def jinc(self, x):
# I'm jury-rigging / hard coding this to work
# to test it
y = numpy.zeros(len(x))
if (x[0] == 0.0):
y[0] = 0.5
y[1:] = j1(x[1:]) / (x[1:])
else:
return j1(x) / x
def obsc_airy(x, h, w, q):
#Search for 0 in array - if not present, pass
try:
#loc = n.argwhere(x==0.)[0]
#x[loc[0],loc[1]] = 1.E-30 #Fix for 0. in array - using 1.E-30 seems to give good result for Airy function
loc = n.argwhere(x==0.)
x[loc] = 1.E-30
except IndexError:
pass
return h*((2*j1(n.sin(x)/w) - q*j1(q*(n.sin(x)/w)))/((n.sin(x)/w)*(1 - q**2)))**2
def cutoffHE1(b):
a = rho * b
i = ivp(1, u1*a) / (u1*a * i1(u1*a))
X = (1 / (u1*a)**2 + 1 / (u2*a)**2)
P = j1(u2*a) * y1(u2*b) - y1(u2*a) * j1(u2*b)
Ps = (jvp(1, u2*a) * y1(u2*b) - yvp(1, u2*a) * j1(u2*b)) / (u2 * a)
return (i * P + Ps) * (n12 * i * P + n22 * Ps) - n32 * X * X * P * P
def wave(self, **kw):
amplitude, r, a, w, f = (
kw.get(key, Human.aawf[key]) for key in (
'amplitude', 'radius', 'aperture', 'wavelength', 'focal'))
u = (pi*r*a)/(w*f)
if isinstance(r,float):
return amplitude*(1.0 if u == 0.0 else 2.0*j1(u)/u)
else:
u[u==0.0] = 1e-32
return amplitude*2.0*j1(u)/u
def wijCoefficients(n, x, y, rad1, rad2, theta, ellipticity):
wid, xcen, ycen = n, (n+1)/2, (n+1)/2
dx, dy = x - xcen, y - ycen
if n%2:
dx, dy = dx + 1, dy + 1
ftWij = numpy.zeros([wid, wid], dtype=complex)
scale = 1.0 - ellipticity
for iy in range(wid):
ky = float(iy - ycen)/wid
for ix in range(wid):
kx = float(ix - xcen)/wid
# rotate and rescale
cosT, sinT = numpy.cos(theta), numpy.sin(theta)
kxr, kyr = kx*cosT + ky*sinT, scale*(-kx*sinT + ky*cosT)
k = numpy.sqrt(kxr**2 + kyr**2)
# compute the airy terms, and apply shift theorem
if k != 0.0:
airy1 = rad1*special.j1(2.0*numpy.pi*rad1*k)/k
airy2 = rad2*special.j1(2.0*numpy.pi*rad2*k)/k
else:
airy1, airy2 = numpy.pi*rad1**2, numpy.pi*rad2**2
airy = airy2 - airy1
phase = numpy.exp(-1.0j*2.0*numpy.pi*(dx*kxr + dy*kyr))
ftWij[iy,ix] = phase*scale*airy
ftWijShift = numpy.fft.fftshift(ftWij)
wijShift = numpy.fft.ifft2(ftWijShift)
wij = numpy.fft.fftshift(wijShift)
return wij.real
def rootAiryIntensity(self, myxaxis, epsilon=0.0, showplot=False):
"""
This function computes 2*J1(x)/x, which can be squared to get an Airy disk.
myxaxis: the x-axis values to use
epsilon: radius of central hole in units of the dish diameter
Migrated from AnalysisUtils.py (revision 1.2204, 2015/02/18)
"""
if (epsilon > 0):
a = (2*spspec.j1(myxaxis)/myxaxis - \
epsilon**2*2*spspec.j1(myxaxis*epsilon)/(epsilon*myxaxis)) / (1-epsilon**2)
else:
a = 2*spspec.j1(myxaxis)/myxaxis # simpler formula for epsilon=0
return(a)
def funcCn(root, b):
"""
First term in the theta function
root = root from the zeta, Bi equation
b = shape factor where 2 sphere, 1 cylinder, 0 slab
"""
if b == 2:
Cn = 4*(np.sin(root)-root*np.cos(root)) / (2*root-np.sin(2*root))
elif b == 1:
Cn = (2/root) * (sp.j1(root) / (sp.j0(root)**2 + sp.j1(root)**2))
elif b == 0:
Cn = (4*np.sin(root)) / (2*root + np.sin(2*root))
return Cn
def jinc(self, x):
# a 'jinc' function
if (len(x) == 1):
if (x == 0.0):
return 0.5
else:
return j1(x)/x
else:
y = numpy.zeros(len(x))
index_0 = numpy.where(x == 0.0)
index_not_0
y[index_0] = 0.5
y[index_not_0] = j1(x[index_not_0])/x[index_not_0]
def j1_x_over_x(x,EPS=1E-8):
x = numpy.asarray(x)
if x.shape==():
if abs(x)<EPS:
return 1./3.
else:
return j1(x)/x
else:
x = numpy.asarray(x)
i = numpy.where(abs(x)<EPS)
x[i] = 1.
ret = j1(x)/x
ret[i] = 1./3.
return ret
def SNRcalc(self, pulsar, pop):
"""Calculate the S/N ratio of a given pulsar in the survey"""
# if not in region, S/N = 0
# if we have a list of pointings, use this bit of code
# haven't tested yet, but presumably a lot slower
# (loops over the list of pointings....)
# otherwise check if pulsar is in entire region
if self.inRegion(pulsar):
# If pointing list is provided, check how close nearest
# pointing is
if self.pointingslist is not None:
# convert offset from degree to arcmin
offset = self.inPointing(pulsar) * 60.0
else:
# calculate offset as a random offset within FWHM/2
offset = self.fwhm * math.sqrt(random.random()) / 2.0
else:
return -2
# Get degfac depending on self.gainpat
if self.gainpat == 'airy':
conv = math.pi/(60*180.) # Conversion arcmins -> radians
eff_diam = 3.0e8/(self.freq*self.fwhm*conv*1.0e6) # Also MHz -> Hz
a = eff_diam/2. # Effective radius of telescope
lamda = 3.0e8/(self.freq*1.0e6) # Obs. wavelength
kasin = (2*math.pi*a/lamda)*np.sin(offset*conv)
degfac = 4*(j1(kasin)/kasin)**2
else:
#### NOTE! HERE I WANT TO CHECK UNITS OF FWHM (ARCMIN???)
degfac = math.exp(-2.7726 * offset * offset / (self.fwhm *self.fwhm))
# Dunc's code here uses a ^-2.6 to convert frequencies
# don't think I need to do this - I'm using the frequency in call
Ttot = self.tsys + self.tskypy(pulsar)
# calc dispersion smearing across single channel
tdm = self._dmsmear(pulsar)
# calculate bhat et al scattering time (inherited from GalacticOps)
# in units of ms
tscat = go.scatter_bhat(pulsar.dm, pulsar.scindex, self.freq)
# Calculate the effective width
weff_ms = math.sqrt(pulsar.width_ms()**2 + self.tsamp**2 + tdm**2 + tscat**2)
# calculate duty cycle (period is in ms)
delt = weff_ms / pulsar.period
#print weff_ms, pulsar.period
# if pulse is smeared out, return -1.0
if delt > 1.0:
#print weff_ms, tscat, pulsar.dm, pulsar.gl, pulsar.gb, pulsar.dtrue
return -1
else:
return self._SNfac(pulsar, pop.ref_freq, degfac, Ttot) \
* math.sqrt((1.0 -delt)/delt)
def vis_disk(u, v, wavels, p): ''' Calculate the visibilities observed by an array on a binary star ---------------------------------------------------------------- p: 3-component vector (+2 optional), the binary "parameters": - p[0] = angular size of star (mas) - p[1] = linear limb-darkening coefficient - not done yet - p[2] = quadratic limb-darkening coefficient - not done yet - u,v: baseline coordinates (meters) - wavel: wavelength (meters) ---------------------------------------------------------------- ''' p = np.array(p) # relative locations th1 = mas2rad(p[0]) if np.size(p)>1: mu = p[1] else: mu = 0. # baselines into number of wavelength pref = 1./((1.-mu)/2. + mu/3.) eps = 1.e-20 x = np.sqrt(u*u+v*v)/wavels+eps v1 = pref*((1.-mu)*j1(np.pi*th1*x)/(np.pi*th1*x) + mu*np.sqrt(np.pi/2.)*jv(1.5,np.pi*th1*x)/((np.pi*th1*x)**1.5)) return np.abs(v1)
def makeAiryArr(maxRad,dd):
from numpy import pi,sqrt,arctan2,zeros,zeros_like
from scipy import where
from scipy.special import j1
pixCtr = maxRad / dd
##make position/radius Arrays
npix = int(2.*maxRad/dd + 1)
xArr = zeros((npix,npix))
yArr = zeros_like(xArr)
for x in range(npix):
xArr[x,:] = -maxRad + x*dd
yArr[:,x] = -maxRad + x*dd
radArr = sqrt(xArr**2 + yArr**2)
azArr = arctan2(yArr,xArr)
##remove zero
radArr = where(radArr == 0.,1.e-12,radArr)
###radius is the angle in units of lambda/d
##Make Airy function
bessArr = j1(pi*radArr)
airyArr = ((2./pi) * (bessArr / radArr))**2
return(xArr,yArr,airyArr)
def airyconv(xArr,yArr,airyArr,wid,radList):
from scipy.special import j1,j0
from numpy import pi,sum,ones,arange
from scipy import where
from scipy.signal import convolve2d
from beam import beam_azsm
nRad=len(radList)
dd=radList[2]-radList[1]
#print 'wid:',wid
#print 'dd:',dd
radListTot=arange(0,300,0.01)
radListTot=where(radListTot == 0.,1.e-12,radListTot)
airyList= ((2./pi) * (j1(pi*radListTot)/radListTot))**2
airyTot=sum(airyList*2.*pi*radListTot*dd)
#print 'airyTot:',airyTot
npKer=wid / dd
kernel=ones((npKer,npKer))
#kernel = kernel/size(kernel)
kernel = kernel * (dd**2.) / airyTot
convSame=convolve2d(airyArr,kernel,mode="same")
convAz=beam_azsm(xArr,yArr,convSame,radList,nsamp=360)
return(convAz)
def calc_ff(self, Q_VALS):
for q in Q_VALS:
temp = np.sin(self.theta)*np.cos(self.phi)+ np.sin(self.theta)*np.sin(self.phi)
if temp < -1:
a = np.floor(temp)
temp = temp - a
if temp > 1:
a = np.ceil(temp)
temp = temp - a
self.alpha = np.arccos(temp)
h,r,v,c, a= self.h,self.r,self.V,self.rho, self.alpha
inside = q*r*np.sin(a)
if a == 0:
J1 = 0
else:
J1 = special.j1(inside)/(inside)
inside = q*r*np.cos(a)
if inside == 0:
j0 = 1
else:
j0 = np.sin(inside)/inside
temp = 2*(2*3.14*r*h)*c*j0*J1
yield temp
def discreteModel(theta, uvsamples, bins):
# retrieve inputs
incl, PA, offset, w = theta
rin, b = bins
u, v = uvsamples # in **lambda** units
# convert angles to radians
inclr = np.radians(incl)
PAr = 0.5*np.pi-np.radians(PA)
offr = offset * np.pi / (180.*3600.)
# coordinate change to deal with projection, rotation, and shifts
#uprime = ((u-offr[0])*np.cos(PAr) + (v-offr[1])*np.sin(PAr)) * np.cos(inclr)
#vprime = (-(u-offr[0])*np.sin(PAr) + (v-offr[1])*np.cos(PAr))
uprime = (u*np.cos(PAr) + v*np.sin(PAr))
vprime = (-u*np.sin(PAr) + v*np.cos(PAr)) * np.cos(inclr)
rho = np.sqrt(uprime**2 + vprime**2) * np.pi / (180.*3600.)
# re-orient arrays
rbin = np.concatenate([np.array([rin]), b])
wbin = np.append(np.concatenate([np.array([0.0]), w]), 0.)
ww = wbin-np.roll(wbin, -1)
intensity = np.delete(ww, b.size+1)
# compute the visibilities
jarg = np.outer(2.*np.pi*rbin, rho)
jinc = sc.j1(jarg)/jarg
vis = np.dot(2.*np.pi*rbin**2*intensity, jinc)
# impart a phase center shift
shift = np.exp(-2.*np.pi*1.0j*((u*-offr[0]) + (v*-offr[1])))
model_vis = vis*shift
return model_vis
def d3sbModel(theta, uvsamples, bins):
incl = 0.
w = theta
#incl = np.deg2rad(theta[0]) #Projected Inclination
#w = theta[1:] #Ring bin amplitudes
u, v = uvsamples
udeproj = u * np.cos(incl) #Deproject
#vdeproj = v * np.cos(incl) #Deproject
rho = 1e3*np.sqrt(udeproj**2+v**2)
#rho = 1e3*np.sqrt(vdeproj**2+u**2)
rin, b = bins
rbin = np.concatenate([np.array([rin]), b])
wbin = np.append(np.concatenate([np.array([0.0]), w]), 0.)
ww = wbin-np.roll(wbin, -1)
wgt = np.delete(ww, b.size+1)
jarg = np.outer(2.*np.pi*rbin, rho/206264.806427)
jinc = sc.j1(jarg)/jarg
vis = np.dot(2.*np.pi*rbin**2*wgt, jinc)
return vis
def jinc(x):
"""Jinc function (J_1(x) / x)."""
j = np.ones(x.shape)
# Handle 0/0 at origin
nonzero_x = abs(x) > 1e-20
j[nonzero_x] = 2 * sp.j1(np.pi * x[nonzero_x]) / (np.pi * x[nonzero_x])
return j
def gravground( x ,
y ,
g ,
G ,
*args ,
**kwargs):
'''
def gravground( x ,
y ,
g ,
G ,
*args ,
**kwargs):
Create the Thomas-Fermi ground state for a gravitational system
'''
X,Y = np.meshgrid(x,y)
R = np.sqrt( X ** 2. + Y ** 2. )
bj0z1 = jn_zeros( 0, 1 ) #First zero of zeroth order besselj
scaling = np.sqrt( 2 * np.pi * G / g )
gr0 = bj0z1 / scaling
Rprime = R * scaling
gtfsol = j0( Rprime ) * np.array( [ map( int,ii ) for ii in map(
lambda rad: rad <= gr0, R ) ] )
gtfsol *= scaling ** 2. / ( 2 * np.pi * j1( bj0z1 ) * bj0z1 )
return gtfsol
def discreteModel(theta, uvsamples, bins):
# retrieve inputs
incl, PA, offset, w = theta
rin, b = bins
u, v = uvsamples # in **lambda** units
# convert angles to radians
inclr = np.radians(incl)
PAr = np.radians(PA)
offr = offset * np.pi / (180.*3600.)
# coordinate change to deal with projection, rotation, and shifts
uprime = (u * np.cos(PAr) + v * np.sin(PAr)) * np.cos(inclr)
vprime = (-u * np.sin(PAr) + v * np.cos(PAr))
rho = np.sqrt(uprime**2 + vprime**2) * np.pi / (180.*3600.)
# re-orient arrays
rbin = np.concatenate([np.array([rin]), b])
wbin = np.append(np.concatenate([np.array([0.0]), w]), 0.)
ww = wbin-np.roll(wbin, -1)
intensity = np.delete(ww, b.size+1)
jarg = np.outer(2.*np.pi*rbin, rho)
jinc = sc.j1(jarg)/jarg
vis = np.dot(2.*np.pi*rbin**2*intensity, jinc)
Re_vis = vis * np.cos(-2*np.pi*(u*offr[0] + v*offr[1]))
Im_vis = vis * np.sin(-2*np.pi*(u*offr[0] + v*offr[1]))
vis_complex = Re_vis + 1j*Im_vis
return vis_complex
def tetmConstants(self, ri, ro, neff, wl, EH, c, idx):
a = numpy.empty((2, 2))
n = self.maxIndex(wl)
u = self.u(ro, neff, wl)
urp = self.u(ri, neff, wl)
if neff < n:
B1 = j0(u)
B2 = y0(u)
F1 = j0(urp) / B1
F2 = y0(urp) / B2
F3 = -j1(urp) / B1
F4 = -y1(urp) / B2
c1 = wl.k0 * ro / u
else:
B1 = i0(u)
B2 = k0(u)
F1 = i0(urp) / B1
F2 = k0(urp) / B2
F3 = i1(urp) / B1
F4 = -k1(urp) / B2
c1 = -wl.k0 * ro / u
c3 = c * c1
a[0, 0] = F1
a[0, 1] = F2
a[1, 0] = F3 * c3
a[1, 1] = F4 * c3
return numpy.linalg.solve(a, EH.take(idx))
def _tmcoeq(self, v0, nu):
u1r1, u2r1, u2r2, s1, s2, n1sq, n2sq, n3sq = self.__params(v0)
if s1 == 0: # e
f11a, f11b = 2, 1
elif s1 > 0: # a, b, d
f11a, f11b = j0(u1r1) * u1r1, j1(u1r1)
else: # c
f11a, f11b = i0(u1r1) * u1r1, i1(u1r1)
if s2 > 0:
f22a, f22b = j0(u2r2), y0(u2r2)
f2a = j1(u2r1) * f22b - y1(u2r1) * f22a
f2b = j0(u2r1) * f22b - y0(u2r1) * f22a
else: # a
f22a, f22b = i0(u2r2), k0(u2r2)
f2a = i1(u2r1) * f22b + k1(u2r1) * f22a
f2b = i0(u2r1) * f22b - k0(u2r1) * f22a
return f11a * n2sq * f2a - f11b * n1sq * f2b * u2r1
def phase_binary(u, v, wavel, p): ''' Calculate the phases observed by an array on a binary star ---------------------------------------------------------------- p: 3-component vector (+2 optional), the binary "parameters": - p[0] = sep (mas) - p[1] = PA (deg) E of N. - p[2] = contrast ratio (primary/secondary) optional: - p[3] = angular size of primary (mas) - p[4] = angular size of secondary (mas) - u,v: baseline coordinates (meters) - wavel: wavelength (meters) ---------------------------------------------------------------- ''' p = np.array(p) # relative locations th = (p[1] + 90.0) * np.pi / 180.0 ddec = mas2rad(p[0] * np.sin(th)) dra = -mas2rad(p[0] * np.cos(th)) # baselines into number of wavelength x = np.sqrt(u*u+v*v)/wavel # decompose into two "luminosity" l2 = 1. / (p[2] + 1) l1 = 1 - l2 # phase-factor phi = np.zeros(u.size, dtype=complex) phi.real = np.cos(-2*np.pi*(u*dra + v*ddec)/wavel) phi.imag = np.sin(-2*np.pi*(u*dra + v*ddec)/wavel) # optional effect of resolved individual sources if p.size == 5: th1, th2 = mas2rad(p[3]), mas2rad(p[4]) v1 = 2*j1(np.pi*th1*x)/(np.pi*th1*x) v2 = 2*j1(np.pi*th2*x)/(np.pi*th2*x) else: v1 = np.ones(u.size) v2 = np.ones(u.size) cvis = l1 * v1 + l2 * v2 * phi phase = np.angle(cvis, deg=True) return np.mod(phase + 10980., 360.) - 180.0
def _jinc(x):
"""Internal implementation of the `jinc` function.
The `jinc(x)` is defined as `J_1(x)/2x` and `jinc(0)=0.5`.
"""
mask = x != 0.0
result = 0.5*_np.ones(x.shape)
result[mask] = _sps.j1(x[mask])/(x[mask])
return result
def _tefield(self, wl, nu, neff, r):
rho = self.fiber.outerRadius(0)
k = wl.k0
nco2 = self.fiber.maxIndex(0, wl)**2
ncl2 = self.fiber.minIndex(1, wl)**2
u = rho * k * sqrt(nco2 - neff**2)
w = rho * k * sqrt(neff**2 - ncl2)
if r < rho:
hz = -Y0 * u / (k * rho) * j0(u * r / rho) / j1(u)
ephi = -j1(u * r / rho) / j1(u)
else:
hz = Y0 * w / (k * rho) * k0(w * r / rho) / k1(w)
ephi = -k1(w * r / rho) / k1(w)
hr = -neff * Y0 * ephi
return numpy.array((0, ephi, 0)), numpy.array((hr, 0, hz))
def airyconv_MG(wid,radList):
from scipy.special import j1,j0
from numpy import pi,sum,ones,arange,zeros,sqrt
from scipy import where
nRad=len(radList)
dd=radList[2]-radList[1]
#print 'wid:',wid
#print 'dd:',dd
radListTot=arange(0,300,0.01)
radListTot=where(radListTot == 0.,1.e-12,radListTot)
airyList= ((2./pi) * (j1(pi*radListTot)/radListTot))**2
airyTot=sum(airyList*2.*pi*radListTot*dd)
conv1=zeros(nRad)
conv2=zeros(nRad)
npKer=wid/dd
xList0=arange(-wid/2.,wid/2.+dd,dd)
npKer=len(xList0)
xArr0=zeros((npKer,npKer))
yArr0=zeros((npKer,npKer))
for x in range(npKer):
xArr0[x,:]=xList0[x]
yArr0[:,x]=xList0[x]
for r in range(nRad):
##move along axis
radArr=sqrt((xArr0+radList[r])**2 + yArr0**2)
radArr=where(radArr == 0.,1.e-12,radArr) ##prevent div/0 errors
bessArr = j1(pi*radArr)
airyArr = ((2./pi) * (bessArr / radArr))**2
conv1[r] = sum(airyArr)*dd**2./airyTot
##move at 45 degrees
radArr=sqrt((xArr0+radList[r]/sqrt(2))**2 + (yArr0+radList[r]/sqrt(2))**2)
radArr=where(radArr == 0.,1.e-12,radArr) ##prevent div/0 errors
bessArr = j1(pi*radArr)
airyArr = ((2./pi) * (bessArr / radArr))**2
conv2[r] = sum(airyArr)*dd**2./airyTot
convAz=(conv1 + conv2)/2.
return(conv1,conv2,convAz)
def _j1(self, x):
if self._nu == -1:
return j0(x)
elif self._nu == 0:
return j1(x)
elif np.floor(self._nu) == self._nu:
return jn(self._nu + 1, x)
else:
return jv(self._nu + 1, x)
def plotCF():
ncl = 1.444
nco = ncl + 0.05
V = numpy.linspace(1, 7)
pyplot.plot(V, j0(V))
pyplot.plot(V, j1(V))
pyplot.plot(V, jn(2, V))
n02 = ncl*ncl / (nco * nco)
a = (1 - n02) / (1 + n02)
pyplot.plot(V, a * jn(2, V) - j0(V))
pyplot.plot(V, a * jn(3, V) - j1(V))
pyplot.plot(V, a * jn(4, V) - jn(2, V))
pyplot.axhline(0, ls='--', color='k')
pyplot.show()
def funcZetaCyl(z, Bi):
"""
zeta, Bi function for cylinder as f(z) = z*J1(z) - Bi*J0(z)
note that J1 and J0 are bessel functions
z = zeta values which are later solved for the positive roots for theta
Bi = Biot number h*L/k, (-)
"""
f = z*sp.j1(z) - Bi*sp.j0(z)
return f
def _tmfield(self, wl, nu, neff, r):
rho = self.fiber.outerRadius(0)
k = wl.k0
nco2 = self.fiber.maxIndex(0, wl)**2
ncl2 = self.fiber.minIndex(1, wl)**2
u = rho * k * sqrt(nco2 - neff**2)
w = rho * k * sqrt(neff**2 - ncl2)
if r < rho:
ez = -u / (k * neff * rho) * j0(u * r / rho) / j1(u)
er = j1(u * r / rho) / j1(u)
hphi = Y0 * nco2 / neff * er
else:
ez = nco2 / ncl2 * w / (k * neff * rho) * k0(w * r / rho) / k1(w)
er = nco2 / ncl2 * k1(w * r / rho) / k1(w)
hphi = Y0 * nco2 / ncl2 * k1(w * r / rho) / k1(w)
return numpy.array((er, 0, ez)), numpy.array((0, hphi, 0))
def _j1(self, x):
if self._nu == -1:
return j0(x)
elif self._nu == 0:
return j1(x)
elif isinstance(self._nu, int):
return jn(self._nu + 1, x)
else:
return jv(self._nu + 1, x)
def plot_temp_ana(name):
# parameters
mu = 0.01
E = 30000 # Young's modulus
v = 0.2 # poisson's ratio
mu_s = E / (2 + 2 * v) # lame coefficient
lambda_s = 2 * mu_s * v / (1 - 2 * v) # lame coefficient
HA = lambda_s + 2 * mu_s # aggregate modulus
r0 = 1.0 # radius 1m
ubc = -0.05 # displacement boundary
epsilon0 = ubc / r0 # strain
kappa = 1e-10 # permeability
vis = 1e-3 # viscosity
K = kappa / vis # hydraulic conductivity
t = 10 # unit second
eta = (1 - v) / (1 - 2 * v)
tol = 1e-10 # tolerance to stop
r = np.arange(0, 1.05, 0.05) # the different locations along the axises
u1 = np.zeros(21)
for i in range(len(r)):
u1[i] = r0 * epsilon0 * (1 - eta) / (2 * eta - 1) * r[i] / r0
u2 = np.zeros(21)
for i in range(len(r)):
error = 1e10 # initial error
n = 1 # iteration calculator
u0 = 1e10 # initial guess for displacement
while error > tol:
a1 = sp.j1(roots[n] * r[i] / r0)
a2 = roots[n] * sp.j0(
roots[n]) * (2 * eta - 1 - eta * eta * roots[n] * roots[n])
a3 = math.exp(-roots[n] * roots[n] * t * HA * K / (r0 * r0))
# u2[i] += (HA - lambda_s) / (
# roots[n] * ((HA + lambda_s) * sp.j0(roots[n]) - roots[n] * HA * sp.j1(roots[n]))) * sp.j1(
# roots[n] * r[i] / r0) * math.exp(-HA * K / (r0 ** 2) * (roots[n] ** 2) * t)
u2[i] += a1 / a2 * a3
error = np.fabs(u2[i] - u0)
u0 = u2[i]
n += 1
u2 = u2 * r0 * epsilon0
u = u1 + u2
# np.savetxt('test',R)
setupMatplotlib()
plt.close('all')
# fig, ((ax1, ax2), (ax3, ax4)) = plt.subplots(2, 2, sharex='col', sharey='row')
# ax.plot(r, s_tt, label='analytical')
fig = plt.figure(figsize=(6, 4))
ax = fig.add_subplot(111)
# lns1 = ax.plot(data.arc_length, data.displacement1, "o", mec="red", mfc="none", label='ogs6')
lns2 = ax.plot(r, u, "b--", label='analytical')
ax.set_xlabel('radius (m)')
ax.set_ylabel('Displacment_x (m)', color='b')
legend = ax.legend(loc='upper right', shadow=True)
plt.title('Unconfined Consolidation')
ax.grid()
'''fig = plt.figure(figsize=(6, 4))
# fig, axes = plt.subplots(nrows=2, ncols=2)
ax = fig.add_subplot(111)
ay = ax.twinx()
ax1 = fig.add_subplot(221)
# fig, ax1 = plt.subplots()
ax2 = ax1.twinx()
ax1.set_title('2 months')
#lns1 = ax1.plot(data1_2m.arc_length, data1_2m.TEMPERATURE1, "b", label='ogs5')
ax1.plot(data2_2m.arc_length, T2months_num, "o", mec="red", mfc="none", label='ogs6')
ax1.plot(x, T2months, "b--", label='analytical')
ax2.plot(x, R1, "black", label='error')
ax3 = fig.add_subplot(222)
ax4 = ax3.twinx()
ax3.set_title('1 year')
#ax3.plot(data1_1y.arc_length, data1_1y.TEMPERATURE1, "b", label='ogs5')
ax3.plot(data2_1y.arc_length, T1year_num, "o", mec="red", mfc="none", label='ogs6')
ax3.plot(x, T1year, "g--", label='analytical')
ax4.plot(x, R2, "black", label='error')
ax5 = fig.add_subplot(223)
ax6 = ax5.twinx()
ax5.set_title('2 years')
#ax5.plot(data1_2y.arc_length, data1_2y.TEMPERATURE1, "b", label='ogs5')
ax5.plot(data2_2y.arc_length, T2years_num, "o", mec="red", mfc="none", label='ogs6')
ax5.plot(x, T2years, "g--", label='analytical')
ax6.plot(x, R3, "black", label='error')
ax7 = fig.add_subplot(224)
ax8 = ax7.twinx()
ax7.set_title('4 years')
#ax7.plot(data1_4y.arc_length, data1_4y.TEMPERATURE1, "b", label='ogs5')
ax7.plot(data2_4y.arc_length, T4years_num, "o", mec="red", mfc="none", label='ogs6')
ax7.plot(x, T4years, "g--", label='analytical')
ax8.plot(x, R4, "black", label='error')
# Turn off axis lines and ticks of the big subplot
ax.spines['top'].set_color('none')
ax.spines['bottom'].set_color('none')
ax.spines['left'].set_color('none')
ax.spines['right'].set_color('none')
ay.spines['top'].set_color('none')
ay.spines['bottom'].set_color('none')
ay.spines['left'].set_color('none')
ay.spines['right'].set_color('none')
ax.tick_params(labelcolor='w', top='off', bottom='off', left='off', right='off')
ay.tick_params(labelcolor='w', top='off', bottom='off', left='off', right='off')
ax.set_xlabel('Distance (m)')
ax.set_ylabel('Temperature (K)')
ay.set_ylabel('Error')
ax1.legend()
ax3.legend()
ax5.legend()
ax7.legend()
# ax1.set_title('2 months')
# commonFormat(ax)
# ax.set_xlabel('Distance', fontsize=16)
# ax.xaxis.set_label_coords(0.5,-0.05)
# ax.set_ylabel('Temperature', fontsize=18)
# ax.set_xlim(right=10)
# ax.set_ylim(bottom=3.e-2)
# ax.set_ylim(top = 7.e-2)
# ax.set_yscale('log')
# ax.legend(loc='lower right')
# ax.yaxis.set_label_coords(-0.05, 0.5) '''
fig.tight_layout()
fig.savefig(name)
return None
def getAiryProbe(
wavelength=0.142e-9, # 8.7 keV
pixel_pitch=30e-9, # 55 micrometers
npix=32,
n_photons=1e6,
beam_diam_pixels=5):
"""Calculates the beam profile given the final beam diameter.
Parameters:
wavelength :
self-explanatory
pixel_pitch :
object/probe pixel pitch. Usually calculated using the Nyquist theorem using the object-detector
distance and the detector pixel pitch.
n_pix:
number of pixels along each axis in the probe view
n_photons:
self-explanatory
beam_diam_pixels:
the diameter (in pixels) of the first central lobe of the probe beam at the object (sample) plane.
Assumption:
- propagation distance (from aperture to sample) and initial aperture width are calculated
assuming a Fresnel number of 0.1
"""
beam_width_dist = beam_diam_pixels * pixel_pitch
radius = beam_width_dist / 2
# Initial Aperture width
w = 0.1 * 2 * np.pi * radius / (special.jn_zeros(1, 1))
# Propagation dist from aperture to sample
z = 0.1 * (2 * np.pi * radius)**2 / (special.jn_zeros(1, 1)**2 *
wavelength)
beam_center = npix // 2
xvals = np.linspace(-beam_center * pixel_pitch, beam_center * pixel_pitch,
npix)
xx, yy = np.meshgrid(xvals, xvals)
k = 2 * np.pi / wavelength
lz = wavelength * z
S = xx**2 + yy**2
jinc_input = np.sqrt(S) * w / lz
mask = (jinc_input != 0)
jinc_term = np.pi * np.ones_like(jinc_input)
jinc_term[mask] = special.j1(
jinc_input[mask] * 2 * np.pi) / jinc_input[mask]
# wavefield
#term1 = np.exp(1j * k * z) / (1j * lz)
term1 = 1 / (1j * lz)
term2 = np.exp(1j * k * S / (2 * z))
term3 = w**2 * jinc_term
field_vals = (term1 * term2 * term3).astype('complex64')
scaling_factor = np.sqrt(n_photons / (np.abs(field_vals)**2).sum())
field_vals = scaling_factor * field_vals
return field_vals
def sombrero():
a0 = 2.
S = lambda r: a0 * scs.j1(a0 * r) / r
H = lambda r: r < a0
return S, H
def SNRcalc(self, pulsar, pop):
"""Calculate the S/N ratio of a given pulsar in the survey"""
# if not in region, S/N = 0
# if we have a list of pointings, use this bit of code
# haven't tested yet, but presumably a lot slower
# (loops over the list of pointings....)
if pulsar.dead:
return 0.
# otherwise check if pulsar is in entire region
if self.inRegion(pulsar):
# If pointing list is provided, check how close nearest
# pointing is
if self.pointingslist is not None:
# convert offset from degree to arcmin
offset = self.inPointing(pulsar) * 60.0
else:
# calculate offset as a random offset within FWHM/2
offset = self.fwhm * math.sqrt(random.random()) / 2.0
else:
return -2
# Get degfac depending on self.gainpat
if self.gainpat == 'airy':
conv = math.pi / (60 * 180.) # Conversion arcmins -> radians
eff_diam = 3.0e8 / (self.freq * self.fwhm * conv * 1.0e6
) # Also MHz -> Hz
a = eff_diam / 2. # Effective radius of telescope
lamda = 3.0e8 / (self.freq * 1.0e6) # Obs. wavelength
kasin = (2 * math.pi * a / lamda) * np.sin(offset * conv)
degfac = 4 * (j1(kasin) / kasin)**2
else:
degfac = math.exp(-2.7726 * offset * offset /
(self.fwhm * self.fwhm))
# calc dispersion smearing across single channel
tdm = self._dmsmear(pulsar)
# calculate bhat et al scattering time (inherited from GalacticOps)
# in units of ms
tscat = go.scatter_bhat(pulsar.dm, pulsar.scindex, self.freq)
# Calculate the effective width
width_ms = pulsar.width_degree * pulsar.period / 360.0
weff_ms = math.sqrt(width_ms**2 + self.tsamp**2 + tdm**2 + tscat**2)
# calculate duty cycle (period is in ms)
delta = weff_ms / pulsar.period
# if pulse is smeared out, return -1.0
if delta > 1.0:
return -1
#radiometer signal to noise
sig_to_noise = rad.calcSNR(self.calcflux(pulsar,
pop.ref_freq), self.beta,
self.tsys, self.tskypy(pulsar), self.gain,
self.npol, self.tobs, self.bw, delta)
# account for aperture array, if needed
if self.AA:
sig_to_noise *= self._AA_factor(pulsar)
# return the S/N accounting for beam offset
return sig_to_noise * degfac
# jn_zeros calculates the first n zeros of the zero-th order Bessel function
# of the first kind
max_zeros = 10
w = jn_zeros(0, max_zeros)
b = 2 * np.sqrt(L / g)
def func(u, n):
"""The integrand for the Fourier-Bessel coefficient A_n."""
return u * f(u) * j0(w[n - 1] / b * u)
# Calculate the Fourier-Bessel coefficients using numerical integration
A = []
for n in range(1, max_zeros + 1):
An = quad(func, 0, b, args=(n, ))[0] * 2 / (b * j1(w[n - 1]))**2
A.append(An)
# Create a labelled plot comparing f(z) with the Fourier-Bessel series
# approximation truncated at 2, 3 and 5 terms.
plt.plot(f(u), z, 'k', lw=2, label='$f(z)$')
for nmax in (3, 5, 10):
ffit = np.zeros(N)
for n in range(nmax):
ffit += A[n - 1] * j0(w[n - 1] / b * u)
plt.plot(ffit, z, label=r'$n_\mathrm{max} = ' + '{}$'.format(nmax))
plt.xlim(-d / 5, d)
plt.xlabel(r'$x$')
plt.ylabel(r'$z$')
plt.legend()
# the loop repeats
# what could be added is a clause that says for r <= 10**-10 r = 0.01? (small)
# can include the line with math.log( ... , 10) to log base 10 the function so it fits better on my
# graph do i need to multiply by the dx here or? not sure
for i in x:
cj = 0
for j in y:
r = x[ci]**2 + y[cj]**2
I_theta_arr[ci][cj] = ((2 * sp.j1(r) / r)**2) * dx
cj = cj + 1
ci = ci + 1
# these help make the lengths more identifiable, can be uncommented if needed
# print(r)
#
# print(I_theta_arr)
# print(I_theta_arr.size)
# print(x.size)
# print(y.size)
# print(r.size)
res_sc_br = minimize_scalar(f, method='brent')
print(res_sc_br.x)
print(res_sc_br.fun)
print(f(res_sc_br.x))
#--------------------------------------------------
# Bounded method: Need to specify an interval.
#--------------------------------------------------
from scipy.special import j1
# Plot this function to show an optimum.
x_grid = np.arange(4, 7, 0.01)
f_grid = x_grid * 0
for i in range(0, len(x_grid)):
f_grid[i] = j1(x_grid[i])
plt.figure()
plt.plot(x_grid, f_grid, label='J_1(x)')
plt.xlabel('x')
plt.ylabel('J_(x)')
plt.show()
# Search for good bounds first:
j1(4)
j1(5)
j1(7)
res_sc_bdd = minimize_scalar(j1, bounds=(4, 7), method='bounded')
print(res_sc_bdd.x)
arg3_DOP = [
Lav, betst, Inc, Ombn, alph, n0, Rde, pp, thetTst, J3, aeff, nu0, nne
]
arg4_DOP = [
Lav, betst, Inc, Ombn, alph, n0, Rde, pp, thetTst, J4, aeff, nu0, nne
]
arg5_DOP = [
Lav, betst, Inc, Ombn, alph, n0, Rde, pp, thetTst, J5, aeff, nu0, nne
]
Dop1_anal = 1. / (2. * ma.pi * frc_a)**2 * np.sin(Inc - np.pi / 2) * (
2. * ma.pi * frc_a * np.cos(2. * ma.pi * frc_a * np.cos(thetTst)) -
np.sin(2. * ma.pi * frc_a * np.cos(thetTst)) / np.cos(thetTst))
from scipy import special as spc
DopRing_anal = spc.j1(2. * ma.pi * frc_a)
DOP1_AIR_over_AUV = np.zeros(len(frc_t))
DOP2_AIR_over_AUV = np.zeros(len(frc_t))
DOP3_AIR_over_AUV = np.zeros(len(frc_t))
DOP4_AIR_over_AUV = np.zeros(len(frc_t))
DOP5_AIR_over_AUV = np.zeros(len(frc_t))
for i in range(len(frc_t)):
DOP1_AIR_over_AUV[i] = DOP_AIR_o_AUV(frc_t[i], numn, numx, Dst,
arg1_DOP, RHS_table, T_table)
DOP2_AIR_over_AUV[i] = DOP_AIR_o_AUV(frc_t[i], numn, numx, Dst,
arg2_DOP, RHS_table, T_table)
DOP3_AIR_over_AUV[i] = DOP_AIR_o_AUV(frc_t[i], numn, numx, Dst,
arg3_DOP, RHS_table, T_table)
DOP4_AIR_over_AUV[i] = DOP_AIR_o_AUV(frc_t[i], numn, numx, Dst,
def calc_P_clad(omega, a, beta, u, w, A):
return ((np.pi / 2) * omega * mu_0 * beta * np.abs(A)**2 * (a**4 / u**2) *
j1(u)**2 * (((k0(w) * kv(2, w)) / k1(w)**2) - 1))
def Hr_core(r, phase, beta, a, u, A):
return 1j * beta * (a / u) * A * j1(u * r / a) * np.exp(1j * phase)
def Ep_core(r, phase, omega, a, u, A):
return -1j * omega * mu_0 * (a / u) * A * j1(u * r / a) * np.exp(
1j * phase)
def eve_lhs(u, w):
return j1(u) / (u * j0(u))
def jinc(x):
return spl.j1(x) / x
G_tot = 72.41
A = np.exp(-tau * G_tot)
m = (DWF * U * dk * tau * 1e-9)**2 * sqrt(N) / (4 * sqrt(2) *
sqrt(A**(-2) - 1))
plt.plot(amps_th,
fit(amps_th, m, 2),
'--',
label=r'{:.3f}*z^({:.3f})'.format(m, 2))
amps_th = np.linspace(1, 3, 500)
pwr = np.linspace(0, 1, 500)
theta_max_2 = np.linspace(2, 2, 300)
amps_th_2 = np.linspace(10, 0.25, 300)
theta_max = np.append(theta_max_2, theta_max)
amps_th = np.append(amps_th_2, amps_th)
Pup = 1 / 2 - 1 / 2 * A * j0(theta_max)
sig_proj = 1 / sqrt(55) * sqrt(Pup * (1 - Pup))
sig_up = (A**2) / 8 * (1 + j0(2 * theta_max) - 2 * j0(theta_max)**2)
delta_J_m = 2 * A**(-1) * sqrt(sig_proj**2 + sig_up)
delta_th_m = delta_J_m / (j1(theta_max))
S_N_m = theta_max / delta_th_m / 2
#plt.plot(amps_th,S_N_m,'--')
plt.legend(loc=(1, 0))
plt.ylim(0, 2)
#plt.xlim(0,.5)
plt.xlabel('Displacement Amplitude (nm)')
plt.ylabel('Signal to noise')
etaV = etaV[0, :]
zetaH = zetaH[0, :]
zetaV = zetaV[0, :]
rtol = htarg['rtol']
atol = htarg['atol']
nquad = htarg['nquad']
maxint = htarg['maxint']
pts_per_dec = htarg['pts_per_dec']
diff_quad = htarg['diff_quad']
a = htarg['a']
b = htarg['b']
limit = htarg['limit']
g_x, g_w = special.roots_legendre(nquad)
b_zero = np.pi * np.arange(1.25, maxint + 1)
for i in range(10):
b_x0 = special.j1(b_zero)
b_x1 = special.jv(2, b_zero)
b_h = -b_x0 / (b_x0 / b_zero - b_x1)
b_zero += b_h
if all(np.abs(b_h) < 8 * np.finfo(float).eps * b_zero):
break
xint = np.concatenate((np.array([1e-20]), b_zero))
dx = np.repeat(np.diff(xint) / 2, nquad)
Bx = dx * (np.tile(g_x, maxint) + 1) + np.repeat(xint[:-1], nquad)
BJ0 = special.j0(Bx) * np.tile(g_w, maxint)
BJ1 = special.j1(Bx) * np.tile(g_w, maxint)
intervals = xint / off[:, None]
lambd = Bx / off[:, None]
ang_fact = kernel.angle_factor(angle, ab, msrc, mrec)
# 1 Spline version
start = np.log(lambd.min())
def airy_fun(x, centre, amp): # , exp): # , amp, bg):
with np.errstate(divide='ignore', invalid='ignore'):
return np.where((x - centre) == 0, amp * .5**2,
amp * (special.j1(x - centre) / (x - centre))**2)
def update_Theta(self, p):
'''
Update the model to the current Theta parameters.
:param p: parameters to update model to
:type p: model.ThetaParam
'''
# durty HACK to get fixed logg
# Simply fixes the middle value to be 4.29
# Check to see if it exists, as well
fix_logg = Starfish.config.get("fix_logg", None)
if fix_logg is not None:
p.grid[1] = fix_logg
print("grid pars are", p.grid)
self.logger.debug("Updating Theta parameters to {}".format(p))
# Store the current accepted values before overwriting with new proposed values.
self.flux_mean_last = self.flux_mean.copy()
self.flux_std_last = self.flux_std.copy()
self.eigenspectra_last = self.eigenspectra.copy()
self.mus_last = self.mus
self.C_GP_last = self.C_GP
# Local, shifted copy of wavelengths
wl_FFT = self.wl_FFT * np.sqrt((C.c_kms + p.vz) / (C.c_kms - p.vz))
# If vsini is less than 0.2 km/s, we might run into issues with
# the grid spacing. Therefore skip the convolution step if we have
# values smaller than this.
# FFT and convolve operations
if p.vsini < 0.0:
raise C.ModelError("vsini must be positive")
elif p.vsini < 0.2:
# Skip the vsini taper due to instrumental effects
eigenspectra_full = self.EIGENSPECTRA.copy()
else:
FF = np.fft.rfft(self.EIGENSPECTRA, axis=1)
# Determine the stellar broadening kernel
ub = 2. * np.pi * p.vsini * self.ss
sb = j1(ub) / ub - 3 * np.cos(ub) / (
2 * ub**2) + 3. * np.sin(ub) / (2 * ub**3)
# set zeroth frequency to 1 separately (DC term)
sb[0] = 1.
# institute vsini taper
FF_tap = FF * sb
# do ifft
eigenspectra_full = np.fft.irfft(FF_tap, self.pca.npix, axis=1)
# Spectrum resample operations
if min(self.wl) < min(wl_FFT) or max(self.wl) > max(wl_FFT):
raise RuntimeError(
"Data wl grid ({:.2f},{:.2f}) must fit within the range of wl_FFT ({:.2f},{:.2f})"
.format(min(self.wl), max(self.wl), min(wl_FFT), max(wl_FFT)))
# Take the output from the FFT operation (eigenspectra_full), and stuff them
# into respective data products
for lres, hres in zip(
chain([self.flux_mean, self.flux_std], self.eigenspectra),
eigenspectra_full):
interp = InterpolatedUnivariateSpline(wl_FFT, hres, k=5)
lres[:] = interp(self.wl)
del interp
# Helps keep memory usage low, seems like the numpy routine is slow
# to clear allocated memory for each iteration.
gc.collect()
# Adjust flux_mean and flux_std by Omega
Omega = 10**p.logOmega
self.flux_mean *= Omega
self.flux_std *= Omega
# Now update the parameters from the emulator
# If pars are outside the grid, Emulator will raise C.ModelError
self.emulator.params = p.grid
self.mus, self.C_GP = self.emulator.matrix
def lnprob(p):
grid = p[:3]
vz, vsini, logOmega = p[3:]
# Local, shifted copy of wavelengths
wl_FFT = wl_FFT_orig * np.sqrt((C.c_kms + vz) / (C.c_kms - vz))
# Holders to store the convolved and resampled eigenspectra
eigenspectra = np.empty((pca.m, ndata))
flux_mean = np.empty((ndata, ))
flux_std = np.empty((ndata, ))
# If vsini is less than 0.2 km/s, we might run into issues with
# the grid spacing. Therefore skip the convolution step if we have
# values smaller than this.
# FFT and convolve operations
if vsini < 0.0:
return -np.inf
elif vsini < 0.2:
# Skip the vsini taper due to instrumental effects
eigenspectra_full = EIGENSPECTRA.copy()
else:
FF = np.fft.rfft(EIGENSPECTRA, axis=1)
# Determine the stellar broadening kernel
ub = 2. * np.pi * vsini * ss
sb = j1(ub) / ub - 3 * np.cos(ub) / (2 * ub**2) + 3. * np.sin(ub) / (
2 * ub**3)
# set zeroth frequency to 1 separately (DC term)
sb[0] = 1.
# institute vsini taper
FF_tap = FF * sb
# do ifft
eigenspectra_full = np.fft.irfft(FF_tap, pca.npix, axis=1)
# Spectrum resample operations
if min(wl) < min(wl_FFT) or max(wl) > max(wl_FFT):
raise RuntimeError(
"Data wl grid ({:.2f},{:.2f}) must fit within the range of wl_FFT ({:.2f},{:.2f})"
.format(min(wl), max(wl), min(wl_FFT), max(wl_FFT)))
# Take the output from the FFT operation (eigenspectra_full), and stuff them
# into respective data products
for lres, hres in zip(chain([flux_mean, flux_std], eigenspectra),
eigenspectra_full):
interp = InterpolatedUnivariateSpline(wl_FFT, hres, k=5)
lres[:] = interp(wl)
del interp
gc.collect()
# Adjust flux_mean and flux_std by Omega
Omega = 10**logOmega
flux_mean *= Omega
flux_std *= Omega
# Now update the parameters from the emulator
# If pars are outside the grid, Emulator will raise C.ModelError
try:
emulator.params = grid
mus, C_GP = emulator.matrix
except C.ModelError:
return -np.inf
# Get the mean spectrum
X = (flux_std * np.eye(ndata)).dot(eigenspectra.T)
mean_spec = flux_mean + X.dot(mus)
R = fl - mean_spec
# Evaluate chi2
lnp = -0.5 * np.sum((R[mask] / sigma[mask])**2)
return lnp
#All parameters are in Ang
#This is for circular annular ap in 1d
import matplotlib.pyplot as plt
import numpy as np
import scipy.special as spl
q1 = np.linspace(-2 * 10**7, 2 * 10**7, 512)
a = 1 * 10**7
k = 2 * 3.14 / 6000
f = 10**10
e = 0.6
m = q1 * k * a / f
n = k * a * e * q1 / f
z = 1 / (1 - e**2) * ((2 * spl.j1(m) / m) - (e) * (2 * spl.j1(n) / m))
plt.xlabel('q1*a*k/f')
plt.ylabel('intensity')
plt.plot(q1 * a * k / f, z**2)
plt.show()
def __init__(self, L, R, n_modes_z, n_modes_r):
self.n_modes_r = n_modes_r
self.n_modes_z = n_modes_z
self.domain_L = L
self.domain_R = R
self.kr = jn_zeros(0, self.n_modes_r) / R
self.kz = np.pi * np.arange(1, self.n_modes_z + 1) / L
self.oneOkr = 1. / self.kr
self.oneOkz = 1. / self.kz
# Needed for the normalization
zero_zeros = jn_zeros(0, self.n_modes_r)
self.omega_coords = np.zeros((self.n_modes_z, self.n_modes_r, 2))
self.dc_coords = np.zeros((self.n_modes_z, self.n_modes_r, 2))
self.mode_mass = np.ones((self.n_modes_z, self.n_modes_r))
self.omega = np.zeros((self.n_modes_z, self.n_modes_r))
for idx_r in range(0, self.n_modes_r):
for idx_z in range(0, self.n_modes_z):
self.omega[idx_z,idx_r]= \
np.sqrt(self.kr[idx_r]**2 +self.kz[idx_z]**2)
# Integral of cos^2(k_z z)*J_z(k_r r)^2 over the domain volume
self.mode_mass[idx_z, idx_r] = R * R * L * (j1(
zero_zeros[idx_r]))**2 / (4. * consts.c)
self.omegaOtwokz = 0.5 * np.einsum('z, zr -> zr', 1. / self.kz,
self.omega)
self.omegaOtwokr = 0.5 * np.einsum('r, zr -> zr', 1. / self.kr,
self.omega)
self.oneOomega = 1. / self.omega
self.kzOomega = einsum('z, zr -> zr', self.kz, self.oneOomega)
self.krOomega = einsum('r, zr -> zr', self.kr, self.oneOomega)
self.delta_P_dc = np.zeros((self.n_modes_z, self.n_modes_r))
self.delta_P_omega = np.zeros((self.n_modes_z, self.n_modes_r))
# Particles are tent functions with widths the narrowest of the
# k-vectors for each direction. Default for now is to have the
# particle widths be half the shortest wavelength, which should
# resolve the wave physics reasonably well.
self.ptcl_width_z = .25 * 2. * np.pi / max(self.kz)
self.ptcl_width_r = .25 * 2. * np.pi / max(self.kr)
self.shape_function_z = np.exp(-0.5 * (self.kz * self.ptcl_width_z)**2)
# With the conducting boundaries, unphysically large fields can
# get trapped on the conducting surfaces. To squelch this, we put
# a tanh-function envelope on the fields to make the fields go to
# zero quickly but smoothly on the boundaries.
self.tanh_width = np.max(self.kz)
self.z_mean = 0.5 * self.domain_L
# Create the mpi communicator
self.comm = mpi.COMM_WORLD
def jinc(x):
j = np.ones(x.shape)
# Handle 0/0 at origin
nonzero_x = abs(x) > 1e-20
j[nonzero_x] = 2 * sp.j1(np.pi * x[nonzero_x]) / (np.pi * x[nonzero_x])
return j
TCM = 72.013
h = 6.582 * 10**(-22)
r0 = 1.44 * 10**-15
k1 = 2. * mr * TCM
k2 = c**2 * h**2
k = np.sqrt(k1 / k2)
R = r0 * (16**(1 / 3) + 12**(1 / 3))
kR = k * R
k2R4 = k**2 * R**4
k2R4mb = k2R4 * 10**31
y = np.array([
400, 180, 70, 28, 20, 10, 18, 27, 45, 70, 80, 90, 80, 70, 50, 25, 11, 6,
1.1, 4.2, 7, 12, 16, 20, 15, 11, 9, 7, 5, 3, 2, 1.8, 2.2, 3.1, 3.8, 4.1,
3.7, 3.1, 2.7, 2.5, 1.1, 1.5, 1.3, 1.0, 0.9, 0.8, 0.9, 1.0, 1.05, 1.2, 1.2,
0.8, 0.6, 0.4, 0.3, 0.2
])
x = np.linspace(np.pi * 15.0 / 180, np.pi * 42.5 / 180, len(y))
theta = np.arange(np.pi * 15.0 / 180, np.pi * 42.5 / 180, 0.005)
y1 = k2R4mb * ((spe.j1(kR * theta) / (kR * theta))**2)
plt.plot(x, y, '.')
plt.plot(theta, y1, '-')
plt.title('Difracción Teoría de Fraunhöfer vs. Pts. Experimentales', size='10')
plt.xlabel(r'$\theta$ (rad)')
plt.ylabel('Sección eficaz diferencial (mbarn/sr)')
plt.legend(('Pts. predichos por Fraunhöfer', 'Valores experimentales'),
prop={'size': 10},
loc='upper right')
import numpy as np
import pickle
import scipy.special as sp
from scipy.optimize import fmin
expect_full = []
expect_with_full = []
#for i in range(1):
# with open("data/expectation_{0}_False.txt".format(i),"rb") as fp:
# expect_full.append(pickle.load(fp))
# with open("data/expectation_{0}_True.txt".format(i),"rb") as fp:
# expect_with_full.append(pickle.load(fp))
expect_theo = 1j * sp.j1(1) / sp.j0(1)
#R = len(expect_full)
#Nr = len(expect_full[0])
#N = Nr*R
with open("data/config_mu_0.1_T_4.0.txt", "rb") as fp:
expect_full = pickle.load(fp)
#with open("data/expectation_Nt20_beta1_step10_True.txt","rb") as fp:
# expect_full = pickle.load(fp)
N = len(expect_full)
print('N=', N)
R = 0
Nr = N
def Moyenne(list_obs):
# # Origen: Propio. # Autor: José MarÃa Herrera-Fernandez, Luis Miguel Sánchez-Brea # # Creación: 18 de Septiembre de 2013 # Historia: # # Dependencias: scipy, matplotlib # Licencia: GPL #---------------------------------------------------------------------- """ Descripción: Cálculo de los coeficientes de orden cero y uno de la función de Bessel contenida en el paquete special. """ from scipy import special # Importamos scipy.special import scipy as sp # Importamos scipy como el alias sp import matplotlib.pyplot as plt # Importamos matplotlib.pyplot como el alias plt. # Creamos el array dimensional x = sp.arange(0, 50, .1) print(x) x1 = sp.arange(0, 1, .1) print(x1) # Calculamos los coeficientes de orden cero. # j0 = special.j0(x) j0 = special.j0(x1) print(j0) # # Calculamos los coeficientes de orden uno. j1 = special.j1(x) print(j1)
def get2DSpheromakFlux(r, z, R=1, L=1, b0=1):
j1_zero1 = jn_zeros(1, 1)[0]
kr = j1_zero1 / R
kz = np.pi / L
return b0 * r * j1(kr * r) * np.sin(kz * z)
def expect_th(beta=default_beta):
# Return the imaginary part of the theoretical expect. value
result = []
for i in beta:
result.append(sp.j1(i) / sp.j0(i))
return result
def __init__(self,
target_res=0.005,
lenslet_CA=0.165,
zsampling='uniform_random',
cross_corr_norm='log_sum_exp',
aberrations=False,
GrinAber='',
GrinDictName='',
zernikes=[],
psf_norm='l1',
logsumparam=1e-2,
psf_scale=1e2): #'log_sum_exp'
# psf_scale: scale all PSFS by this constant after normalizing
super(Model, self).__init__()
target_option = 'airy'
#self.samples = (512,512) #Grid for PSF simulation
self.samples = (768, 768) #Grid for PSF simulation
self.lam = 510e-6
if GrinAber:
#'/media/hongdata/Kristina/MiniscopeData/GrinAberrations.mat'
# 'GrinAberrations'
file = scipy.io.loadmat(GrinAber)
GrinAber = file[GrinDictName]
self.Grin = []
for i in range(len(GrinAber)):
Grinpad = pad_frac_tf(GrinAber[i, :, :] * self.lam,
padfrac=0.5)
Grinresize = cv2.resize(Grinpad.numpy(),
(self.samples[1], self.samples[1]))
self.Grin.append(Grinresize)
# min and max lenslet focal lengths in mm
self.fmin = 6.15
self.fmax = 25.
self.ior = 1.515
self.lam = 510e-6
self.psf_norm = psf_norm
# Min and max lenslet radii
self.Rmin = self.fmin * (self.ior - 1.)
self.Rmax = self.fmax * (self.ior - 1.)
self.psf_scale = tf.constant(psf_scale)
# Convert to curvatures
self.cmin = 1 / self.Rmax
self.cmax = 1 / self.Rmin
self.xgrng = np.array((-1.8, 1.8)).astype(
'float32'
) #Range, in mm, of grid of the whole plane (not just grin)
self.ygrng = np.array((-1.8, 1.8)).astype('float32')
self.t = 10. #Distance to sensor from mask in mm
#Compute depth range of virtual image that mask sees (this is assuming an objective is doing some magnification)
self.zmin_virtual = 1. / (1. / self.t - 1. / self.fmin)
self.zmax_virtual = 1. / (1. / self.t - 1. / self.fmax)
self.CA = .9
#semi clear aperature of GRIN
self.mean_lenslet_CA = lenslet_CA #average lenslest semi clear aperture in mm.
#Getting number of lenslets and z planes needed as well as defocus list
self.ps = (self.xgrng[1] - self.xgrng[0]) / self.samples[0]
self.Nlenslets = np.int(
np.floor((self.CA**2) / (self.mean_lenslet_CA**2)))
self.Nz = 20
self.zsampling = zsampling
self.grid_z_planes = 20
#self.defocus_grid= 1./(np.linspace(1/self.zmin_virtual, 1./self.zmax_virtual, self.grid_z_planes)) #mm or dioptres
#if self.zsampling is 'fixed':
# self.defocus_list = 1./(np.linspace(1/self.zmin_virtual, 1./self.zmax_virtual, self.Nz)) #mm or dioptres
self.min_offset = 0 #-10e-3
self.max_offset = 50e-3
#self.lenslet_offset=tfe.Variable(tf.zeros(self.Nlenslets),name='offset', dtype = tf.float32)
#self.lenslet_offset=tfe.Variable(tf.zeros(self.Nlenslets),name='offset', dtype = tf.float32,constraint=lambda t: tf.clip_by_value(t,self.min_offset, self.max_offset))
self.lenslet_offset = tf.zeros(self.Nlenslets)
#initializing the x and y positions
[xpos, ypos, rlist] = poissonsampling_circular(self)
self.min_r = self.Rmin
self.max_r = self.Rmax
self.rlist = tf.Variable(
rlist,
name='rlist',
dtype=tf.float32,
constraint=lambda t: tf.clip_by_value(t, self.min_r, self.max_r))
#self.xpos = tfe.Variable(xpos, name='xpos', dtype = tf.float32, constraint=lambda t: tf.clip_by_value(t,-self.CA, self.CA))
#self.ypos = tfe.Variable(ypos, name='ypos', dtype = tf.float32, constraint=lambda t: tf.clip_by_value(t,-self.CA, self.CA))
self.xpos = tf.Variable(xpos, name='xpos', dtype=tf.float32)
self.ypos = tf.Variable(ypos, name='ypos', dtype=tf.float32)
#parameters for making the lenslet surface
self.yg = tf.constant(np.linspace(self.ygrng[0], self.ygrng[1],
self.samples[0]),
dtype=tf.float32)
self.xg = tf.constant(np.linspace(self.xgrng[0], self.xgrng[1],
self.samples[1]),
dtype=tf.float32)
self.px = tf.constant(self.xg[1] - self.xg[0], tf.float32)
self.py = tf.constant(self.yg[1] - self.yg[0], tf.float32)
self.xgm, self.ygm = tf.meshgrid(self.xg, self.yg)
# Normalized coordinates
self.xnorm = self.xgm / np.max(self.xgm)
self.ynorm = self.ygm / np.max(self.ygm)
#PSF generation parameters
self.lam = tf.constant(510. * 10.**(-6.), dtype=tf.float32)
self.k = np.pi * 2. / self.lam
fx = tf.constant(np.linspace(-1. / (2. * self.ps), 1. / (2. * self.ps),
self.samples[1]),
dtype=tf.float32)
fy = tf.constant(np.linspace(-1. / (2. * self.ps), 1. / (2. * self.ps),
self.samples[0]),
dtype=tf.float32)
self.Fx, self.Fy = tf.meshgrid(fx, fy)
self.field_list = tf.constant(np.array((0., 0.)).astype('float32'))
M = 6
self.corr_pad_frac = 0
self.target_res = target_res * M # micron
# sig = 2*self.target_res/(2.355) * 1e-3
# real_target = tf.exp(-(tf.square(self.xgm) + tf.square(self.ygm))/(2*tf.square(sig)))
# real_target = pad_frac_tf(real_target / tf.reduce_max(real_target), self.corr_pad_frac)
# self.target_F = tf.abs(tf.fft2d(tf.complex(tf_fftshift(real_target), 0.)))
D = 1.22 * self.lam / (tf.sin(2 * tf.atan(self.target_res /
(2 * self.t)))
) #0.514 for half_max, 1.22 for first zero.
x_airy = self.k * D * tf.sin(tf.atan(self.xgm / self.t)) / 2
y_airy = self.k * D * tf.sin(tf.atan(self.ygm / self.t)) / 2
#Xa, Ya = tf.meshgrid(x_airy, y_airy)
Ra = tf.sqrt(tf.square(x_airy) + tf.square(y_airy))
# self.target_airy=((2*scsp.j1(x_airy)/x_airy)**2) *((2*scsp.j1(y_airy)/y_airy)**2)
target_airy = (2 * scsp.j1(Ra) / Ra)**2
self.target_airy = target_airy / tf.sqrt(
tf.reduce_sum(tf.square(target_airy)))
self.target_airy_pad = pad_frac_tf(self.target_airy,
self.corr_pad_frac)
sig_aprox = 0.42 * self.lam * self.t / D
self.airy_aprox_target = tf.exp(
-(tf.square(self.xgm) + tf.square(self.ygm)) /
(2 * tf.square(sig_aprox)))
self.airy_aprox_pad = pad_frac_tf(
self.airy_aprox_target / tf.reduce_max(self.airy_aprox_target),
self.corr_pad_frac)
if target_option == 'airy':
self.target_F = tf.square(
tf.abs(
tf.signal.fft2d(
tf.complex(tf_fftshift(self.target_airy_pad), 0.))))
elif target_option == 'gaussian':
self.target_F = tf.abs(
tf.signal.fft2d(
tf.complex(tf_fftshift(self.airy_aprox_pad), 0.)))
# Set regularization. Problem will be treated as l1 of spectral error at each depth + tau * l_inf(cross_corr)
self.cross_corr_norm = cross_corr_norm
self.logsumexp_param = tf.constant(
logsumparam, tf.float32
) #lower is better l-infinity approximation, higher is worse but smoother
self.tau = tf.constant(
30, tf.float32) #Extra weight for cross correlation terms
self.ignore_dc = True #Ignore DC in autocorrelations when computing frequency domain loss
dc_mask = np.ones_like(
self.target_F.numpy()
) #Mask for DC. Set to zero anywhere we want to ignore loss computation of power spectrum
dc_mask[:3, :3] = 0
dc_mask[-2:, :1] = 0
dc_mask[:1, -2:] = 0
dc_mask[-2:, -2:] = 0
dc_mask = dc_mask * self.target_F.numpy() > (
.001 * np.max(self.target_F.numpy()))
self.dc_mask = tf.constant(dc_mask, tf.float32)
# Use Zernike aberrations with random initialization
# Zernike
self.zernikes = zernikes
self.numzern = len(zernikes)
if aberrations == True:
zern_init = []
for i in range(self.Nlenslets):
aberrations = np.zeros(self.numzern)
zern_init.append(aberrations)
self.zernlist = tf.Variable(zern_init,
dtype='float32',
name='zernlist')
else:
self.zernlist = 0
def airy(r, sigma):
from scipy.special import j1
r = r / sigma * np.sqrt(2)
a = (2 * j1(r) / r)**2
a[r == 0] = 1
return a
def hqwe(zsrc, zrec, lsrc, lrec, off, angle, depth, ab, etaH, etaV, zetaH,
zetaV, xdirect, qweargs, use_spline, use_ne_eval, msrc, mrec):
"""Hankel Transform using Quadrature-With-Extrapolation.
*Quadrature-With-Extrapolation* was introduced to geophysics by
[Key_2012]_. It is one of many so-called *ISE* methods to solve Hankel
Transforms, where *ISE* stands for Integration, Summation, and
Extrapolation.
Following [Key_2012]_, but without going into the mathematical details
here, the QWE method rewrites the Hankel transform of the form
.. math:: F(r) = \int^\infty_0 f(\lambda)J_v(\lambda r)\
\mathrm{d}\lambda
as a quadrature sum which form is similar to the FHT (equation 15),
.. math:: F_i \\approx \sum^m_{j=1} f(x_j/r)w_j g(x_j) =
\sum^m_{j=1} f(x_j/r)\hat{g}(x_j) \ ,
but with various bells and whistles applied (using the so-called Shanks
transformation in the form of a routine called :math:`\epsilon`-algorithm
([Shanks_1955]_, [Wynn_1956]_; implemented with algorithms from
[Trefethen_2000]_ and [Weniger_1989]_).
This function is based on `get_CSEM1D_FD_QWE.m`, `qwe.m`, and
`getBesselWeights.m` from the source code distributed with [Key_2012]_.
In the spline-version, `hqwe` checks how steep the decay of the
wavenumber-domain result is, and calls QUAD for the very steep interval,
for which QWE is not suited.
The function is called from one of the modelling routines in :mod:`model`.
Consult these modelling routines for a description of the input and output
parameters.
Returns
-------
fEM : array
Returns frequency-domain EM response.
kcount : int
Kernel count.
conv : bool
If true, QWE/QUAD converged. If not, <htarg> might have to be adjusted.
"""
# Input params have an additional dimension for frequency, reduce here
etaH = etaH[0, :]
etaV = etaV[0, :]
zetaH = zetaH[0, :]
zetaV = zetaV[0, :]
# Get rtol, atol, nquad, maxint, and pts_per_dec
rtol, atol, nquad, maxint, pts_per_dec = qweargs[:5]
# 1. PRE-COMPUTE THE BESSEL FUNCTIONS
# at fixed quadrature points for each interval and multiply by the
# corresponding Gauss quadrature weights
# Get Gauss quadrature weights
g_x, g_w = special.p_roots(nquad)
# Compute n zeros of the Bessel function of the first kind of order 1 using
# the Newton-Raphson method, which is fast enough for our purposes. Could
# be done with a loop for (but it is slower):
# b_zero[i] = optimize.newton(special.j1, b_zero[i])
# Initial guess using asymptotic zeros
b_zero = np.pi*np.arange(1.25, maxint+1)
# Newton-Raphson iterations
for i in range(10): # 10 is more than enough, usually stops in 5
# Evaluate
b_x0 = special.j1(b_zero) # j0 and j1 have faster versions
b_x1 = special.jv(2, b_zero) # j2 does not have a faster version
# The step length
b_h = -b_x0/(b_x0/b_zero - b_x1)
# Take the step
b_zero += b_h
# Check for convergence
if all(np.abs(b_h) < 8*np.finfo(float).eps*b_zero):
break
# 2. COMPUTE THE QUADRATURE INTERVALS AND BESSEL FUNCTION WEIGHTS
# Lower limit of integrand, a small but non-zero value
xint = np.concatenate((np.array([1e-20]), b_zero))
# Assemble the output arrays
dx = np.repeat(np.diff(xint)/2, nquad)
Bx = dx*(np.tile(g_x, maxint) + 1) + np.repeat(xint[:-1], nquad)
BJ0 = special.j0(Bx)*np.tile(g_w, maxint)
BJ1 = special.j1(Bx)*np.tile(g_w, maxint)
# 3. START QWE
# Intervals and lambdas for all offset
intervals = xint/off[:, None]
lambd = Bx/off[:, None]
# Angle dependent factors
factAng = kernel.angle_factor(angle, ab, msrc, mrec)
# Call and return QWE, depending if spline or not
if use_spline: # If spline, we calculate all kernels here
# New lambda, from min to max required lambda with pts_per_dec
start = np.log10(lambd.min())
stop = np.log10(lambd.max())
ilambd = np.logspace(start, stop, (stop-start)*pts_per_dec + 1)
# Call the kernel
PJ0, PJ1, PJ0b = kernel.wavenumber(zsrc, zrec, lsrc, lrec, depth,
etaH[None, :], etaV[None, :],
zetaH[None, :], zetaV[None, :],
np.atleast_2d(ilambd), ab, xdirect,
msrc, mrec, use_ne_eval)
# Interpolation : Has to be done separately on each PJ,
# in order to work with multiple offsets which have different angles.
sPJ0r = iuSpline(np.log10(ilambd), PJ0.real)
sPJ0i = iuSpline(np.log10(ilambd), PJ0.imag)
sPJ1r = iuSpline(np.log10(ilambd), PJ1.real)
sPJ1i = iuSpline(np.log10(ilambd), PJ1.imag)
sPJ0br = iuSpline(np.log10(ilambd), PJ0b.real)
sPJ0bi = iuSpline(np.log10(ilambd), PJ0b.imag)
# Get quadargs: diff_quad, a, b, limit
diff_quad, a, b, limit = qweargs[5:]
# Set quadargs if not given:
if not limit:
limit = maxint
if not a:
a = intervals[:, 0]
else:
a = a*np.ones(off.shape)
if not b:
b = intervals[:, -1]
else:
b = b*np.ones(off.shape)
# Check if we use QWE or SciPy's QUAD
# If there are any steep decays within an interval we have to use QUAD,
# as QWE is not designed for these intervals.
check0 = np.log10(intervals[:, :-1])
check1 = np.log10(intervals[:, 1:])
doqwe = np.all((np.abs(sPJ0r(check0) + 1j*sPJ0i(check0) +
sPJ1r(check0) + 1j*sPJ1i(check0) +
sPJ0br(check0) + 1j*sPJ0bi(check0)) /
np.abs(sPJ0r(check1) + 1j*sPJ0i(check1) +
sPJ1r(check1) + 1j*sPJ1i(check1) +
sPJ0br(check1) + 1j*sPJ0bi(check1)) < diff_quad), 1)
# Pre-allocate output array
fEM = np.zeros(off.size, dtype=complex)
conv = True
# Carry out SciPy's Quad if required
if np.any(~doqwe):
# Loop over offsets that require Quad
for i in np.where(~doqwe)[0]:
# Input-dictionary for quad
iinp = {'a': a[i], 'b': b[i], 'epsabs': atol, 'epsrel': rtol,
'limit': limit}
fEM[i], tc = quad(sPJ0r, sPJ0i, sPJ1r, sPJ1i, sPJ0br, sPJ0bi,
ab, off[i], factAng[i], iinp)
# Update conv
conv *= tc
# Return kcount=1 in case no QWE is calculated
kcount = 1
if np.any(doqwe):
# Get EM-field at required offsets
sPJ0 = sPJ0r(np.log10(lambd)) + 1j*sPJ0i(np.log10(lambd))
sPJ1 = sPJ1r(np.log10(lambd)) + 1j*sPJ1i(np.log10(lambd))
sPJ0b = sPJ0br(np.log10(lambd)) + 1j*sPJ0bi(np.log10(lambd))
# Carry out and return the Hankel transform for this interval
sEM = np.sum(np.reshape(sPJ1*BJ1, (off.size, nquad, -1),
order='F'), 1)
if ab in [11, 12, 21, 22, 14, 24, 15, 25]: # Because of J2
# J2(kr) = 2/(kr)*J1(kr) - J0(kr)
sEM /= np.atleast_1d(off[:, np.newaxis])
sEM += np.sum(np.reshape(sPJ0b*BJ0, (off.size, nquad, -1),
order='F'), 1)
sEM *= factAng[:, np.newaxis]
sEM += np.sum(np.reshape(sPJ0*BJ0, (off.size, nquad, -1),
order='F'), 1)
getkernel = sEM[doqwe, :]
# Get QWE
fEM[doqwe], kcount, tc = qwe(rtol, atol, maxint, getkernel,
intervals[doqwe, :], None, None, None)
conv *= tc
else: # If not spline, we define the wavenumber-kernel here
def getkernel(i, inplambd, inpoff, inpfang):
"""Return wavenumber-domain-kernel as a fct of interval i."""
# Indices and factor for this interval
iB = i*nquad + np.arange(nquad)
# PJ0 and PJ1 for this interval
PJ0, PJ1, PJ0b = kernel.wavenumber(zsrc, zrec, lsrc, lrec, depth,
etaH[None, :], etaV[None, :],
zetaH[None, :], zetaV[None, :],
np.atleast_2d(inplambd)[:, iB],
ab, xdirect, msrc, mrec,
use_ne_eval)
# Carry out and return the Hankel transform for this interval
gEM = inpfang*np.dot(PJ1[0, :], BJ1[iB])
if ab in [11, 12, 21, 22, 14, 24, 15, 25]: # Because of J2
# J2(kr) = 2/(kr)*J1(kr) - J0(kr)
gEM /= np.atleast_1d(inpoff)
gEM += inpfang*np.dot(PJ0b[0, :], BJ0[iB])
gEM += np.dot(PJ0[0, :], BJ0[iB])
return gEM
# Get QWE
fEM, kcount, conv = qwe(rtol, atol, maxint, getkernel, intervals,
lambd, off, factAng)
return fEM, kcount, conv
from scipy import special, r_, optimize
import pylab as plt
x = r_[2:7.1:.1]
j1x = special.j1(x)
plt.plot(x, j1x, '-')
plt.hold('on')
x_min = optimize.fminbound(special.j1, 4, 7)
plt.plot([x_min], [special.j1(x_min)], 'ro')
plt.hold('off')
plt.show()
