def test_chebpts2(self):
#test exceptions
assert_raises(ValueError, cheb.chebpts2, 1.5)
assert_raises(ValueError, cheb.chebpts2, 1)
#test points
tgt = [-1, 1]
assert_almost_equal(cheb.chebpts2(2), tgt)
tgt = [-1, 0, 1]
assert_almost_equal(cheb.chebpts2(3), tgt)
tgt = [-1, -0.5, .5, 1]
assert_almost_equal(cheb.chebpts2(4), tgt)
tgt = [-1.0, -0.707106781187, 0, 0.707106781187, 1.0]
assert_almost_equal(cheb.chebpts2(5), tgt)
def test_chebpts2(self):
#test exceptions
assert_raises(ValueError, cheb.chebpts2, 1.5)
assert_raises(ValueError, cheb.chebpts2, 1)
#test points
tgt = [-1, 1]
assert_almost_equal(cheb.chebpts2(2), tgt)
tgt = [-1, 0, 1]
assert_almost_equal(cheb.chebpts2(3), tgt)
tgt = [-1, -0.5, .5, 1]
assert_almost_equal(cheb.chebpts2(4), tgt)
tgt = [-1.0, -0.707106781187, 0, 0.707106781187, 1.0]
assert_almost_equal(cheb.chebpts2(5), tgt)
def cheb_fitcurve(x, y, order):
x = cheb.chebpts2(len(x))
order = 64
coef = legend.legfit(x, y, order)
assert_equal(len(coef), order + 1)
y1 = legend.legval(x, coef)
err_1 = np.linalg.norm(y1 - y) / np.linalg.norm(y)
coef = cheb.chebfit(x, y, order)
assert_equal(len(coef), order + 1)
thrsh = abs(coef[0] / 1000)
for i in range(len(coef)):
if abs(coef[i]) < thrsh:
coef = coef[0:i + 1]
break
y2 = cheb.chebval(x, coef)
err_2 = np.linalg.norm(y2 - y) / np.linalg.norm(y)
plt.plot(x, y2, '.')
plt.plot(x, y, '-')
plt.title("nPt={} order={} err_cheby={:.6g} err_legend={:.6g}".format(
len(x), order, err_2, err_1))
plt.show()
assert_almost_equal(cheb.chebval(x, coef), y)
#
return coef
def __cheb_quad(n): ''' weights for Chebyshev–Gauss quadrature, https://keisan.casio.com/exec/system/1282551763 ''' W = np.zeros(n) a = np.pi / (n-1) for i,y in enumerate(cb.chebpts2(n)): if 0 < i < n-1: W[i] = a * np.sin(a*i)**2 / (1 - y**2)**.5 return W / 2 # /2 for normalizing the integration on range [-1,1]
def points_and_weights(self, N):
self.N = N
if self.quad == "GL":
points = n_cheb.chebpts2(N)[::-1]
weights = np.zeros((N)) + np.pi / (N - 1)
weights[0] /= 2
weights[-1] /= 2
elif self.quad == "GC":
points, weights = n_cheb.chebgauss(N)
return points, weights
def points_and_weights(self, N):
self.N = N
if self.quad == "GC":
points = n_cheb.chebpts2(N)[::-1]
weights = np.zeros((N))+np.pi/(N-1)
weights[0] /= 2
weights[-1] /= 2
elif self.quad == "GL":
points, weights = n_cheb.chebgauss(N)
return points, weights
def simulate(Xpred):
' Generates y from x data '
lambda_0 = 240
lambda_f = 2000
n_spct = 256
cheb_plot = cheb.chebpts2(n_spct)
scale = (lambda_f - lambda_0) / 2
offset = lambda_0 + scale
scaled_cheb = [i * scale + offset for i in cheb_plot]
# Load refractive indices of materials at different sizes
n_dict = N_Dict()
n_dict.Load("Si3N4", "./Si3N4_310nm-14280nm.txt", scale=1000)
n_dict.Load("Graphene", "./Graphene_240nm-30000nm.txt")
n_dict.InitMap2(["Si3N4", "Graphene"], scaled_cheb)
map2 = n_dict.map2
Ypred = np.zeros((Xpred.shape[0], n_spct))
for c, x in enumerate(Xpred):
dataY = np.zeros((n_spct, 3))
t_layers = np.array([])
for val in x:
# Graphene is preset to always be 0.35 thick
t_layers = np.concatenate((t_layers, np.array([0.35, val])))
t_layers = np.concatenate((np.array([np.nan]),t_layers, np.array([np.nan])))
# Each y-point has different ns in each layer
for row, lenda in enumerate(scaled_cheb):
n_layers = np.array([])
for i in range(len(t_layers) - 2):
# -2 added to subtract added substrate parameters from length
if i % 2 == 0:
n_layers = np.concatenate((n_layers, np.array([map2['Graphene', lenda]])))
else:
n_layers = np.concatenate((n_layers, np.array([map2['Si3N4', lenda]])))
# Sandwich by substrate
n_layers = np.concatenate((np.array([1.46 + 0j]), n_layers, np.array([1 + 0j])))
# TARGET PARAMETERS OF THE PROBLEM!
xita = 0
polar = 0
r, t, R, T, A = jreftran_rt(lenda, t_layers, n_layers, xita, polar)
dataY[row, 0] = R
dataY[row, 1] = T
dataY[row, 2] = A
# Because graphs are of absorbance 2nd value is selected
Ypred[c, :] = dataY[0:n_spct, 2]
return Ypred
def points_and_weights(self, N):
self.N = N
if self.quad == "GL":
points = (n_cheb.chebpts2(N)[::-1]).astype(float)
weights = zeros(N)+pi/(N-1)
weights[0] /= 2
weights[-1] /= 2
elif self.quad == "GC":
points, weights = n_cheb.chebgauss(N)
points = points.astype(float)
weights = weights.astype(float)
return points, weights
def assemble(self):
k = arange(self.N).astype(float)
Am = Amat(k)
Bm = Bmat(k, "GC")
Cm = Cmat(k, "GC")
weights = zeros((self.N))+pi/(self.N-1)
weights[0] /= 2
weights[-1] /= 2
A = Am.diags().toarray()
B = Bm.diags().toarray()
C = Cm.diags().toarray()
points = n_cheb.chebpts2(self.N)[::-1]
self.B = -1j*self.alfa*self.Re*(C+self.alfa**2*B)
self.A = (A + (2*self.alfa**2+1j*self.alfa*self.Re*(1-points**2))*C +
(self.alfa**4+1j*self.alfa**3*self.Re*(1-points**2)-2*1j*self.alfa*self.Re)*B)
def assemble(self):
k = arange(self.N).astype(float)
Am = Amat(k)
Bm = Bmat(k, "GC")
Cm = Cmat(k, "GC")
weights = zeros((self.N))+pi/(self.N-1)
weights[0] /= 2
weights[-1] /= 2
A = Am.diags().toarray()
B = Bm.diags().toarray()
C = Cm.diags().toarray()
points = n_cheb.chebpts2(self.N)[::-1]
self.B = -1j*self.alfa*self.Re*(C+self.alfa**2*B)
self.A = (A + (2*self.alfa**2+1j*self.alfa*self.Re*(1-points**2))*C +
(self.alfa**4+1j*self.alfa**3*self.Re*(1-points**2)-2*1j*self.alfa*self.Re)*B)
def points_and_weights(self, N, scaled=False):
if self.quad == "GL":
points = -(n_cheb.chebpts2(N)).astype(float)
weights = np.full(N, np.pi/(N-1))
weights[0] /= 2
weights[-1] /= 2
elif self.quad == "GC":
points, weights = n_cheb.chebgauss(N)
points = points.astype(float)
weights = weights.astype(float)
if scaled is True:
points = self.map_true_domain(points)
return points, weights
def points_and_weights(self, N):
self.N = N
if self.quad == "GL":
points = (n_cheb.chebpts2(N)[::-1]).astype(float)
#points = (n_cheb.chebpts2(N)).astype(float)
weights = zeros(N) + pi / (N - 1)
weights[0] /= 2
weights[-1] /= 2
elif self.quad == "GC":
points, weights = n_cheb.chebgauss(N)
#points = points[::-1]
points = points.astype(float)
weights = weights.astype(float)
return points, weights
def _fit_fft(self,func,N):
""" Get the chebyshev coefficients using fft
inspired by: http://www.scientificpython.net/1/post/2012/4/the-fast-chebyshev-transform.html
*Doesn't seem to work right now*
"""
pts = cheb.chebpts2(N)
y = func(pts)
A = y[:,np.newaxis]
m = np.size(y,0)
k = m-1-np.arange(m-1)
V = np.vstack((A[0:m-1,:],A[k,:]))
F = ifft(V,n=None,axis=0)
B = np.vstack((F[0,:],2*F[1:m-1,:],F[m-1,:]))
if A.dtype != 'complex':
return np.real(B)
return B
def Init_lenda_tic(self ):
l_0 = self.lenda_0
self.lenda_tic=[]
if self.tic_type=="cheb_":
pt_1 = cheb.chebpts1(256)
pt_2 = cheb.chebpts2(256)
scale = (self.lenda_1-l_0)/2
off = l_0+scale
self.lenda_tic = [i * scale + off for i in pt_2]
assert(self.lenda_tic[0]==l_0 and self.lenda_tic[-1]==self.lenda_1)
else:
if( self.lenda_0<500 ):
self.lenda_tic = list(range(self.lenda_0,500,1))
l_0 = 500
self.lenda_tic.extend( list(range(l_0,self.lenda_1+self.lenda_step,self.lenda_step)) )
def collocation_matrix(n, bcs):
x = 0.5 * (1 + chebpts2(n))
b = numpy.zeros((n, ))
A = numpy.zeros((n, n))
# Left boundary
b[0] = bcs[0]
A[0, 0] = 1
# Right boundary
b[-1] = bcs[-1]
A[-1, :] = 1
for nx in range(1, n - 1):
for j in range(n):
A[nx, j] = A[nx, j] - x[nx]**(j)
for j in range(2, n):
A[nx, j] = A[nx, j] + (j) * (j - 1) * x[nx]**(j - 2)
return numpy.linalg.solve(A, b)
def chebpts2(npts):
from numpy.polynomial.chebyshev import chebpts2
return chebpts2(npts)
plt.legend() plt.show() plt.close() if __name__ == '__main__': Ny = 201 Ly = 1. # ys = Ly * np.linspace(0, 1, Ny) # ys = Ly * (1 - np.cos(.5*np.pi*np.linspace(0,1,Ny))) ys = cb.chebpts2(Ny) # D1, D2, I2 = Schemes.compact6(ys) # D1, D2, I2 = Schemes.center2(ys) D1, D2, I1 = Schemes.cheb(ys) # us = np.sin(2*np.pi*ys) * np.cos(np.pi*ys**2) us = (ys - ys[0]) * (ys[-1] - ys) print(cb.chebval(ys, cb.chebint(cb.chebfit(ys, us, len(ys)-1), lbnd=ys[0]))[-1]) print(np.diag(I1) @ us * (ys[-1]-ys[0])) Schemes.check(ys, us, D1, D2)
def _fit_builtin(self,func,N):
""" Return the chebyshev coefficients using the builtin chebfit """
pts = cheb.chebpts2(N)
y = func(pts)
coeffs = cheb.chebfit(pts,y,N)
return coeffs
def chebpts2(npts):
from numpy.polynomial.chebyshev import chebpts2
return chebpts2(npts)
"""
# Use sympy to compute a rhs, given an analytical solution
x = Symbol("x")
u = (1 - x ** 2) ** 2 * cos(np.pi * x) * (x - 0.25) ** 2
f = u.diff(x, 2)
# Choices
banded = True
fast_transform = False
N = 20
k = np.arange(N - 2)
# Get points and weights for Chebyshev weighted integrals
points = n_cheb.chebpts2(N)[::-1]
weights = np.zeros((N)) + np.pi / (N - 1)
weights[0] /= 2
weights[-1] /= 2
# Note! N points in real space gives N-2 bases in spectral space
# Build Vandermonde matrix for Lobatto points and weights
V = n_cheb.chebvander(points, N - 3).T - n_cheb.chebvander(points, N - 1)[:, 2:].T
# Gauss-Chebyshev quadrature to compute rhs
fj = np.array([f.subs(x, j) for j in points], dtype=float) # Get f on quad points
def fastChebScalar(fj):
return dct(fj, 1) * np.pi / (2 * (len(fj) - 1))
weighted L_w norm (u, v)_w = \int_{-1}^{1} u v / \sqrt(1-x^2) dx
(\nabla^ u, \phi_k)_w = (f, \phi_k)_w
"""
# Use sympy to compute a rhs, given an analytical solution
x = Symbol("x")
u = cos(np.pi * x)
f = u.diff(x, 2)
N = 50
k = np.arange(N - 2).astype(np.float)
# Get points and weights for Chebyshev weighted integrals
points = n_cheb.chebpts2(N)[::-1]
weights = np.zeros((N)) + np.pi / (N - 1)
weights[0] /= 2
weights[-1] /= 2
# Build Vandermonde matrix for Lobatto points and weights
V = n_cheb.chebvander(points, N - 3).T - (
(k /
(k + 2))**2)[:, np.newaxis] * n_cheb.chebvander(points, N - 1)[:, 2:].T
V = V[1:, :] # Not using k=0
def fastShenTransNeumann(fj):
"""Fast Shen transform on cos(j*pi/N).
"""
cj = fastShenScalarNeumann(fj)