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chop.py
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chop.py
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from sympy import symbols, Rational
from sympy.simplify.simplify import simplify
from sympy.simplify.ratsimp import ratsimp
from sympy import cancel
from sympy.simplify.simplify import radsimp
import scipy.optimize
import sympy
from sympy.polys.polytools import div as polydiv, rem as polyrem
def chop_1_over_s( P, Q, s):
print( P.subs( { s: 0}))
print( Q.subs( { s: 0}))
factor = s
Q_prime, Q_prime_rem = polydiv( Q.as_poly(s,domain='RR'), factor)
print( f"Q_prime: {Q_prime}")
print( f"Q_prime rem: {Q_prime_rem}")
A = sympy.limit( P/Q_prime, s, 0)
print(f"A: {A}")
P_new = P - A*Q_prime
print( f"P_new: {P_new}")
P_prime, P_prime_rem = polydiv( P_new.as_poly(s,domain='RR'), factor)
print( f"P_prime: {P_prime}")
print( f"P_prime rem: {P_prime_rem}")
return P_prime, Q_prime
def remove_leading_coeff( P):
coeff = sympy.LC(P)
if np.abs(coeff) < 1.e-7:
print(f"P (before removing leading term): {P} {coeff}")
P = P-coeff*sympy.LM(P)
print(f"P (after removing leading term): {P}")
return P
def chop_s( P, Q, s):
degreeP = sympy.degree(P)
print(f"P: {P}")
print(f"Q: {Q}")
B = sympy.limit( P/(s*Q), s, sympy.oo)
print(f"B: {B}")
P_prime = P - B*s*Q
print( f"degrees: {degreeP} {sympy.degree(P_prime)}")
if degreeP == sympy.degree(P_prime):
P_prime = remove_leading_coeff( P_prime)
return P_prime, Q
def chop_linear( P, Q, s, s0):
print( P.subs( { s: s0}))
print( Q.subs( { s: s0}))
A = sympy.limit( (s-s0)*P/Q, s, s0)
print(f"A: {A}")
factor = s - s0
Q_prime, Q_prime_rem = polydiv( Q.as_poly(s,domain='RR'), factor)
print( f"Q_prime: {Q_prime}")
print( f"Q_prime rem: {Q_prime_rem}")
P_new = P - A*Q_prime
print( f"P_new: {P_new}")
P_prime, P_prime_rem = polydiv( P_new.as_poly(s,domain='RR'), factor)
print( f"P_prime: {P_prime}")
print( f"P_prime rem: {P_prime_rem}")
return P_prime, Q_prime
def quadratic_residue( P, Q, s, w0):
factor = s**2 + w0**2
Q_prime, Q_prime_rem = polydiv( Q.as_poly(s,domain='RR'), factor)
print( f"Q_prime: {Q_prime}")
print( f"Q_prime rem: {Q_prime_rem}")
Ap = sympy.limit( P, s, sympy.I*w0)
Aq = sympy.limit( (s-sympy.I*w0)/Q, s, sympy.I*w0)
Aqq = sympy.limit( 1/(Q_prime*(s+sympy.I*w0)), s, sympy.I*w0)
A = Ap*Aqq
print(f"Ap, Aq, Aqq, A: {Ap} {Aq} {Aqq} {A}")
Ahat = 2*(s*sympy.re(A)-w0*sympy.im(A))
print(f"Ahat: {Ahat}")
return Ahat, Q_prime, factor
def chop_quadratic( P, Q, s, w0):
print( P.subs( { s: sympy.I*w0}))
print( P.subs( { s: -sympy.I*w0}))
print( Q.subs( { s: sympy.I*w0}))
print( Q.subs( { s: -sympy.I*w0}))
Ahat, Q_prime, factor = quadratic_residue( P, Q, s, w0)
P_new = P - Ahat*Q_prime
print( f"P_new: {P_new}")
P_prime, P_prime_rem = polydiv( P_new.as_poly(s,domain='RR'), factor)
print( f"P_prime: {P_prime}")
print( f"P_prime rem: {P_prime_rem}")
return P_prime, Q_prime
def rp( Z, s):
w = symbols( "w")
real_part = sympy.re(Z.subs({s:sympy.I*w}))
return sympy.lambdify( w, real_part, "numpy")
def remove_r( P, Q, s, bracket=None, at_infinity=False, at_zero=False):
Z = P/Q
f = rp(Z, s)
if True:
plot( f)
if at_infinity:
degreeP = sympy.degree(P)
R = sympy.limit( Z, s, sympy.oo)
w0 = sympy.oo
Z = Z-R
P, Q = Z.as_numer_denom()
if degreeP == sympy.degree(P):
P = remove_leading_coeff(P)
return P, Q, w0, R
elif at_zero:
R = sympy.limit( Z, s, 0)
w0 = 0
else:
result = scipy.optimize.minimize_scalar( f, method="brent", bracket=bracket)
assert result.success
R = result.fun
w0 = result.x
Z = Z-R
P, Q = Z.as_numer_denom()
return P, Q, w0, R
def bott_duffin( P, Q, s, w0):
X = (P/Q).subs({s : sympy.I*w0})
print( f"X: {X.evalf()}")
target0 = X/(sympy.I*w0)
print( f"target: {target0.evalf()}")
target0 = sympy.re(target0).evalf()
assert target0 > 0
k = symbols('k')
k0_result = scipy.optimize.root_scalar( sympy.lambdify( k, (P/Q).subs({s:k}) - k*target0, "numpy"), method="brentq", bracket=[0,1000])
assert k0_result.converged
k0 = k0_result.root
print( f"k0: {k0}")
Z_k0 = (P/Q).subs({s:k0})
print( f"Z_k0: {Z_k0.evalf()}")
num = k0*P - s*Z_k0*Q
den = Q*k0*Z_k0 - s*P
eta_num,eta_den = num.as_poly(s,domain='RR'),den.as_poly(s,domain='RR')
factor = sympy.poly( s-k0)
eta_num, eta_num_rem = polydiv( eta_num, factor)
eta_den, eta_den_rem = polydiv( eta_den, factor)
print( f"eta_num: {eta_num}")
print( f"eta_den: {eta_den}")
print( f"eta_num rem: {eta_num_rem}")
print( f"eta_den rem: {eta_den_rem}")
if False:
num_roots = eta_num.nroots()
den_roots = eta_den.nroots()
print( f"roots for eta_num: {num_roots}")
print( f"roots for eta_den: {den_roots}")
fuzz = 0.000001
def find_pure_imaginary( roots):
result = []
for r in roots:
f = sympy.re(r)
if np.abs(f) < fuzz:
result.append(sympy.im(r)*sympy.I)
return result
imag_num_roots = find_pure_imaginary(num_roots)
imag_den_roots = find_pure_imaginary(den_roots)
print( f"imaginary numer roots: {imag_num_roots}")
print( f"imaginary denom roots: {imag_den_roots}")
assert len(imag_num_roots) % 2 == 0
assert len(imag_den_roots) % 2 == 0
assert len(imag_num_roots) == 2
assert np.isclose( np.array(imag_num_roots).astype(np.complex128)[0], 1j*w0) or np.isclose( np.array(imag_num_roots).astype(np.complex128)[1], 1j*w0)
P,Q = eta_num, eta_den
return P, Q