def in_monomial_basis(self):
res = poly.Polynomial({poly.MonomialVector({}): 1})
for v, p in self:
c = cheb2poly([0] * p + [1])
basis = poly.MonomialVector.construct_basis([v], p)
res *= poly.Polynomial(dict(zip(basis, c)))
return res
def chebfun_to_poly(fun, text=False):
low, high = fun._domain
# Reverse the coefficients, and use cheb2poly to make it in the polynomial domain
poly_coeffs = cheb2poly(fun.coefficients())[::-1].tolist()
if not text:
return poly_coeffs
s = 'coeffs = %s\n' % poly_coeffs
delta = high - low
delta_sum = high + low
# Generate the expression
s += 'horner(coeffs, %.18g*(x - %.18g))' % (2.0 / delta, 0.5 * delta_sum)
# return the string
return s
def chebfun_to_poly(fun, text=False):
from numpy.polynomial.chebyshev import cheb2poly
low, high = fun._domain
# Reverse the coefficients, and use cheb2poly to make it in the polynomial domain
poly_coeffs = cheb2poly(fun.coefficients())[::-1].tolist()
if not text:
return poly_coeffs
s = 'coeffs = %s\n' %poly_coeffs
delta = high - low
delta_sum = high + low
# Generate the expression
s += 'horner(coeffs, %.18g*(x - %.18g))' %(2.0/delta, 0.5*delta_sum)
# return the string
return s
def chebyshev_transpose_mult_slow(v):
"""Naive multiplication P^T v where P is the matrix of coefficients of
Chebyshev polynomials.
Parameters:
v: (batch_size, n)
Return:
P^T v: (batch_size, n)
"""
n = v.shape[-1]
# Construct the coefficient matrix P for Chebyshev polynomials
P = np.zeros((n, n), dtype=np.float32)
for i, coef in enumerate(np.eye(n)):
P[i, :i + 1] = chebyshev.cheb2poly(coef)
P = torch.tensor(P)
return v @ P
def conv_cheb(T):
"""
Convert a chebyshev polynomial to the power basis representation.
Args:
T (): The chebyshev polynomial to convert.
Returns:
new_conv (): The chebyshev polynomial converted to the power basis representation.
"""
conv = C.cheb2poly(T)
if conv.size == T.size:
return conv
else:
pad = T.size - conv.size
new_conv = np.pad(conv, ((0, pad)), 'constant')
return new_conv
def cheb_to_poly(coeffs_or_fun, domain=None):
'''Just call horner on the outputs!
'''
from fluids.numerics import horner as horner_poly
if isinstance(coeffs_or_fun, Chebfun):
coeffs = coeffs_or_fun.coefficients()
domain = coeffs_or_fun._domain
else:
coeffs = coeffs_or_fun
low, high = domain
coeffs = cheb2poly(coeffs)[::-1].tolist() # Convert to polynomial basis
# Mix in limits to make it a normal polynomial
my_poly = Polynomial(
[-0.5 * (high + low) * 2.0 / (high - low), 2.0 / (high - low)])
poly_coeffs = horner_poly(coeffs, my_poly).coef[::-1].tolist()
return poly_coeffs
def cheb_to_poly(coeffs_or_fun, domain=None):
"""Just call horner on the outputs!"""
from fluids.numerics import horner as horner_poly
if isinstance(coeffs_or_fun, Chebfun):
coeffs = coeffs_or_fun.coefficients()
domain = coeffs_or_fun._domain
elif hasattr(
coeffs_or_fun, '__class__'
) and coeffs_or_fun.__class__.__name__ == 'ChebyshevExpansion':
coeffs = coeffs_or_fun.coef()
domain = coeffs_or_fun.xmin(), coeffs_or_fun.xmax()
else:
coeffs = coeffs_or_fun
low, high = domain
coeffs = cheb2poly(coeffs)[::-1].tolist() # Convert to polynomial basis
# Mix in limits to make it a normal polynomial
my_poly = Polynomial(
[-0.5 * (high + low) * 2.0 / (high - low), 2.0 / (high - low)])
poly_coeffs = horner_poly(coeffs, my_poly).coef[::-1].tolist()
return poly_coeffs
def test_chebyshev_arithmetic(self):
print("chebdomain:\t" + str(cbs.chebdomain))
print("chebzero:\t" + str(cbs.chebzero))
print("chebone\t:" + str(cbs.chebone))
print("chebx\t:" + str(cbs.chebx))
p1 = cbs.chebval(0.2, 4)
print("P(0.5) = " + str(p1))
N = 100
chebyshevMatrix = np.zeros([N, N], dtype=float)
for zmc in np.arange(N):
vec = np.zeros([zmc + 1], dtype=float)
vec[zmc] = 1.
chebyshevMatrix[zmc, 0:zmc + 1] = cbs.cheb2poly(vec)
print("Size of Matrix:")
print(chebyshevMatrix.shape)
# for zmc in np.arange(chebyshevMatrix.shape[0]):
# print(chebyshevMatrix[zmc])
np.save("chebyshevCoeficientsMatrix100", chebyshevMatrix)
def conv_cheb(T):
"""
Convert a Chebyshev polynomial to the power basis representation in one dimension.
Parameters
----------
T : array_like
A one dimensional array_like object that represents the coeff of a
Chebyshev polynomial.
Returns
-------
ndarray
A one dimensional array that represents the coeff of a power basis polynomial.
"""
conv = cheb.cheb2poly(T)
if conv.size == T.size:
return conv
else:
pad = T.size - conv.size
new_conv = np.pad(conv, ((0, pad)), 'constant')
return new_conv
def chebfun_to_poly(coeffs_or_fun, domain=None, text=False):
if isinstance(coeffs_or_fun, Chebfun):
coeffs = coeffs_or_fun.coefficients()
domain = coeffs_or_fun._domain
elif hasattr(
coeffs_or_fun, '__class__'
) and coeffs_or_fun.__class__.__name__ == 'ChebyshevExpansion':
coeffs = coeffs_or_fun.coef()
domain = coeffs_or_fun.xmin(), coeffs_or_fun.xmax()
else:
coeffs = coeffs_or_fun
low, high = domain
# Reverse the coefficients, and use cheb2poly to make it in the polynomial domain
poly_coeffs = cheb2poly(coeffs)[::-1].tolist()
if not text:
return poly_coeffs
s = 'coeffs = %s\n' % poly_coeffs
delta = high - low
delta_sum = high + low
# Generate the expression
s += 'horner(coeffs, %.18g*(x - %.18g))' % (2.0 / delta, 0.5 * delta_sum)
# return the string
return s
def test_cheb2poly(self) :
for i in range(10) :
assert_almost_equal(cheb.cheb2poly([0]*i + [1]), Tlist[i])
def cheb2poly(cs):
from numpy.polynomial.chebyshev import cheb2poly
return cheb2poly(cs)
def fit_cheb_poly(func, low, high, n,
interpolation_property=None, interpolation_property_inv=None,
interpolation_x=lambda x: x, interpolation_x_inv=lambda x: x,
arg_func=None):
r'''Fit a function of one variable to a polynomial of degree `n` using the
Chebyshev approximation technique. Transformations of the base function
are allowed as lambdas.
Parameters
----------
func : callable
Function to fit, [-]
low : float
Low limit of fitting range, [-]
high : float
High limit of fitting range, [-]
n : int
Degree of polynomial fitting, [-]
interpolation_property : None or callable
When specified, this callable will transform the output of the function
before fitting; for example a property like vapor pressure should be
`interpolation_property=lambda x: log(x)` because it rises
exponentially. The output of the evaluated polynomial should then have
the reverse transform applied to it; in this case, `exp`, [-]
interpolation_property_inv : None or callable
When specified, this callable reverses `interpolation_property`; it
must always be provided when `interpolation_property` is set, and it
must perform the reverse transform, [-]
interpolation_x : None or callable
Callable to transform the input variable to fitting. For example,
enthalpy of vaporization goes from a high value at low temperatures to zero at
the critical temperature; it is normally hard for a chebyshev series
to match this, but by setting this to lambda T: log(1. - T/Tc), this
issue is resolved, [-]
interpolation_x_inv : None or callable
Inverse function of `interpolation_x_inv`; must always be provided when
`interpolation_x` is set, and it must perform the reverse transform,
[-]
arg_func : None or callable
Function which is called with the value of `x` in the original domain,
and that returns arguments to `func`.
Returns
-------
coeffs : list[float]
Polynomial coefficients in order for evaluation by `horner`, [-]
Notes
-----
This is powered by Ian Bell's ChebTools.
'''
global ChebTools
if ChebTools is None:
import ChebTools
low_orig, high_orig = low, high
cheb_fun = None
low, high = interpolation_x(low_orig), interpolation_x(high_orig)
if arg_func is not None:
if interpolation_property is not None:
def func_fun(T):
arg = interpolation_x_inv(T)
if arg > high_orig:
arg = high_orig
if arg < low_orig:
arg = low_orig
return interpolation_property(func(arg, *arg_func(arg)))
else:
def func_fun(T):
arg = interpolation_x_inv(T)
if arg > high_orig:
arg = high_orig
if arg < low_orig:
arg = low_orig
return func(arg, *arg_func(arg))
else:
if interpolation_property is not None:
def func_fun(T):
arg = interpolation_x_inv(T)
if arg > high_orig:
arg = high_orig
if arg < low_orig:
arg = low_orig
return interpolation_property(func(arg))
else:
def func_fun(T):
arg = interpolation_x_inv(T)
if arg > high_orig:
arg = high_orig
if arg < low_orig:
arg = low_orig
return func(arg)
func_fun = np.vectorize(func_fun)
if n == 1:
coeffs = [func_fun(0.5*(low + high)).tolist()]
else:
cheb_fun = ChebTools.generate_Chebyshev_expansion(n-1, func_fun, low, high)
coeffs = cheb_fun.coef()
coeffs = cheb2poly(coeffs)[::-1].tolist() # Convert to polynomial basis
# Mix in low high limits to make it a normal polynomial
if high != low:
# Handle the case of no transformation, no limits
my_poly = Polynomial([-0.5*(high + low)*2.0/(high - low), 2.0/(high - low)])
coeffs = horner(coeffs, my_poly).coef[::-1].tolist()
return coeffs
def test_chebint(self) :
# check exceptions
assert_raises(ValueError, cheb.chebint, [0], .5)
assert_raises(ValueError, cheb.chebint, [0], -1)
assert_raises(ValueError, cheb.chebint, [0], 1, [0,0])
# test integration of zero polynomial
for i in range(2, 5):
k = [0]*(i - 2) + [1]
res = cheb.chebint([0], m=i, k=k)
assert_almost_equal(res, [0, 1])
# check single integration with integration constant
for i in range(5) :
scl = i + 1
pol = [0]*i + [1]
tgt = [i] + [0]*i + [1/scl]
chebpol = cheb.poly2cheb(pol)
chebint = cheb.chebint(chebpol, m=1, k=[i])
res = cheb.cheb2poly(chebint)
assert_almost_equal(trim(res), trim(tgt))
# check single integration with integration constant and lbnd
for i in range(5) :
scl = i + 1
pol = [0]*i + [1]
chebpol = cheb.poly2cheb(pol)
chebint = cheb.chebint(chebpol, m=1, k=[i], lbnd=-1)
assert_almost_equal(cheb.chebval(-1, chebint), i)
# check single integration with integration constant and scaling
for i in range(5) :
scl = i + 1
pol = [0]*i + [1]
tgt = [i] + [0]*i + [2/scl]
chebpol = cheb.poly2cheb(pol)
chebint = cheb.chebint(chebpol, m=1, k=[i], scl=2)
res = cheb.cheb2poly(chebint)
assert_almost_equal(trim(res), trim(tgt))
# check multiple integrations with default k
for i in range(5) :
for j in range(2,5) :
pol = [0]*i + [1]
tgt = pol[:]
for k in range(j) :
tgt = cheb.chebint(tgt, m=1)
res = cheb.chebint(pol, m=j)
assert_almost_equal(trim(res), trim(tgt))
# check multiple integrations with defined k
for i in range(5) :
for j in range(2,5) :
pol = [0]*i + [1]
tgt = pol[:]
for k in range(j) :
tgt = cheb.chebint(tgt, m=1, k=[k])
res = cheb.chebint(pol, m=j, k=list(range(j)))
assert_almost_equal(trim(res), trim(tgt))
# check multiple integrations with lbnd
for i in range(5) :
for j in range(2,5) :
pol = [0]*i + [1]
tgt = pol[:]
for k in range(j) :
tgt = cheb.chebint(tgt, m=1, k=[k], lbnd=-1)
res = cheb.chebint(pol, m=j, k=list(range(j)), lbnd=-1)
assert_almost_equal(trim(res), trim(tgt))
# check multiple integrations with scaling
for i in range(5) :
for j in range(2,5) :
pol = [0]*i + [1]
tgt = pol[:]
for k in range(j) :
tgt = cheb.chebint(tgt, m=1, k=[k], scl=2)
res = cheb.chebint(pol, m=j, k=list(range(j)), scl=2)
assert_almost_equal(trim(res), trim(tgt))
def test_chebint(self):
# check exceptions
assert_raises(ValueError, cheb.chebint, [0], .5)
assert_raises(ValueError, cheb.chebint, [0], -1)
assert_raises(ValueError, cheb.chebint, [0], 1, [0, 0])
# test integration of zero polynomial
for i in range(2, 5):
k = [0]*(i - 2) + [1]
res = cheb.chebint([0], m=i, k=k)
assert_almost_equal(res, [0, 1])
# check single integration with integration constant
for i in range(5):
scl = i + 1
pol = [0]*i + [1]
tgt = [i] + [0]*i + [1/scl]
chebpol = cheb.poly2cheb(pol)
chebint = cheb.chebint(chebpol, m=1, k=[i])
res = cheb.cheb2poly(chebint)
assert_almost_equal(trim(res), trim(tgt))
# check single integration with integration constant and lbnd
for i in range(5):
scl = i + 1
pol = [0]*i + [1]
chebpol = cheb.poly2cheb(pol)
chebint = cheb.chebint(chebpol, m=1, k=[i], lbnd=-1)
assert_almost_equal(cheb.chebval(-1, chebint), i)
# check single integration with integration constant and scaling
for i in range(5):
scl = i + 1
pol = [0]*i + [1]
tgt = [i] + [0]*i + [2/scl]
chebpol = cheb.poly2cheb(pol)
chebint = cheb.chebint(chebpol, m=1, k=[i], scl=2)
res = cheb.cheb2poly(chebint)
assert_almost_equal(trim(res), trim(tgt))
# check multiple integrations with default k
for i in range(5):
for j in range(2, 5):
pol = [0]*i + [1]
tgt = pol[:]
for k in range(j):
tgt = cheb.chebint(tgt, m=1)
res = cheb.chebint(pol, m=j)
assert_almost_equal(trim(res), trim(tgt))
# check multiple integrations with defined k
for i in range(5):
for j in range(2, 5):
pol = [0]*i + [1]
tgt = pol[:]
for k in range(j):
tgt = cheb.chebint(tgt, m=1, k=[k])
res = cheb.chebint(pol, m=j, k=list(range(j)))
assert_almost_equal(trim(res), trim(tgt))
# check multiple integrations with lbnd
for i in range(5):
for j in range(2, 5):
pol = [0]*i + [1]
tgt = pol[:]
for k in range(j):
tgt = cheb.chebint(tgt, m=1, k=[k], lbnd=-1)
res = cheb.chebint(pol, m=j, k=list(range(j)), lbnd=-1)
assert_almost_equal(trim(res), trim(tgt))
# check multiple integrations with scaling
for i in range(5):
for j in range(2, 5):
pol = [0]*i + [1]
tgt = pol[:]
for k in range(j):
tgt = cheb.chebint(tgt, m=1, k=[k], scl=2)
res = cheb.chebint(pol, m=j, k=list(range(j)), scl=2)
assert_almost_equal(trim(res), trim(tgt))
def test_cheb2poly(self):
for i in range(10):
assert_almost_equal(cheb.cheb2poly([0]*i + [1]), Tlist[i])
coeff = np.float64(coeff)
t0 = np.float64(t0 - stt_date) * 86400.0
t1 = np.float64(t1 - stt_date) * 86400.0
dt = (np.array([0, nbin0]) * tsamp + stt_sec - t0) / (t1 - t0) * 2 - 1
coeff1 = coeff.sum(1)
phase = nc.chebval(dt, coeff1) + dispc / freq_end**2
phase0 = np.ceil(phase[0])
nperiod = int(
np.floor(phase[-1] - phase0 + dispc / freq_start**2 -
dispc / freq_end**2))
coeff[0, 0] -= phase0
coeff1 = coeff.sum(1)
roots = nc.chebroots(coeff1)
roots = np.real(roots[np.isreal(roots)])
root = roots[np.argmin(np.abs(roots - dt[0]))]
period = 1. / np.polyval(np.polyder(nc.cheb2poly(coeff1)[::-1]),
root) / 2.0 * (t1 - t0)
stt_sec = (root + 1) / 2.0 * (t1 - t0) + t0
stt_date = stt_date + stt_sec // 86400
stt_sec = stt_sec % 86400
info['phase0'] = int(phase0) + tmp
phase -= phase0
#
info['nperiod'] = nperiod
info['period'] = period
#
nbin_max = (nbin0 - 1) / (np.max(phase) - np.min(phase))
if args.nbin:
nbin = args.nbin
if nbin > nbin_max:
temp_multi = 1
def cheb2poly(cs) :
from numpy.polynomial.chebyshev import cheb2poly
return cheb2poly(cs)
def normalized_dct(Y):
Z = dct(Y)
# print(Z)
Z = Z / len(Z)
# print(Z)
Z[0] = Z[0] / 2
# print(Z)
return Z
C = normalized_dct(Y)
# print('C:', C)
#### Compute the coefficients of the Chebyshev expansion in the canonical basis.
P = cheb2poly(C)
# print('P:', P)
#### Plot f and its chebyshev interpolation approximation
nb_plot_points = 100
X_plot = [a + i * (b - a) / nb_plot_points for i in range(nb_plot_points + 1)]
Y_f_plot = [f(x) for x in X_plot]
Y_P_plot = [polyval(inverse_affine_transformation(x), P) for x in X_plot]
# plt.plot(X_plot, Y_f_plot, label='f')
# plt.plot(X_plot, Y_P_plot, label = 'P')
# plt.legend()
# plt.show()
import matplotlib.pyplot as plt
import numpy as np
from numpy.polynomial.chebyshev import Chebyshev, cheb2poly
mindeg, maxdeg = 0, 5
cmap = plt.get_cmap('rainbow')
colors = cmap(np.linspace(0, 1, maxdeg - mindeg + 1))
print(colors)
l = list(np.zeros(mindeg, int)) + [1]
xx = np.linspace(-1, 1, 100)
tx = 0.2
for i, col in zip(range(mindeg, maxdeg + 1), colors):
c = Chebyshev(l)
print('T({}) = {}'.format(i, cheb2poly(c.coef)))
yy = [c(x) for x in xx]
#col = colors.pop()
plt.gcf().text(tx, 0.9, 'T({})'.format(i), color=col)
tx += 0.1
plt.plot(xx, yy, color=col)
l.insert(0, 0) # increment the degree
plt.grid(True)
plt.show()
tmp=int(coeff[0,0]) coeff[0,0]-=tmp coeff=np.float64(coeff) time0=file_time[0] t0=np.float64(t0-time0[-1])*86400.0 t1=np.float64(t1-time0[-1])*86400.0 dt=(np.array([0,nbin0])*tsamp+time0[:-1].sum()-delay-t0)/(t1-t0)*2-1 coeff1=coeff.sum(1) phase=nc.chebval(dt,coeff1)+dispc/freq_end**2 phase0=np.ceil(phase[0]) nperiod=int(np.floor(phase[-1]-phase0+dispc/freq_start**2-dispc/freq_end**2)) coeff[0,0]-=phase0 roots=nc.chebroots(coeff1) roots=np.real(roots[np.isreal(roots)]) root=roots[np.argmin(np.abs(roots-dt[0]))] period=1./np.polyval(np.polyder(nc.cheb2poly(coeff1)[::-1]),root)/2.0*(t1-t0) stt_sec=(root+1)/2.0*(t1-t0)+t0 stt_date=time0[-1]+stt_sec//86400 stt_sec=stt_sec%86400 info['phase0']=int(phase0)+tmp phase-=phase0 print(coeff) # info['stt_sec']=stt_sec info['stt_date']=stt_date info['stt_time']=stt_date+stt_sec/86400.0 # info['nperiod']=nperiod info['period']=period # nbin_max=(nbin0-1)/(np.max(phase)-np.min(phase))