def poly_decimation(p, t):
"""Decimates polynomial and returns decimated polynomial.
:type p: sympy.Poly
:type t: int
:rtype: sympy.Poly
"""
from sympy.abc import x
from operator import mul
n = sympy.degree(p)
s = seq_decimation(p, t)
while len(s) < 2 * n:
s += s
cd = sympy.Poly(1, x, modulus=2)
l, m, bd = 0, -1, 1
for i in range(2 * n):
sub_cd = list(reversed(cd.all_coeffs()))[1:l + 1]
sub_s = list(reversed(s[i - l:i]))
sub_cd += [0] * (len(sub_s) - len(sub_cd))
disc = s[i] + sum(map(mul, sub_cd, sub_s))
if disc % 2 == 1:
td = cd
cd += bd * sympy.Poly(x**(i - m), x, modulus=2)
if l <= i / 2:
l = i + 1 - l
m = i
bd = td
if sympy.degree(cd) == n:
cd = sympy.Poly(reversed(cd.all_coeffs()), x, modulus=2)
return cd
else:
return None
def normalize_poly(poly):
poly = sp.simplify(poly)
num = sp.Poly(sp.expand(sp.numer(poly)))
den = sp.Poly(sp.expand(sp.denom(poly)))
num /= den.coeffs()[0]
den /= den.coeffs()[0]
return num / den
def symToTransferFn(Y):
Y = sympy.expand(sympy.simplify(Y))
n, d = sympy.fraction(Y)
n, d = sympy.Poly(n, s), sympy.Poly(d, s)
num, den = n.all_coeffs(), d.all_coeffs()
num, den = [float(f) for f in num], [float(f) for f in den]
return num, den
def poly_slicer(poly, first_n_terms=None, show_zeros=True, ghost_terms=True, underline=False,
x=sym.symbols('x')):
"""Helper Function for Polynomial long division"""
if show_zeros:
terms = sym.Poly(poly, x).all_terms()
else:
terms = sym.Poly(poly, x).terms()
first_n_terms = len(terms) if (first_n_terms is None or first_n_terms > len(terms)) else first_n_terms
if underline:
poly_string = f"\\underline{{ \\left({pmsign(terms[0][1], leading=True)}{sym.latex(x ** terms[0][0][0])}"
for term in terms[1:first_n_terms]:
poly_string += f"{constant_sign(term[1])}" \
f"{sym.latex(x ** term[0][0]) if term[0][0] != 0 else ''}"
poly_string += " \\right)}"
else:
poly_string = f"{pmsign(terms[0][1], leading=True)}" \
f"{sym.latex(x ** terms[0][0][0]) if terms[0][0][0] != 0 else ''}"
for term in terms[1:first_n_terms]:
poly_string += f"{constant_sign(term[1])}" \
f"{sym.latex(x ** term[0][0]) if term[0][0] != 0 else ''}"
if ghost_terms:
poly_string += f"\\phantom{{ { '{{}}' if not underline else ''} "
for term in terms[first_n_terms:]:
poly_string += f"{constant_sign(term[1])}" \
f"{sym.latex(x ** term[0][0]) if term[0][0] != 0 else ''}"
poly_string += " }"
return poly_string
def get_big_poly(p, t):
n = sympy.degree(p)
d = high_decimation(p, t)
while len(d) < (2 * n):
d += d
cd = sympy.Poly(1, sym_x, modulus=2)
l, m, bd = 0, -1, 1
for i in range(2 * n):
sub_cd = list(reversed(cd.all_coeffs()))[1:l + 1]
sub_s = list(reversed(d[i - l:i]))
sub_cd += [0] * (len(sub_s) - len(sub_cd))
disc = d[i] + sum(map(mul, sub_cd, sub_s))
if disc % 2 == 1:
td = cd
cd += bd * sympy.Poly(sym_x**(i - m), sym_x, modulus=2)
if l <= i / 2:
l = i + 1 - l
m = i
bd = td
if sympy.degree(cd) == n:
cd = sympy.Poly(reversed(cd.all_coeffs()), sym_x, modulus=2)
return cd
else:
return None
def e2nd(expression):
""" basic helper function that accepts a sympy expression, expands it,
attempts to simplify it, and returns a numerator and denomenator pair for the instantiation of a scipy LTI system object. """
expression = expression.expand().cancel()
n = sympy.Poly(sympy.numer(expression), s).all_coeffs()
d = sympy.Poly(sympy.denom(expression), s).all_coeffs()
return ([float(x) for x in n], [float(x) for x in d])
def __Mult_Poli(self):
x, y, z, w = sp.symbols('x,y,z,w')
print("_________________________________________________")
print(
" ###### MÓDULO PARA MULTIPLICAR POLINOMIOS ####### \n \n Ejemplo: 2x³+4x²+5y²+y = 2*x**3+4*x**2+5y**2+y"
)
print("_________________________________________________")
self.Poly1 = (input("Ingrese f(x,y,z,w) = "))
print("-------------------------------------------------")
self.Poly2 = (input("Ingrese g(x,y,z,w) = "))
print("\n" * 100)
print("_________________________________________________")
self.P1 = sp.Poly(self.Poly1)
self.P2 = sp.Poly(self.Poly2)
self.Mult = self.P1 * self.P2
print("_________________________________________________")
print(" ###### MÓDULO PARA MULTIPLICAR POLINOMIOS ####### ")
print("-------------------------------------------------")
print(" f(x,y,z,w) = ", self.Poly1)
print("-------------------------------------------------")
print(" g(x,y,z,w) = ", self.Poly2)
print("-------------------------------------------------")
print(" f x g = ", self.Mult)
print("_________________________________________________")
def Matv(v, base, indep_prompt=True):
if isinstance(v, (list)):
return [Matv(i, base, indep_prompt=True) for i in v]
if re.match(r'P.', base[0].space):
n = len(base)
c = sym.symbols('c0:{}'.format(n))
x = sym.symbols('x')
C = np.array(c)[:, np.newaxis]
B = np.array(base)
Base_Poly = sym.Poly(str(invMatv(C, base)), x)
Vec_Poly = sym.Poly(v.vec, x)
Eq = (Base_Poly - Vec_Poly).all_coeffs()
Coef = sym.linsolve(Eq, c)
if len(Coef) < 1 or len(Coef.free_symbols) > 0:
if indep_prompt:
print('Warning: Improper base! Returning 0')
return 0
dtype = 'float'
if any(abs(sym.im(c)) > tol for c in Coef.args[0]): dtype = 'complex'
res = np.array(Coef.args[0])[:, np.newaxis].astype(dtype)
if re.match(r'F.', base[0].space):
B = np.array([(b.vec).transpose() for b in base]).transpose()
res = np.array(la.pinv(B).dot(v.vec[:, np.newaxis]))
chk = B.dot(res) - v.vec[:, np.newaxis]
if res.dtype != 'object' and la.norm(chk) > tol:
if indep_prompt:
print('Warning: Improper base! Returning 0')
return 0
return res
def canonical(self):
"""Convert rational function to canonical form with unity
highest power of denominator.
See also general, partfrac, mixedfrac, and ZPK"""
try:
N, D, delay = self.as_ratfun_delay()
except ValueError:
# TODO: copy?
return self.expr
var = self.var
Dpoly = sym.Poly(D, var)
Npoly = sym.Poly(N, var)
K = sym.cancel(Npoly.LC() / Dpoly.LC())
if delay != 0:
K *= sym.exp(self.var * delay)
# Divide by leading coefficient
Nm = Npoly.monic()
Dm = Dpoly.monic()
expr = K * (Nm / Dm)
return expr
def keygen(m_val, t_val, files):
global x
# Random irreducible polynomial of degree m
k_vec = get_irreducible(m_val)
if args.v: print("k(x) = ", sympy.Poly(k_vec, x, domain='FF(2)'))
# Random irreducible polynomial of degree t
g_vec = get_irreducible(t_val)
if args.v: print("g(x) = ", sympy.Poly(g_vec, x, domain='FF(2)'))
# Produce k*n generator matrix G for the code
H_matrix = get_H(m_val, t_val, k_vec, g_vec)
if args.v: print('\nH ='); sympy.pprint(H_matrix); print(H_matrix.shape)
G_matrix = get_G(H_matrix[:,:])
if args.v: print('\nG ='); sympy.pprint(G_matrix); print(G_matrix.shape)
k_val = G_matrix.shape[0]
# Select a random k*k binary non-singular matrix S
S_matrix = get_nonsingular(k_val)
if args.v: print('\nS ='); sympy.pprint(S_matrix)
# Select a random n*n permutation matrix P
n_val = G_matrix.shape[1]
permutation = random.sample(range(n_val), n_val)
P_matrix = sympy.Matrix(n_val, n_val,
lambda i, j: int((permutation[i]-j)==0))
if args.v: print('\nP ='); sympy.pprint(P_matrix)
# Compute k*n matrix G_pub = SGP
G_pub = (S_matrix * G_matrix * P_matrix).applyfunc(lambda x: mod(x,2))
if args.v: print('\nG_pub ='); sympy.pprint(G_pub); print(G_pub.shape)
# Public key is (G_pub, t)
# Private key is (S, G, P)
# length of the Goppa Code
if args.v: print("\nn = ", n_val)
if args.v: print("k =", k_val)
writeKeys(G_pub, t_val, S_matrix, H_matrix, P_matrix, files)
def Chebyshev_gen(N, poles):
"""Function to evaluate the numerator and denominator polynomials of the
generalized Chebyshev (i.e., with/without poles) filtering function."""
if(len(poles) == 0):
Num = sp.polys.orthopolys.chebyshevt_poly(N)
Num = sp.Poly(Num).all_coeffs()
Den = np.array([[1]])
else:
# evaluation of the polynomial P (i.e., the denominator)
P = np.poly(poles); Den = np.reshape(P, (len(P), -1)) # denominator
# placing all other poles at infinity
poles = np.hstack((poles, np.inf * np.ones(N - poles.size)))
# R. J. Cameron's recursive algorithm
x = sp.Symbol('x')
tmp1 = x - 1 / poles[0]
tmp2 = sp.sqrt(x ** 2 - 1) * np.sqrt(1 - 1 / poles[0] ** 2)
for j in range(1, len(poles)):
if(np.isinf(poles[j])):
U = x * tmp1 + sp.sqrt(x ** 2 - 1) * tmp2
V = x * tmp2 + sp.sqrt(x ** 2 - 1) * tmp1
tmp1 = sp.simplify(U); tmp2 = sp.simplify(V)
else:
U = x * tmp1 - tmp1 / poles[j] + sp.sqrt(x ** 2 - 1) * np.sqrt(1 - 1 / poles[j] ** 2) * tmp2
V = x * tmp2 - tmp2 / poles[j] + sp.sqrt(x ** 2 - 1) * np.sqrt(1 - 1 / poles[j] ** 2) * tmp1
tmp1 = sp.simplify(U); tmp2 = sp.simplify(V)
U = tmp1; Num = sp.Poly(U, x).all_coeffs() # numerator
Num = I_to_i(Num)
norm = np.polyval(Num, 1) / np.polyval(Den, 1)
Num = np.reshape(Num, (len(Num), -1)) / norm # so that abs(polynomial) value becomes 1 at +1
return Num, Den
def __new__(self, *polynomials, var=None, floater=False):
"""takes a symbolic polynomial and returns
a vector of the leading coefficients
args:
polynomials: Expression(s) for which you wish to get the
coefficients
var: Polynomial variable such as s in s**2 + s**1 + s**0
must be specified if polynomial has symbolic coefficients
"""
results = []
for poly in polynomials:
p = sym.simplify(poly)
# if the variable is defined
if var:
p = sym.Poly(p, var)
c = p.all_coeffs()
results.append(c)
# if variable is not defined
else:
try:
p = sym.Poly(p)
c = p.all_coeffs()
# handles s**0 poly's
except sym.polys.GeneratorsNeeded:
c = [sym.Float(p)]
finally:
results.append(c)
# try to convert to floats if floater set to true
if floater:
results = [[float(x) for x in item] for item in results]
return results
def s_to_w(poly_ip, coef_norm=False):
r"""
Arraytool mostly uses polynomials defined in 's' domain, i.e., the Laplace domain.
However, sometimes we may need polynomials in 'w' (omega) domain, i.e., lowpass
prototype frequency domain. So, this function can be used to convert a given
polynomial from 's' to 'w' domain (s=jw).
:param poly_ip: Input polynomial
:param coef_norm: If `true`, the output polynomial will be normalized such
that the coefficient of the highest degree term becomes 1.
:rtype: Returns a polynomial of the same order, however defined
in `s=jw` domain
"""
if len(poly_ip) == 1:
poly_op = np.array([[1]])
else:
s = sp.Symbol('s');
w = sp.Symbol('w')
poly_ip = sp.Poly(poly_ip.ravel().tolist(), s)
poly_op = sp.simplify(poly_ip.subs(s, 1j * w)) # substitution
poly_op = sp.Poly(poly_op, w).all_coeffs()
poly_op = I_to_i(poly_op)
poly_op = np.reshape(poly_op, (len(poly_op), -1))
if (coef_norm): poly_op = poly_op / poly_op[0]
return poly_op
def poly_E(eps, eps_R, F, P):
r"""
Function to obtain the polynomial E and its roots in the s-domain.
:param eps: Constant term associated with S21
:param eps_R: Constant term associated with S11
:param F: Polynomial F, i.e., numerator of S11 (in s-domain)
:param P: Polynomial P, i.e., numerator of S21 (in s-domain)
For further explanation, see "filter theory notes" by the author.
:rtype: [polynomial_E, roots_E] ... where polynomial E is the
denominator of S11 and S21
"""
N = len(F);
nfz = len(P)
if (((N + nfz) % 2 == 0) and (abs(eps.real) > 0)):
warnings.warn("'eps' value should be pure imaginary when (N+nfz) is an even number")
elif (((N + nfz + 1) % 2 == 0) and (abs(eps.imag) > 0)):
warnings.warn("'eps' value should be pure real when (N+nfz) is an odd number")
s = sp.Symbol('s')
poly_P = sp.Poly(P.ravel().tolist(), s)
poly_F = sp.Poly(F.ravel().tolist(), s)
poly_E = eps_R * poly_P + eps * poly_F
roots_E = I_to_i(sp.nroots(poly_E))
roots_E = np.reshape(roots_E, (len(roots_E), -1))
roots_E = -abs(roots_E.real) + 1j * roots_E.imag # all roots to the LHS
# create the polynomial from the obtained LHS roots
poly_E = np.poly(roots_E.ravel().tolist())
poly_E = np.reshape(poly_E, (len(poly_E), -1))
return poly_E, roots_E
def compileParseMoment(x,
objectiveFunction,
equalityConstraints,
inequalityConstraints,
omega=None):
# Useful constants
n = len(x)
xID = [x for x in range(n)]
# Find maximum degree of each constraint and objective function
equalityConstraintsDegrees = [
sym.Poly(constraint, *x).total_degree()
for constraint in equalityConstraints
]
inequalityConstraintsDegrees = [
sym.Poly(constraint, *x).total_degree()
for constraint in inequalityConstraints
]
objectiveFunctionDegree = sym.Poly(objectiveFunction, *x).total_degree()
maxDegree = math.ceil(
max(max(equalityConstraintsDegrees), max(inequalityConstraintsDegrees),
objectiveFunctionDegree) / 2)
# Corect omega if required
omega = maxDegree if omega is None or omega < maxDegree else omega
# Find cliques
CD = corrSparsityCliques(x, objectiveFunction,
equalityConstraints + inequalityConstraints)
# TODO: Continuing from here
return (1, 1, 1, 1)
def poly_euclid(a, b):
if b.degree() > a.degree():
bigger, smaller = b, a
else:
bigger, smaller = a, b
prev_coeff_bigger = sympy.Poly(1, x, modulus=3)
prev_coeff_smaller = sympy.Poly(0, x, modulus=3)
coeff_bigger = sympy.Poly(0, x, modulus=3)
coeff_smaller = sympy.Poly(1, x, modulus=3)
while smaller.degree() >= 0:
quotient, remainder = bigger.div(smaller)
bigger = smaller
smaller = remainder
next_coeff_bigger = prev_coeff_bigger - quotient * coeff_bigger
next_coeff_smaller = prev_coeff_smaller - quotient * coeff_smaller
prev_coeff_bigger = coeff_bigger
prev_coeff_smaller = coeff_smaller
coeff_bigger = next_coeff_bigger
coeff_smaller = next_coeff_smaller
leading_coeff = sympy.Poly(bigger.coeffs()[0], x, modulus=3)
gcd = leading_coeff * bigger
cofactor_bigger = leading_coeff * prev_coeff_bigger
cofactor_smaller = leading_coeff * prev_coeff_smaller
return gcd, cofactor_bigger, cofactor_smaller
def scale_filter_dbN(n):
pol_string, pol_aux = pol_db(n)
roots_less_1 = roots_less_one(pol_string, 'z')
string = f'(z+1)**({n}/2)*'
for root in roots_less_1:
string_aux = f'(z-{root})*'
string = string + string_aux
string = string[:-1]
pol1 = sym.sympify(string)
pol1 = sym.Poly(pol1, z)
coefs1 = pol1.coeffs()
string = '0+'
for k in range(n):
string_aux = f'z**({n}-{k}-1)+'
string = string + string_aux
string = string[:-1]
pol2 = sym.sympify(string)
pol2 = sym.Poly(pol2, z)
coefs2 = pol2.coeffs()
ck = [i/j for i,j in zip(coefs1, coefs2)]
ck_real = np.array([sym.re(i) for i in ck])
sum_ck = np.dot(ck_real,ck_real)
norm = math.sqrt(sum_ck)
ck_norm = [i/norm for i in ck_real]
ck_norm = ck_norm[::-1]
return np.array(ck_norm)
def extend(self, y: sp.Symbol, n: int, ext_irr: sp.Poly):
self.y = y
self.ext_irr = ext_irr
self.ext_elements = [sp.Poly(0, self.ext_irr.gens[::-1], modulus=self.p)] + \
[sp.Poly(self.y ** i, self.ext_irr.gens[::-1], modulus=self.p) for i in range(n)]
for i in range(n, self.dim**n):
el = sp.rem(sp.Poly(self.y**i, self.y),
self.ext_irr,
modulus=self.p,
symmetric=False)
el = sp.rem(sp.Poly(el, el.gens[::-1]),
self.irr,
modulus=self.p,
symmetric=False)
if el not in self.ext_elements:
self.ext_elements.append(el)
else:
if el == 1 and len(self.elements) == self.dim:
print(
'Last element is 1. Irreducible polynomial is really primitive.'
)
else:
raise ValueError(
'Repeated field element detected; please make sure your irreducible polynomial is primitive.'
)
def __init__(self, x: sp.Symbol, p: int, n: int, irr: sp.Poly):
# Extension parameters
self.ext_elements = None
self.ext_irr = None
self.y = None
self.p = p
self.n = n
self.irr = irr
self.x = x
self.dim = p**n
# Initial elements
self.elements = [sp.Poly(0, self.x, modulus=self.p)] + \
[sp.Poly(self.x ** i, self.x, modulus=self.p) for i in range(n)]
for i in range(n, self.dim):
el = sp.rem(sp.Poly(self.x**i, x),
self.irr,
modulus=p,
symmetric=False)
if el not in self.elements:
self.elements.append(el)
else:
if el == 1 and len(self.elements) == self.dim:
print(
'Last element is 1. Irreducible polynomial is really primitive.'
)
else:
raise ValueError(
'Repeated field element detected; please make sure your irreducible polynomial is primitive.'
)
def main():
print "Root Locus Extras"
print "Methods to add functionality to control.matlab.rlocus"
print "requires control.matlab package"
print "suggested usage: import homework8.root_locus_extras as rlx"
_s = sympy.Symbol("s")
_S = 1 / ((_s + 1) * (_s + 2) * (_s + 10))
_numerS = map(float, sympy.Poly(_S.as_numer_denom()[0], _s).all_coeffs())
_denomS = map(float, sympy.Poly(_S.as_numer_denom()[1], _s).all_coeffs())
_tf = matlab.tf(_numerS, _denomS)
_gain = 164.5
_damping_ratio = 0.174
_gain = gainFromDampingRatio(_tf, _damping_ratio)
_poles = polesFromTransferFunction(_tf)
_overshoot = overshootFromDampingRatio(_tf, _damping_ratio)
_damping_ratio = dampingRatioFromGain(_tf, _gain)
print "example: "
print "system = ", _tf
print "Damping Ratio provided: ", _damping_ratio
print "Calculated values:"
print "Gain: ", _gain
print "Poles: ", _poles
print "Overshoot: ", _overshoot
print "Damping Ratio: ", dampingRatioFromGain(_tf, _gain)
print "Overshoot: ", overshootFromGain(_tf, _gain)
print "Frequency:", frequencyFromGain(_tf, _gain)
def rational_inequality_solve(eqn):
lhs = eqn.lhs
numerator, denominator = lhs.as_numer_denom()
numerator, denominator = sympy.Poly(numerator), sympy.Poly(denominator)
relation_operator = eqn.rel_op
return sympy.solve_rational_inequalities([[((numerator, denominator),
relation_operator)]])
def get_legendre_poly(n, sym, r):
a = r[0]
b = r[1]
c = (a + b) / 2
m = 2 / (b - a)
print sym, n
if n == 0:
const_val = np.sqrt(1.0 / (b - a))
return SampledPolynomial(sympy.Poly(const_val,
*base_syms,
domain='RR'),
None,
inputs=inputs)
sym_n = (sym - c) * m
p = sympy.expand((sym * sym - 1)**n)
for i in range(n):
p = p.diff()
p = p.subs(sym, sym_n)
return SampledPolynomial(sympy.Poly(
(1 / ((2**n) * scipy.misc.factorial(n))) * np.sqrt(n + 0.5) * p,
*base_syms,
domain='RR'),
None,
inputs=inputs)
def symbolic_transfer_function(
eq: Union[sp.Expr, int, float]) -> ct.TransferFunction:
"""
Transform a symbolic equation to a transfer function (sympy -> control)
:param eq: your symbolic sympy based equation
:return: a control Transfer Function class
"""
s = sp.var('s')
try:
if eq.is_real:
pass
except:
if not isinstance(eq, float) and not isinstance(eq, int):
used_symbols = [str(sym) for sym in eq.free_symbols]
if not len(used_symbols) == 1 or "s" not in used_symbols:
raise Exception(
"invalid equation, please use correct transfer function equation (e.g. 1/(s**2+3))"
)
n, d = sp.fraction(sp.factor(eq))
num = sp.Poly(sp.expand(n), s).all_coeffs()
den = sp.Poly(sp.expand(d), s).all_coeffs()
num: [float] = [float(v) for v in num]
den: [float] = [float(v) for v in den]
return ct.TransferFunction(num, den)
def polynomial_gcd(a, b):
"""Function to find gcd of two poly1d polynomials.
Return gcd, s, t, u, v
with a s + bt = gcd (Bezout s theorem)
a = u gcd
b = v gcd
Hence
s u + t v = 1
These are used in diagimalize procedure
"""
s = sp.Poly(0, x, domain='QQ').as_expr()
old_s = sp.Poly(1, x, domain='QQ').as_expr()
t = sp.Poly(1, x, domain='QQ').as_expr()
old_t = sp.Poly(0, x, domain='QQ').as_expr()
r = b
old_r = a
while not is_zero_polynomial(r):
quotient, remainder = sp.div(old_r, r, x)
(old_r, r) = (r, remainder)
(old_s, s) = (s, old_s - quotient * s)
(old_t, t) = (t, old_t - quotient * t)
# output "Bézout coefficients:", (old_s, old_t)
# output "greatest common divisor:", old_r
# output "quotients by the gcd:", (t, s)
u, _ = sp.div(a, old_r, x, domain='QQ')
v, _ = sp.div(b, old_r, x, domain='QQ')
return old_r.as_expr(), old_s.as_expr(),\
old_t.as_expr(), u.as_expr(), v.as_expr()
def w_to_s(poly_ip, coef_norm=False):
r"""
Arraytool mostly uses polynomials defined in 's' domain, i.e., Laplace domain.
So, this function can be used to convert a given polynomial from 'w' to 's'
domain (w=-js).
:param poly_ip: Input polynomial
:param coef_norm: If `true`, the output polynomial will be normalized such
that the coefficient of the highest degree term becomes 1.
:rtype: Returns a polynomial of the same order, however defined
in `w=-js` domain
"""
if (len(poly_ip) == 1):
poly_op = np.array([[1]])
else:
s = sp.Symbol('s');
w = sp.Symbol('w')
poly_ip = sp.Poly(poly_ip.ravel().tolist(), w)
poly_op = sp.simplify(poly_ip.subs(w, -1j * s)) # substitution
poly_op = sp.Poly(poly_op, s).all_coeffs()
poly_op = I_to_i(poly_op)
poly_op = np.reshape(poly_op, (len(poly_op), -1))
if (coef_norm): poly_op = poly_op / poly_op[0]
return poly_op
def _complete_helper(n): if n == 1: return sympy.Poly(1, p) else: s = sympy.Poly(0, p) for j in range(1, n): s += nCr(n-1, j-1)*_complete_helper(j)*sympy.Poly(1-p)**(j*(n-j)) return 1-sympy.simplify(s)
def test_finite_field(self):
""" Ensures finite fields work in the finite field case. """
p1 = sympy.Poly([1, 5, 3, 7, 3], sympy.abc.x, domain=sympy.GF(11))
p2 = sympy.Poly([0], sympy.abc.x, domain=sympy.GF(11))
p3 = sympy.Poly([10], sympy.abc.x, domain=sympy.GF(11))
self.assertEqual(symbaudio.utils.poly.max_coeff(p1), 7)
self.assertEqual(symbaudio.utils.poly.max_coeff(p2), 0)
self.assertEqual(symbaudio.utils.poly.max_coeff(p3), 10)
def test_integer(self):
""" Ensures Z works in the integer case. """
p1 = sympy.Poly([1, 5, 3, 7, 3], sympy.abc.x, domain="ZZ")
p2 = sympy.Poly([0], sympy.abc.x, domain="ZZ")
p3 = sympy.Poly([10], sympy.abc.x, domain="ZZ")
self.assertEqual(symbaudio.utils.poly.max_coeff(p1), 7)
self.assertEqual(symbaudio.utils.poly.max_coeff(p2), 0)
self.assertEqual(symbaudio.utils.poly.max_coeff(p3), 10)
def convert(Vs):
Vs = sy.simplify(Vs)
num, den = sy.fraction(Vs)
num = np.array(sy.Poly(num, s).all_coeffs(), dtype=np.complex64)
den = np.array(sy.Poly(den, s).all_coeffs(), dtype=np.complex64)
Vo_sig = sp.lti(num, den)
print(Vo_sig)
return Vo_sig
def findLine(x, y, num):
#from second cam to first cam
if num == 2:
med = np.dot(K2.I, np.transpose([np.array([x, y, 1])]))
xs = sp.symbols('xs:4')
EQ1 = T2[0][0] * xs[0] + T2[0][1] * xs[1] + T2[0][2] * xs[2] + T2[0][
3] - xs[3] * med[0]
EQ2 = T2[1][0] * xs[0] + T2[1][1] * xs[1] + T2[1][2] * xs[2] + T2[1][
3] - xs[3] * med[1]
EQ3 = T2[2][0] * xs[0] + T2[2][1] * xs[1] + T2[2][2] * xs[2] + T2[2][
3] - xs[3] * med[2]
solution = sp.solve([EQ1, EQ2, EQ3], xs[:-1])
#print(solution)
med2 = sp.Matrix(
np.dot(K1, np.transpose([np.array([xs[0], xs[1], xs[2]])])) /
xs[2]).subs(solution)
med2.subs(solution)
#print(med2)
ab = sp.symbols('ab:2')
EQ5 = ab[1] - med2[1]
EQ4 = ab[0] - med2[0]
solution2 = sp.solve([EQ4, EQ5], {ab[0], xs[3]})
line = solution2[0][ab[0]]
ploy = sp.Poly(line, ab[1])
#print (a.coeffs())
return ploy.coeffs()
#from third cam to first cam
if num == 3:
med = np.dot(K3.I, np.transpose([np.array([x, y, 1])]))
xs = sp.symbols('xs:4')
EQ1 = T3[0][0] * xs[0] + T3[0][1] * xs[1] + T3[0][2] * xs[2] + T3[0][
3] - xs[3] * med[0]
EQ2 = T3[1][0] * xs[0] + T3[1][1] * xs[1] + T3[1][2] * xs[2] + T3[1][
3] - xs[3] * med[1]
EQ3 = T3[2][0] * xs[0] + T3[2][1] * xs[1] + T3[2][2] * xs[2] + T3[2][
3] - xs[3] * med[2]
solution = sp.solve([EQ1, EQ2, EQ3], xs[:-1])
#print(solution)
med2 = sp.Matrix(
np.dot(K1, np.transpose([np.array([xs[0], xs[1], xs[2]])])) /
xs[2]).subs(solution)
med2.subs(solution)
#print(med2)
ab = sp.symbols('ab:2')
EQ5 = ab[1] - med2[1]
EQ4 = ab[0] - med2[0]
solution2 = sp.solve([EQ4, EQ5], {ab[0], xs[3]})
line = solution2[0][ab[0]]
ploy = sp.Poly(line, ab[1])
#print (a.coeffs())
return ploy.coeffs()