def gauss_laguerre(f, a, b, n):
if a != 0 and b != float('int'):
raise ValueError('Gauss-Laguerre Quadrature integrates over [a,0).')
w = lambda x: math.exp(x) # inverse weighting function
[roots,weights] = sp.l_roots(n,0)
return sum([weights[i]*w(roots[i])*f(roots[i]) for i in range(n)])
def GaussianQuadrature():
########## PART A ##########
# Array to carry integrands
n = 30
intArr = np.zeros(n,)
# Loop over all numbers of nodes
intFact = FD_factor(20.) # kT = 20MeV
for i in range(1,n+1):
# Find roots and approximate integral
[xi,wi] = sp.l_roots(i)
intArr[i-1] = intFact*np.sum(wi*fermi_dirac_mod(xi))
# Save integral array to file
np.savetxt('Q2_numdensity.dat',intArr,fmt='%10.10e')
############################
########## PART B ##########
# nInt-1 intervals of width dE = 5MeV
# in range [0,150] MeV
nInt = 30
dE = 5.
dndE = np.zeros(nInt,)
# Find roots
[xi,wi] = sp.p_roots(n)
# Transformations for xi,wi in each interval
xTemp = np.linspace(0.,150.,num=nInt+1)
bMa = 2.5 # b-a/2
bPa = 0.5*(xTemp[1:] + xTemp[:-1]) # b+a/2
# Loop over all intervals
for i in range(0,nInt):
# Transform variables
w = wi*bMa
x = bMa*xi + bPa[i]
# Evaluate intgral and differential
dndE[i] = (intFact/dE)*np.sum(w*fermi_dirac(x))
# Save spectral information to file
outArr1 = np.zeros((nInt,2.),float)
outArr1[:,0] = bPa
outArr1[:,1] = dndE
np.savetxt('Q2_energyspectrum.dat',outArr1,fmt='%10.10e')
# Verify method and save to file
outArr2 = [intArr[-1],np.sum(dndE*dE)]
np.savetxt('Q2_verifymethod.dat',outArr2,fmt='%10.10e')
############################
return
def gaussLaguerre(g, n):
f = lambda x: eval(g) # define function to evaluate
# get abscissas (xi), and weights (wi) from l_roots(n) function below
# scipy.special.l_roots(n) returns abscissas & weights of Laguerre polynomials
abscissa, weight = l_roots(n)
intSum = 0
# Gauss-Laguerre method is valid over interval [0,infty]
for i in range(n):
# evaluate integral
intSum = intSum + weight[i]*f(abscissa[i])*np.exp(abscissa[i])
return intSum
def quest2a():
"Routine for answering question 2a"
# setting kT = 20 MeV
nodes = range(1,11)
F = 8*pi*np.power(20e6 * const.EV / (const.C*const.H),3)
integral = np.array([])
for k in nodes:
[roots,weights] = spy.l_roots(k,0) #pulls out
integral = np.append(integral,sum(weights*funca(roots)))
print("Integral Node Convergence: {0}".format(integral))
print "NumberDensity: {0}".format(F*integral[-1])
return F*integral[-1]
def twoa(n): kT=3.2*10**(-5)#cgs hc=3.16*10**(-17)#cgs #use Python's built-in Gaussian-Laguerre roots and weights [la_r,la_w]=sp.l_roots(n) f=lambda x:x**2*np.exp(x)/(1.+np.exp(x))#f(x) Q=np.zeros(len(la_r)) #calculate each Qi for i in range(len(Q)): Q[i]=la_w[i]*f(la_r[i]) #and the sum Qtot=np.sum(Q) #then put the constants back in ne=(Qtot*8.*np.pi*(kT)**3)/(2.*np.pi*hc)**3 print "The integral is %.11f"%Qtot print "ne is %g cm^-3"%ne
def gauss_laguerre(f,n):
[laguerre_roots, laguerre_weights] = sp.l_roots(n,0)
return sum( f(laguerre_roots) * laguerre_weights )
# -*- coding: utf-8 -*-
"""
Created on Fri Jan 17 11:44:09 2014
@author: davidvartanyan
"""
from numpy import *
from scipy import special as sp
[laguerre_roots,laguerre_weights]=sp.l_roots(2,0)
#from Wolfram Alpha
k=20 #in MeV
h=1.97327*10**-11 #in Mev cm
def f(x):
return x**2*exp(x)/(exp(x)+1)
print sum(laguerre_weights*f(laguerre_roots))
import matplotlib.pyplot as pl
import plot_defaults
# Part A
# parameters
max_exponent = 8
number_density_coeff = 1.05495e35 # cm^(-3)
ns = [2**p for p in range(1,max_exponent)]
def f(x):
return x * x * np.exp(x) / (np.exp(x) + 1)
[xs, ws] = sp.l_roots(ns[-1], 0)
Qs = np.array([np.sum(ws[:n] * f(xs)[:n]) for n in ns])
number_densities = number_density_coeff * Qs
print("\nPart A\n")
print("Number of nodes:\n{}".format(ns))
print("Number density [cm^(-3)]:\n{}".format(number_densities))
print("Change in number density [cm^(-3)]:\n{}".format(number_densities[1:]
- number_densities[:-1]))
# Answer: 1.902*10^35 cm^(-3)
# Part B
# parameters
from __future__ import division from pylab import * from scipy import special # constants in cgs units c = 2.998e10 # speed of light kB = 1.38065e-16 # Boltzmann constant hbar = 1.054572e-27 # reduced Planck constant MeV_to_ergs = 1.602177e-6 # factor to convert units of energy from MeV to ergs T = 20*MeV_to_ergs/kB # temperature (kT = 20MeV) integrand = lambda x: x**2/(exp(x)+1) # (a) Use Gauss-Laguerre Quadrature to determine the total # number density of electrons [laguerre_roots,laguerre_weights] = special.l_roots(20) weight = lambda x: exp(-x) integral = sum(laguerre_weights*integrand(laguerre_roots)/weight(laguerre_roots)) n_electrons = 8*pi*(kB*T)**3/(2*pi*hbar*c)**3 * integral print 'The number density of electrons is %.4e/cm^3' % n_electrons # (b) Use Gauss-Legendre Quadrature to determine the spectral # distribution of the electrons [legendre_roots,legendre_weights] = special.p_roots(20) dE = 5; Emax = 200 E_bins = linspace(0,Emax,Emax/dE+1) # MeV x_bins = E_bins*MeV_to_ergs/(kB*T) n_bins = zeros(len(E_bins)-1) # number density of electrons in each energy bin
def lag(n): [lagroots, lagweights] = sp.l_roots(n,0) integ = 0.0 for j in range(0,n): integ +=lagweights[j]*g(lagroots[j]) return integ
def integrate_laguerre(f, args=(), n=6):
x, c = l_roots(n)
I = sum(np.exp(x) * f(x, *args) * c)
return I
def GaussLaguerre(func, n):
[laguerre_roots, laguerre_weights] = sp.l_roots(n, 0.)
return np.sum(laguerre_weights*func(laguerre_roots))
def laguerre(n,f):
roots,weights=sp.l_roots(n);
return np.sum(weights*f(roots))
def gla(n):
[lag_roots, lag_weights] = sp.l_roots(n,0)
func = map(f, lag_roots)
prod = np.dot(lag_weights, func)
return prod
