i], nDown[i] = nUpTmp, nDownTmp
else:
energy = energyTmp
energies[it] = energy
if invTemp == 'infty':
if energy < bestGrandpotential and beta >= betaThreshold:
bestGrandpotential = energy
nUpBest, nDownBest = nUp, nDown
eUpBest, eDownBest = eUp, eDown
wUpBest, wDownBest = wUp, wDown
else:
if energy < bestGrandpotential and beta == invTemp:
bestGrandpotential = energy
nUpBest, nDownBest = nUp, nDown
eUpBest, eDownBest = eUp, eDown
wUpBest, wDownBest = wUp, wDown
# Move to the next iteration
it += 1
# Save the last iteration
lastIt = it
return nUpBest, nDownBest, energies, bestGrandpotential, itSwitch,\
lastIt, mu, abs_t0, eUpBest, eDownBest,\
np.absolute(wUpBest.flatten('C'))**2, np.absolute(wDownBest.flatten('C'))**2
def solveForPrice(ap, goal):
fun = lambda p: sum(analyze(ap, price=p)[1].values()) - goal
if not fun(0) * fun(ap['startPrice'] * 1.5) < 0:
return 0
return bisect(fun, 0, ap['startPrice'] * 1.5)
#!/usr/bin/env python3
import math
from scipy.optimize import bisect
refr_idx = [1.0, 10/9, 10/8, 10/7, 10/6, 10/5]
delta_ys = [100/math.sqrt(2)-50] + [10]*5
total_delta_x_target = 100.0 / math.sqrt(2)
def total_delta_x_error(alpha0):
global delta_xs # save for time calculations
sin0 = math.sin(alpha0)
delta_xs = [delta_x(sin0, refr_idx[i], delta_ys[i]) for i in range(6)]
return sum(delta_xs) - total_delta_x_target
def delta_x(sin0, ior, dy):
# delta_y / slope
return dy / (1/(sin0 / math.sqrt(1 - (sin0/ior)**2) / ior))
alpha = bisect(total_delta_x_error, 0.1, math.pi/2-0.1)
sin = math.sin(alpha)
velocities = (10/refr_idx[i] for i in range(6))
lengths = ((dx**2 + dy **2) ** 0.5 for dx, dy in zip(delta_xs, delta_ys))
total_time = sum(l/v for l, v in zip(lengths, velocities))
print(f"{total_time:.10f}")
def get_energy(self, k, m, bracket, xtol):
start, stop = bracket
return bisect(self.equation, start, stop, args=(k, m),
xtol=xtol) # noqa
#!/usr/bin/env python3
# ----------------------------------------- #
#
import numpy as np
from scipy.linalg import solve
from scipy.sparse import diags
from scipy.optimize import bisect
L = 16
beta = bisect(f=(lambda x: np.cosh(x) * np.cos(x) + 1), a=5, b=10)
k = beta / L
print(k, k * L, np.cosh(k * L) * np.cos(k * L))
R = (np.cos(k * L) + np.cosh(k * L)) / (np.sin(k * L) + np.sinh(k * L))
print(R)
def y(x):
return 0.5 * ((np.cosh(k * x) - np.cos(k * x)) - R *
(np.sinh(k * x) - np.sin(k * x)))
print(y(L))
n = L
x = np.linspace(0.5, n - 0.5, n)
A4 = np.array([
np.ones(n - 2),
-4 * np.ones(n - 1),
6 * np.ones(n),
-4 * np.ones(n - 1),
