def __init__(self,N=100,L=10.0,d=1):
self.N = N
self.L = L
self.d = d
self.onedgrid = np.linspace(-L,L,N)
self.x,self.y = np.meshgrid(self.onedgrid, self.onedgrid)
self.r = np.sqrt(self.x**2 + self.y**2)
self.theta = np.arctan2(self.x,self.y)
self.delta = self.onedgrid[1] - self.onedgrid[0]
Delta1D = functions.tridiag(N,-1,2,-1)/self.delta**2
eye = functions.tridiag(N,0,1,0)
self.Delta = scipy.sparse.kron(Delta1D,eye,'csr') + scipy.sparse.kron(eye,Delta1D,'csr')
def fem_mult_op(self,V,l,Mass):
# Vsqrtop = scipy.sparse.spdiags(np.sqrt(np.abs(V)),0,V.shape[0],V.shape[0],'csr')
# return np.sign(V[1]) * np.dot(np.dot(Vsqrtop,Mass),Vsqrtop)
params = self.fem_params(l)
if not params['phi']:
if not params['weight']:
raise Exception("Not implemented")
# the grid has been shifted
x = np.zeros_like(self.x)
x[1:] = self.x[:-1]
deltap = self.deltam
deltam = np.zeros_like(deltap)
deltam[1:] = deltap[:-1]
deltam[0] = np.NaN
h = deltam[5]
diag = V / 2 * x
diag[1:] += 1/12*V[:-1] * (x[1:] - 2/5*h)
diag[:-1] += 1/12*V[1:] * (x[:-1] + 2/5*h)
diag[0] = h*(1/20*V[0] + 1/30 * V[1])
offdiag = 1/12*V * (x + 2/5*h)
offdiag[:-1] += 1/12*V[1:] * (x[:-1] + 3/5*h)
# debug
diag = V/4*(deltam*(x-1/5*deltam) + deltap*(x+1/5*deltap))
diag[1:] += 1/12*V[:-1] * deltam[1:]*(x[1:] - 2/5*deltam[1:])
diag[:-1] += 1/12*V[1:] * deltap[:-1]*(x[:-1] + 2/5*deltap[:-1])
diag[0] = deltap[0]**2*(1/20*V[0] + 1/30 * V[1])
offdiag = 1/12*V * deltap*(x + 2/5*deltap)
offdiag[:-1] += 1/12*V[1:] * deltap[:-1]*(x[:-1] + 3/5*deltap[:-1])
return functions.tridiag(self.N,offdiag,diag,offdiag)
else:
if params['fd']:
return functions.tridiag(self.N,0,V,0)
else:
diag = V/4*(self.deltap + self.deltam)
diag[1:] += 1/12*V[:-1]*self.deltam[1:]
diag[:-1] += 1/12*V[1:]*self.deltap[:-1]
offdiag = 1/12*V*self.deltap
offdiag[:-1] += 1/12*V[1:]*self.deltap[:-1]
return functions.tridiag(self.N,offdiag,diag,offdiag)
def build_rho_2D(g, V, gamma, sigma, numeigs=4):
'''Construit rho à partir des premières fonctions propres de -Delta+V'''
bigN = g.N**g.d
H = g.Delta + functions.tridiag(bigN,0,V.flat,0)
w,vs = scipy.sparse.linalg.eigsh(H,numeigs,sigma=sigma,which='LM')
if w[-1] < 0:
return build_rho_2D(g,V,gamma, sigma, numeigs*2)
elif w[0] > 0:
raise Exception('No negative eigenvalue')
print 'Eigs', w
vecs = []
for i in range(0,len(w)):
if w[i] < 0:
vecs.append(vs[:,i].reshape((g.N,g.N)))
rho = np.zeros_like(V)
for i in range(0,len(vecs)):
rho = rho + np.abs(w[i])**(gamma-1)*vecs[i]**2
return rho, w[w < 0], vecs
# -*- coding: utf-8 -*-
"""
Standalone program for comparing Jacobi's Method to the Bisection Method for
finding eigenvalues
"""
from functions import tridiag, jacobi_solver, bisect_eigens
import numpy as np
import matplotlib.pyplot as plt
import time
Ns = [5,10,15,20]
t = np.zeros((2,len(Ns),1))
j = 0
for i in Ns:
tt = time.perf_counter()
jacobi_solver(tridiag(2,5,i))
t[0,j,0] = time.perf_counter() - tt
tt = time.perf_counter()
bisect_eigens(2,5,i)
t[1,j,0] = time.perf_counter() - tt
j += 1
plt.plot(Ns, t[0,:])
plt.plot(Ns, t[1,:])
plt.legend(['Jacobi','Bisection'])
plt.xlabel('Square Matrix Length')
plt.ylabel('Time to calculate eigenvalues [s]')
plt.title('Time to calculate eigenvalues given a matrix length N')
plt.show()
# -*- coding: utf-8 -*-
"""
Standalone program for plotting N versus Rotations
Runtime is pretty long, beware! The "soft-cap" seems to be at around N = 100,
so remove N = 150 if its taking too long
"""
from functions import tridiag, jacobi_solver
import numpy as np
import matplotlib.pyplot as plt
Ns = [10, 20, 30, 50, 100, 150] #N values
data = np.zeros((len(Ns), 2))
j = 0
for i in Ns:
data[j][0] = i #x-axis
data[j][1] = jacobi_solver(tridiag(-1, 2,
i))[2] #y-axis, returns iterations
j += 1
plt.xlabel('Square Matrix Length')
plt.ylabel('Number of iterations needed')
plt.title(
'Iterations needed with the Jacobi Rotation Algorithm given a length N')
plt.plot(data[:, 0], data[:, 1])
plt.show()
"""
Standalone program for testing and validating some functions used in this
project
"""
"""
Let us test if all our methods return the correct eigenvalues
Let's assume we have the Matrix
5 2 0 0 0
2 5 2 0 0
0 2 5 2 0
0 0 2 5 0
This should return the eigenvalues:
lambda = 7, 5, 3, 5 + 2*sqrt(3), 5 - 2*sqrt(3)
"""
rLambdas = np.sort([7,5,3,5 + 2*np.sqrt(3), 5 - 2*np.sqrt(3)])
A = tridiag(2,5,5)
"""
Let's now test each of our solutions
"""
nEigs = np.sort(np.linalg.eig(A)[0])
aEigs = np.sort(eigenvalues(2,5,5))
jEigs = np.sort(jacobi_solver(A)[0])
if np.allclose(rLambdas,nEigs) == True:
print('Numpy Validated')
if np.allclose(rLambdas,nEigs) == True:
print('Analytical Validated')
if np.allclose(rLambdas,nEigs) == True:
print('Jacobi Validated')
"""
This can also be verified with any other Matrix
def fem_mats(self,l):
# returns Delta, Mass
N = self.N
params = self.fem_params(l)
if not params['phi']:
if self.d != 2:
raise Exception('Not implemented')
# the grid has been shifted
x = np.zeros_like(self.x)
x[1:] = self.x[:-1]
deltap = self.deltam
deltam = np.zeros_like(deltap)
deltam[1:] = deltap[:-1]
deltam[0] = np.NaN
if params['weight']:
# integrate against r
# Mdiag = h * 2/3 * x
# Moffdiag = h * 1/6 * (x + h/2)
# Mdiag[0] = h**2 / 12
# Ddiag = 2 / h * x #OK
# Doffdiag = -1 / h * (x + h/2) #OK
# Ddiag[0] = 1/2 #OK
Mdiag = deltam/3*(x-deltam/4) + deltap/3*(x+deltap/4)
Moffdiag = 1/6*deltap*(x+deltap/2)
Mdiag[0] = deltap[0]**2/12
Ddiag = x*(1/deltam+1/deltap)
Doffdiag = -1/deltap*(x+deltap/2)
Ddiag[0] = 1/2
Delta = functions.tridiag(N,Doffdiag,Ddiag,Doffdiag)
Mass = functions.tridiag(N,Moffdiag,Mdiag,Moffdiag)
else:
if l != 0:
raise Exception('Not implemented')
Delta = functions.tridiag(N,-1,2,-1) / self.delta
Mass = functions.tridiag(N,1,4,1) * self.delta / 6
#neumann
Delta[0,0] = 1/self.delta
Mass[0,0] = 1/3*self.delta
else:
if params['fd']:
Delta = functions.tridiag(N,-1,2,-1) / self.delta**2
# Mass = functions.tridiag(N,0,1,0)
Mass = None
# Delta[0,0] = 1/self.delta**2
else:
# Delta = functions.tridiag(N,-1,2,-1) / self.delta
# Mass = functions.tridiag(N,1,4,1) * self.delta / 6
# deltamat = functions.tridiag(N,0, np.sqrt(self.delta),0)
# deltainvmat = functions.tridiag(N,0, 1/np.sqrt(self.delta),0)
# Delta = np.dot(np.dot(deltainvmat,functions.tridiag(N,-1,2,-1)),deltainvmat)
# Mass = np.dot(np.dot(deltamat,functions.tridiag(N,1,4,1)),deltamat) / 6
# if neumann:
# Delta[0,0] = 1/self.delta
# Mass[0,0] = 1/3*self.delta
Ddiag = (self.deltap+self.deltam)/self.deltap/self.deltam
Doffdiag = -1/self.deltap
Mdiag = 1/3*(self.deltap + self.deltam)
Moffdiag = 1/6 * self.deltap
Delta = functions.tridiag(N,Doffdiag,Ddiag,Doffdiag)
Mass = functions.tridiag(N,Moffdiag,Mdiag,Moffdiag)
return Delta,Mass
# -*- coding: utf-8 -*-
"""
Standalone program for calculating runtime for three different algorithms for
finding eigenvalues
"""
from functions import tridiag, eigenvalues, jacobi_solver
import numpy as np
import matplotlib.pyplot as plt
import time
Ns = [5, 10, 30, 50]
t = np.zeros((3, len(Ns), 1))
j = 0
for i in Ns:
tt = time.perf_counter()
eigenvalues(-5, 10, i)
t[0, j, 0] = time.perf_counter() - tt
tt = time.perf_counter()
np.linalg.eig(tridiag(-5, 10, i))
t[1, j, 0] = time.perf_counter() - tt
tt = time.perf_counter()
jacobi_solver(tridiag(-5, 10, i))
t[2, j, 0] = time.perf_counter() - tt
j += 1
plt.plot(Ns, t[0, :])
plt.plot(Ns, t[1, :])
plt.plot(Ns, t[2, :])
plt.legend(['Analytical', 'Numpy', 'Jacobi'])
plt.xlabel('Square Matrix Length')
plt.ylabel('Time to calculate eigenvalues [s]')
plt.title('Time to calculate eigenvalues given a matrix length N')
plt.show()