def hermit_interpol_polynomial(nodes, f):
def divided_difference(row, col):
# row = j, col = i
# Если знаменатель не ноль
if nodes[row + col] != nodes[row]:
return (dif_matrix[row + 1, col - 1] -
dif_matrix[row, col - 1]) / (nodes[row + col] - nodes[row])
# Если знаменатель ноль
else:
deriv = derivative(f, col)
return deriv(nodes[row]) / factorial(col)
n = len(nodes)
dif_matrix = np.zeros((n, n))
for i in range(n):
dif_matrix[i, 0] = f(nodes[i])
for i in range(1, n):
for j in range(0, n - i):
dif_matrix[j, i] = divided_difference(j, i)
result = Polynomial(dif_matrix[0, 0])
for i in range(n - 1):
result = pl.polyadd(
result,
Polynomial(np.poly(nodes[0:i + 1])[::-1]) *
dif_matrix[0, i + 1])[0]
return result, dif_matrix
def newton_interpol_polynomial(nodes, f):
n = len(nodes)
dif_matrix = np.zeros((n, n))
for i in range(n):
dif_matrix[i, 0] = f(nodes[i])
# Считаем разделённые разности. Храним их в верхнетреугольной матрице
for i in range(1, n):
for j in range(0, n - i):
dif_matrix[j,
i] = (dif_matrix[j + 1, i - 1] -
dif_matrix[j, i - 1]) / (nodes[j + i] - nodes[j])
# Строим интерпол. полином Ньютона, испоьзуя полученные разности
result = Polynomial(dif_matrix[0, 0])
for i in range(n - 1):
result = pl.polyadd(
result,
Polynomial(np.poly(nodes[0:i + 1])[::-1]) *
dif_matrix[0, i + 1])[0]
# Помимо полинома возвращаем ещё и матрицу разностей, т.к. её, вообще говоря, можно использовать,
# если мы захотим добавить ещё узлов интерполяции
return result, dif_matrix
def zernike_dict(nmax):
zerDic = {}
for n in range(nmax+1):
for l in range(n,-1,-2):
# print(n,l)
key = (n,l)
if l == n:
# coef = [0 for _ in range(n+1)]
# coef[n] = 1
poly = xn(n)
# zerDic[key] = poly
elif l == n-2:
if n >=2:
poly = (n+0.5)*zerDic[(n,n)] - (n-0.5)*zerDic[(n-2,n-2)]
else:
poly = (n+0.5)*zerDic[(n,n)]
# zerDic[key] = poly
elif l == 0:
n2 = 2*n
M1 = (n2+1)*(n2-1)/(n+l+1)/(n-l)
M2 = -0.5*((2*l+1)**2*(n2-1) + (n2+1)*(n2-1)*(n2-3))/(n+l+1)/(n-l)/(n2-3)
M3 = -1*(n2+1)*(n+l-1)*(n-l-2)/(n+l+1)/(n-l)/(n2-3)
poly = (M1*Polynomial([0,0,1]) + M2)*zerDic[(n-2,l)] + M3*zerDic[(n-4,l)]
# zerDic[key] = poly
else:
L1 = (2*n+1)/(n+l+1)
L2 = -1*(n-l)/(n+l+1)
poly = L1*Polynomial([0,1])*zerDic[(n-1,l-1)] + L2*zerDic[(n-2,l)]
zerDic[key] = poly
return zerDic
def construct_polynomials(self):
"""Construct polynomials up to nth degree. Save the results to the instance.
"""
from numpy.polynomial.polynomial import Polynomial
p = [Polynomial([1])] # polynomials
a = []
b = [0]
innerprod = [.5
] # constructed such that inner product of f(x)=1 is 1/2
# first polynomial is special
a.append(
quad(lambda x: self.K(x) * x, self.x0, self.x1)[0] / innerprod[0])
p.append(Polynomial([-a[0], 1]) * p[0])
for i in range(1, self.n + 1):
innerprod.append(
quad(lambda x: p[i](x)**2 * self.K(x), self.x0, self.x1)[0])
a.append(
quad(lambda x: x * p[i]
(x)**2 * self.K(x), self.x0, self.x1)[0] / innerprod[i])
b.append(innerprod[i] / innerprod[i - 1])
p.append(Polynomial([-a[i], 1]) * p[i] - b[i] * p[i - 1])
self.p = p
self.a = a
self.b = b
self.innerprod = innerprod
def evaluate(self, par, mjd, t0par=None, integrate=False):
"""Evaluate 'par' at given MJD(s), with zero point t0par
Parameters
----------
par : string
key into dictionary = parameter to evaluate. Takes into
account possible par+'DOT' (for 'F': F0, F1, F2, F3)
mjd : float or Time object
MJD at which to evaluate (or Time with .tdb.mjd attribute)
t0par : string or None
key into dictionary with zero point for par
default: None -> PEPOCH for 'F', 'TASC' for all others
integrate : bool
Whether to integrate the polynomial (e.g., to get mean
anomaly out of 'FB' or pulse phase out of 'F')
"""
if par == 'F':
t0par = t0par or 'PEPOCH'
parpol = Polynomial((self['F0'], self.get('F1', 0.),
self.get('F2', 0.), self.get('F3', 0.)))
else:
t0par = t0par or 'TASC'
parpol = Polynomial((self[par], self.get(par + 'DOT', 0.)))
if integrate:
parpol = parpol.integ()
# given time can be Time object
if hasattr(mjd, 'tdb'):
mjd = mjd.tdb.mjd
return parpol((mjd - self[t0par]) * 86400.)
def n_k(order, coef_func):
if order < 0:
raise ValueError()
if order == 0:
return Polynomial([0, 1])
elif order > 0:
return pl.polymul(n_k(order - 1, coef_func),
Polynomial([coef_func(order), 1]))[0] / (order + 1)
def lagrange(nodes):
result = Polynomial([0])
w = Polynomial(np.poly(nodes)[::-1])
deriv = w.deriv(1)
for i in range(len(nodes)):
result = pl.polyadd(result,
make_l_k(i, nodes, w, deriv) * f(nodes[i]))[0]
return result
def _pq_completion(P):
"""
Find polynomial Q given P such that the following matrix is unitary.
[[P(a), i Q(a) sqrt(1-a^2)],
i Q*(a) sqrt(1-a^2), P*(a)]]
Args:
P: Polynomial object P.
Returns:
Polynomial object Q giving described unitary matrix.
"""
pcoefs = P.coef
P = Polynomial(pcoefs)
Pc = Polynomial(pcoefs.conj())
roots = (1. - P * Pc).roots()
# catagorize roots
real_roots = np.array([], dtype=np.float64)
imag_roots = np.array([], dtype=np.float64)
cplx_roots = np.array([], dtype=np.complex128)
tol = 1e-6
for root in roots:
if np.abs(np.imag(root)) < tol:
# real roots
real_roots = np.append(real_roots, np.real(root))
elif np.real(root) > -tol and np.imag(root) > -tol:
if np.real(root) < tol:
imag_roots = np.append(imag_roots, np.imag(root))
else:
cplx_roots = np.append(cplx_roots, root)
# remove root r_i = +/- 1
ridx = np.argmin(np.abs(1. - real_roots))
real_roots = np.delete(real_roots, ridx)
ridx = np.argmin(np.abs(-1. - real_roots))
real_roots = np.delete(real_roots, ridx)
# choose real roots in +/- pairs
real_roots = np.sort(real_roots)
real_roots = real_roots[::2]
# include negative conjugate of complex roots
cplx_roots = np.r_[cplx_roots, -cplx_roots]
# construct Q
Q = Polynomial(
polyfromroots(np.r_[real_roots, 1j * imag_roots, cplx_roots]))
Qc = Polynomial(Q.coef.conj())
# normalize
norm = np.sqrt(
(1 - P * Pc).coef[-1] / (Q * Qc * Polynomial([1, 0, -1])).coef[-1])
return Q * norm
def _fitfunc(self, p, chip, x):
"""ThreeChipLineTable fit function:
y = x - par[0] + par[1]*(chip-refchip) + par[2]*(chip-refchip)**2
w = refwave + par[3]*y + par[4]*y**2 + ... + par[n]*y**(n-2)
where refchip, refwave are taken from the Table meta data.
"""
pchip = Polynomial([-p[0], p[1], p[2]])
px = Polynomial(np.hstack((self.meta['refwave'], p[3:])))
return px(x + pchip(chip - self.meta['refchip']))
def lagrange(nodes):
result = Polynomial([0])
# np.poly строит полином по набору корней. Но порядок коэффициетов противоположный тому, который
# принимает конструктор класса Polynomial
w = Polynomial(np.poly(nodes)[::-1])
deriv = w.deriv(1)
for i in range(len(nodes)):
result = pl.polyadd(result,
make_l_k(i, nodes, w, deriv) * f(nodes[i]))[0]
# возвращается не только сумма
return result
def _coeffs2poly(self, coeffs):
poly_coeffs = np.zeros((len(self.init_coeffs) + 2, ))
poly_coeffs[0] = 0.5 * coeffs[0] # v1
poly_coeffs[1] = coeffs[0] # v1
poly_coeffs[2] = coeffs[1] # v2
poly_coeffs[3] = 2 * coeffs[1] - coeffs[0]
poly_coeffs[4:] = coeffs[2:]
p = Polynomial(poly_coeffs) * Polynomial([1, -1])**2
assert abs(p.coef[3]) < 1e-10
assert abs(p.coef[1]) < 1e-10
assert abs(p(1.)) < 1e-10
return p
def poles_to_rational_rep(poles, residues, d, h):
"""
Take the parameters produced by vectfit and construct a representation
of the form f(s) / g(s), returning f and g
"""
pole_ps = [Polynomial([-p, 1]) for p in poles]
denom = product(pole_ps)
num = sum(r * denom / p for r, p in zip(residues, pole_ps))
num += Polynomial([d, h]) * denom
# Since we're a PR function, should be safe to convert these to real values
num = Polynomial(num.coef.real)
denom = Polynomial(denom.coef.real)
return num, denom
def hermite5(x0, x1, v0, v1, a0, a1):
return Polynomial([x0,
v0,
0.5 * a0,
-10 * x0 + 10 * x1 - 6 * v0 - 4 * v1 - 1.5 * a0 + 0.5 * a1,
15 * x0 - 15 * x1 + 8 * v0 + 7 * v1 + 1.5 * a0 - a1,
-6 * x0 + 6 * x1 - 3 * v0 - 3 * v1 + -.5 * a0 + 0.5 * a1])
def _fitfunc(self, par, x):
"""LineTable fit function
meta['refwave'] + \
par[1]*(x-par[0])+par[2]*(x-par[0])**2+... +par[n]*(x-par[0])**n
"""
px = Polynomial(np.hstack((self.meta['refwave'], par[1:])))
return px(x - par[0])
def __init__(self, order, monom=None):
self.monom = monom if monom else Polynomial(
np.identity(order + 1)[order])
assert isinstance(self.monom, Polynomial)
self.order = order
self.derivative = self.monom.deriv()
self.linear = order == 1
def __init__(self,
data,
fs=2 * np.pi,
win_size=256,
overlap=None,
poly_order=6,
initial_alpha=None):
""" Initialize class.
"""
if initial_alpha is None:
alpha = np.zeros(poly_order + 1)
else:
alpha = initial_alpha
self.alpha = alpha
self.z = data
self.fs = fs
self.poly_order = poly_order
self._polynomial = Polynomial(alpha)
# transform frequency
num_time_bins = self.z.size // win_size
num_freq_bins = win_size // 2
self.win_size = win_size
if overlap is None:
self.overlap = num_freq_bins // 2
else:
self.overlap = overlap
self.tfd = np.zeros((num_freq_bins, num_time_bins), dtype='complex')
def Newton(pts, prec):
título("Newton", "=")
# Definir Precisão
mm.mp.dps = prec
nv = len(pts[0])
matX = pts[0]
matY = [pts[1]]
for i in range(1, nv):
matY.append([])
for j in range(nv - i):
matY[i].append((matY[i - 1][j + 1] - matY[i - 1][j]) /
(matX[j + i] - matX[j]))
print("\nTabela\n\n", mm.matrix(matY))
# matP = [matY[1][0]]
# for i in range(nv-1):
# mult = 1
# for n in range(i):
# mult *= -matX[n]
# matP.append(mult)
matX = pts[0]
matY = pts[1]
matP = mm.matrix(list(reversed(Polynomial(lagrange(matX, matY)).coef)))
# Transformar Resposta em String Polinômio
sPol = pol2str(matP)
print("\nPolinômio: ", sPol)
return sPol
def _get_pp_poly(self):
p = self.ppsi_polynomial
dp = p.deriv(1)
dp_o_r = p.deriv(1) / Polynomial([0.,1.])
ddp = p.deriv(2)
pp_poly = self.calibration_energy + 2*dp_o_r + (ddp + dp**2)
return pp_poly
def fit(cls, x, y):
if 0:
# scipy version, numerically unstable
from scipy.interpolate import lagrange
final = lagrange(x, y)
if 0:
# numpy version
import numpy as np
x_ = np.array(x)
x_[:, None] - x_[None, :]
js, ms = np.triu_indices(len(x), 1)
x_[js] - x_[ms]
else:
import operator as op
from functools import reduce
basis_polys = []
for j, x_j in enumerate(x):
parts_j = []
for m, x_m in enumerate(x):
if m != j:
part_m_j = Polynomial([-1 * x_m, 1]) / (x_j - x_m)
parts_j.append(part_m_j)
basis_j = reduce(op.mul, parts_j, 1)
basis_polys.append(basis_j)
final = sum(y_j * basis_j for y_j, basis_j in zip(y, basis_polys))
if __debug__:
# Final lagrance polynomial must have this property
for j in range(len(x)):
assert final(x[j]) == y[j]
def test_response_4(self):
pcoefs = [0., -2 + 1j, 0., 2.]
poly = Polynomial(pcoefs)
QuantumSignalProcessingPhases(poly,
signal_operator="Wx",
measurement="z")
def test_response_1(self):
pcoefs = [0, 1]
poly = Polynomial(pcoefs)
QuantumSignalProcessingPhases(poly, signal_operator="Wx")
QuantumSignalProcessingPhases(poly, signal_operator="Wz")
def sturms_sequence(p: Polynomial):
"""
Returns Sturm's sequence of a given polynomial.
"""
# setting up the Sturm's sequence.
sturms_seq = []
sturms_seq.append(p)
sturms_seq.append(p.deriv())
# filling the Sturm's sequence list.
f = -Polynomial(poly.polydiv(sturms_seq[-2].coef, sturms_seq[-1].coef)[1])
while f.degree() != 0 or f.coef[0] != 0:
sturms_seq.append(f.copy())
f = -Polynomial(
poly.polydiv(sturms_seq[-2].coef, sturms_seq[-1].coef)[1])
return sturms_seq
def div_xn(poly, n):
"""
Divide the polynomial by x^n
n <= m, m is the minimal index
"""
c = poly.coef
assert np.all(c[:n] == 0)
return Polynomial(c[n:])
def test_response_5(self):
pcoefs = [-1., 0., 50., 0., -400., 0., 1120., 0., -1280., 0., 512.]
poly = Polynomial(pcoefs)
QuantumSignalProcessingPhases(poly, signal_operator="Wx")
QuantumSignalProcessingPhases(poly, signal_operator="Wz")
QuantumSignalProcessingPhases(poly,
signal_operator="Wx",
measurement="z")
def make_nodes(n, p, a, b):
mu_list = np.array(make_mu_k(n, p, a, b), dtype=np.float64)
left_part = np.array([mu_list[i:n+i][::-1] for i in range(n)], dtype=np.float64)
right_part = -1 * mu_list[n:]
coef = solve(left_part, right_part)
coef = np.append(coef[::-1], 1)
w = Polynomial(coef)
return w.roots()
def __init__(self, x, y, dy):
"""
Spline initialization
Parameters
----------
x, y : array_like float
The (x, y) values at nodes that defines the spline
dy : array_like float with two elements
The first derivatives of the left and right ends of the nodes
"""
from numpy.linalg import solve
from numpy.polynomial.polynomial import Polynomial
self.x = np.asarray(x)
self.y = np.asarray(y)
self.dy = np.asarray(dy)
n = len(self.y)
h = self.x[1:]-self.x[:-1]
r = (self.y[1:]-self.y[:-1])/(self.x[1:]-self.x[:-1])
B = np.zeros((n-2, n))
for i in range(n-2):
k = i+1
B[i, i:i+3] = [h[k], 2*(h[k-1]+h[k]), h[k-1]]
C = np.empty((n-2, 1))
for i in range(n-2):
k = i+1
C[i] = 3*(r[k-1]*h[k]+r[k]*h[k-1])
C[0] = C[0]-self.dy[0]*B[0, 0]
C[-1] = C[-1]-self.dy[1]*B[-1, -1]
B = B[:, 1:n-1]
dys = solve(B, C)
dys = np.array(
[self.dy[0]] + [tmp for tmp in dys.flatten()] + [self.dy[1]])
A0 = self.y[:-1]
A1 = dys[:-1]
A2 = (3*r-2*dys[:-1]-dys[1:])/h
A3 = (-2*r+dys[:-1]+dys[1:])/h**2
self.coef = np.array([A0, A1, A2, A3]).T
self.polys = [Polynomial(c) for c in self.coef]
self.polys.insert(0, Polynomial(
[self.y[0]-self.x[0]*self.dy[0], self.dy[0]]))
self.polys.append(Polynomial(
[self.y[-1]-self.x[-1]*self.dy[-1], self.dy[-1]]))
def encode(self, message):
poly_coeffs = strencode(message, self.k)
polynomial = Polynomial(poly_coeffs)
shares = [(x, int(polyval(x, polynomial.coef))) for x in range(1, (self.k+self.e+1))]
signed_shares = [(i, sign(str(i[1]), self.pri_key, self.bitlen, self.generator, self.prime)) for i in shares]
return signed_shares
def expectedStockValue(p, x_test, days):
poly = Polynomial(p)
results = [poly(num) for num in x_test]
for i in range(days - 1):
results.append(poly(results[-1]))
return np.array(results)
def test_response_6(self):
pcoefs = [
-1., 0., (1 / 2) * (4 + 3j - (1 - 2j) * np.sqrt(3)), 0.,
(1 - 1j) * (-1j + np.sqrt(3))
]
poly = Polynomial(pcoefs)
QuantumSignalProcessingPhases(poly,
signal_operator="Wx",
measurement="z")
def __init__(self):
#distance from encoders
self.distance = 0
#polynomial
self.c_poly = Polynomial(0)
self.r_poly = Polynomial(0)
self.l_poly = Polynomial(0)
self.lookahead_c = 0.2
#obstacles polygon
self.polygons = []
self.start_distance = 0
self.start_back_distance = 0
self.change_distance = 0
#distance from center lane
self.center_dis = 0
self.once_detected =0
self.get_change_distance = 0
self.border_distance_x = 1.1
self.border_distance_y = 0.7
self.fraction = 0.1
self.normal_drive = 0
self.box_right = 0
self.get_call = 0
self.threshold_normal = 2
self.threshold_anormal = 2
self.tests = False
self.stop_call = False
